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A TREATISE THEORY OF FUNCTIONS .mm A TREATISE THEORY OF FUNCTIONS BY JAMES HARKNESS, M.A. ASSOCIATE PROFESSOR OF MATHEMATICS IN BRTN MAWR COLLEGE, PA. LATE SCHOLAR OF TRINITY COLLEGE, CAMBRIDGE FRANK MORLEY, M.A. PROFESSOR OF PURE MATHEMATICS IN HAVBRFORD COLLEGE, PA. LATE SCHOLAR OF KING'S COLLEGE, CAMBRIDGE Neto gork MACMILLAN AND CO. AND LONDON 1893 AU rights reserved COPTKIGHT, 1893, Bt MACillLLAN AIv'D CO. Xortoooti JSrtas: J. S. Gushing & Co. — Berwick & Smith. Boston, Mass., U.S.A. PREFACE. In this book we have sought to give an account of a department of mathematics which is now generally regarded as fundamental. A list of the men to whom the successive advances of the subject are due, includes, with few exceptions, the names of the greatest French and German mathematicians of the century, from Cauchy and Gauss onward. And in line with these advances lie the chief fields of mathematical activity at the present day. The most legitimate extensions of elementary analysis lead so directly into the Theory of Functions, that recent writers on Algebra, Trigonometry, the Calculus, etc., give theories which are indispen- sable parts of our subject. But since these theories are not found in many current text-books, it appears most convenient for the generality of readers to make the earlier chapters complete in themselves. Thus an account is given in eh. i. of the geometric representation of elementary operations; and in ch. iii., before the introduction of Weierstrass's theory of the analytic function, the theory of convergence is discussed at some length. "We have aimed at a full presentation of the standard parts of the subject, with certain exceptions. Of these exceptions, three must be stated. In ch. ii., the theory of real functions of a real variable is given only so far as seems necessary as a basis for what follows. In the account of Abelian integrals (ch. x.), our object is to induct the reader as simply and rapidly as possible into what is itself a suitable theme for more than one large volume. And we have entirely passed over the automorphic functions, since it was not possible to give even an introductory sketch within the space at our disposal. However, an account of some of Kronecker's work, which is necessary for the study of Klein's recent developments of the theory of Abelian functions, is included in ch. vi. ; and ch. viii. is devoted to a somewhat condensed treatment of double theta- VI PREFACE. functions, which goes further than is necessary for our immediate purpose, for the reason that the subject is not very accessible in the English language. Progress is intentionally slow in some places, where what is required is a formation of certain new concepts, rather than an enlargement of ideas that pre-exist. As to the place of the Theory of Functions in the order of those mathematical studies which appear in all curricula, its more ele- mentary parts can be attacked with advantage so soon as a sound knowledge of the Integral Calculus is gained. It will be found that, though collateral subjects such as the theory of Algebraic Equations and the analytic theory of Plane Curves are freely referred to, a previous knowledge of them is rarely necessary for the understanding of what follows. It may be added that an acquaintance with the present subject is requisite for the study of the modern theory of Differential Equations. It is presupposed, for example, in Dr. Craig's treatise. The many writers to whom we are indebted for theories or for elucidations are, of course, referred to in the text. There has appeared quite recently the very important treatise of Dr. Forsyth, unfortunately too late to be included in these references. A Glossary is added, which gives the principal technical terms employed by German and French writers, with the adopted equiva- lents. The page on which these equivalents are defined will be found by consulting the Index. To our respective colleagues. Professor C. A. Scott and Professor E. W. Brown, our hearty thanks are due for valuable assistance with the proof-sheets. May, 1893. CONTENTS. The reader to whom the subject is new may omit chapter ii., chapter vi. from § 184 to the end, chapter viii. from § 233 to the end, and chapter ix. In the following table the numbers refer to the pages. CHAPTER I. Geometric iNXRODrcTiox. Argand Diagram, 1. Fundamental operations, 2. Strokes, 3. Continuity, 8 Algebraic function, 9. Monogenic function, 13. Conform representation, 14. Neumann's sphere, 18. Bilinear transformation, 19. Anharmonic ratios, 21. Covariants of cubic, 26. Covariants of quartic, 32. Examples of many-valued functions, 30. CHAPTER II. Real Fcjjctioss of a Real Variable. Nimibers, 41. Sequences, 42. Correspondence of points and numbers, 46. Variables, 47. Meaning of word Function, 51. Upper and lower limits of function, 54. Continuous function without differential quotient, 58. Functions of two real variables, 61. References, 62. CHAPTER III. The Theory of Infinite Series. Series with real terms, 63. Series with complex terms, 67. Uniform con- vergence, 69. Multiple series, 72. Infinite products, 78. Integral series, 85. Circle of convergence, 88. Cauchy's extension of Taylor's theorem, 91. Weier- strass's preliminary theorem, 94. Behaviour of series on circle of convergence, 96. Theory of analj'tic function, 101. Singular points, 106. Transcendental integral function, 112. Reversion of an integral series, 116. Lacunary spaces, 119. Exponential function, 120. Logarithm, 121. Power, 126. References, 126. CHAPTER IV. Algebraic Flxctioxs. Critical points, 127. Continuity of branches, 128. Fractional series, 129. One-element, 131. Two-element, 132. Determination of coefiBcients in the series for a branch, 134. t-element, 138. Rfisume, 143. Resolution of the singularity, 144. Examples from the theory of plane curves, 145. Newton's parallelogram, 147. Theory of loops, 151. Prepared equation, 154. Liiroth and Clebsch's theorems, 155. References, loO. VUl CONTENTS. CHAPTER V. IXTEGRATIOX. Holomorphic and meromorphic functions, 101. Curvilinear integrals, 161. Cauoiiy's fundamental theorem, 163. Goursat's proof, 164. Sufficient condition for the vanishing of an integral taken round a small circle, 108. Cauchy's second theorem on integrals, 173. Taylor's theorem, 173. Laurent's theorem, 175. Extension of Taylor's theorem, 178. Meromorphic function as sum of rational fraction and holomorphic function, 180. Residues, 181. Essential singularities, 18?. Weierstrass and Jlittag-Leffler's theorems stated, 186. Proof of Mittag-LefHer's theorem, 188. Proof of Weierstrass's fac- tor-theorem, 189. Examples, 192. Applications of Cauchy's theorems to real definite integrals, 196. Differential equations, 199. References, 204. CHAPTER VI. RiEMANN Surfaces. Simple example of a Riemann surface, 205. Riemann sphere, 208. Surface for algebraic function, 211. Examples, 213. The simpler correspondences, 215. Further examples, 220. Connexion of surfaces, 227. Cross-cuts, 228. Deter- mination of index, 230. Index of Riemann surface, 232. The number p, 233. Canonical dissection, 2.34. Klein's normal surface, 236. Delimiting curves, 239. Algebraic functions on a Riemann surface, 243. Integration, 245. Riemann's extension of Cauchy's theorem, 246. Abelian integrals, 249. Introduction of periods, 252. Examples of the theory of integration, 256. Birational transfor- mations, 260. Integral algebraic functions of the surface, 263. Minimal basis, 205. Essential and unessential divisors of the discriminant, 267. Klein's new kind of Riemann surface, 274. References, 276. CHAPTER VII. Elliptic Fukctioxs. Fourier's theorem, 277. Double periodicity, 278. Primitive pairs of periods, 281. Fundamental elliptic function, 283. Lionville's theory, 287. Doubly periodic functions with common periods are algebraically related, 291. Theory of the p-funotion, 295. Its addition theorem, 298. Theory of the f-function, 302. Expression of any elliptic function by means of f, 304. Theory of the (T-function, 305. The allied functions, 308. Expression of any elliptic function by means of , Avhere p and q are integers, call its absolute value p', its amplitude <\>. By 2'"^" is to be understood a quantity which, when multiplied by itself q — 1 times, gives 2''. Therefore p'' = f^,q<^—pd + 2 Xtt, where A is any integer. The absolute value p being positive, there is a real positive value of p''^', and this we choose for p'. The amplitude has q distinct values, got by giving to \ the values 0,1,2, ■■■,q — 1. Xo two of these are congruent with regard to 2 ir, but any other value of the integer A. gives an angle congruent to one of these q amplitudes. The absolute values of the q roots are equal to p', and the amplitudes form an arithmetic progression of q terms, for which the common difference is 2 Tr/q. If X = and = 6„, then o = p9o/q gives the chief root. The representative points form a regular g-gon. A negative power is the quotient of 1 by a positive power. § 14. The Arithmetic Mean of n points z, (r — 1, 2, ••-, n) is the point defined by the equation n 2 = l/?i • 2z,. n n Its co-ordinates are 1/n • 2j,, 1/n ■ Ix/r ; accordingly the point is the centroid of the n points. A Geometric Mean of n points 2^ ()" = 1, 2, •■•, 11) is a point defined by the equation n 2" = Zj2., • • • z,. = nz,. 1 Thus the geometric means are n in number, and form a regular polygon. In particular, the geo- metric means of 2,, z.2 are the points 2, z', which lie on the bisectors of the angles ZjOzj, ZjOZ], at a distance from which is the posi- tive square root of |zi| |z,|. These points are more prop- erly the geometric means loith regard to the origin. A geometric mean with regard to any point Zo is defined by the equation n (z-z„)" = n(z,-2o). 8 GEOMETRIC INTRODUCTION. The Harmonic Mean of n points (with regard to the origin) is the point defined by n n/z = 2 1/2,. The harmonic mean with regard to any point z^, is defined by n n/ (z-Zo) = 21/(2,-2:0) ■ The arithmetic mean is peculiii^in tiah it is independent of the origin. . / § 15. An extension of the Argand representation to points in space is not consistent with the maintenance of the rules of ordinary algebra. If such an extension be possible, a point (?, -q, f) will have for its affix ^ + i-q +jf. The product of two such expressions, (a + ib + jc){a + ip + J7)i must be of the same form. Hence P, ij, fi must be linear functions of i, j. Let i^ =Pi + qii + rj, ij = ji = p.^ + q.,i + rj, f = P3 + q^i + rj, and let the product in question be zero. This requires that the constant term and the coefficients of i, j shall vanish separately. Let the resulting equations be Dr^ + ErP + Fry = (r=l, 2, 3). These equations can be satisfied by values of o, p, y other than a=/3=7=0, provided A = 0, where B, -^, F, A E, F, z>, E, F, Now A is a cubic expression in a, 6, c. If real values be given to 6, c, the equation A = must have one real root ; choose this for a. Thus the product (a + ib + jc)(o + ip +jy) can be made to vanish in cases when neither of the factors vanishes, a result contrary to the laws of ordinary algebra. [See Konigsberger, Elliptische Functionen, p. 10 ; Stolz, AUgemeine Arithmetik, t. ii. ch. 1.] § 16. Continuity of one-valued functions. When to each value of a complex variable z another complex variable w is assigned, so that when z is given, w can be constructed uniquely, w is a one-valued function of z. When, for a given z. n values of w exist, w is an n-valued function of z* A one-valued function f{z) is continuous at a point « = a of a region if to every positive quantity c, however small, a positive quantity 8 can be assigned, such that for all points z of the region for which | z — a | < 8, \f{z)-f{a)\. 8x Sy Sy Sx * The deflDitioD of cODtinuity assigns no meaning to the phrase ' continuous function * except when for the region in question the function is one-valued. When the phrase occurs, the atten- tion is filed on a continuous branch of the function ; and the points where there is coufusiua of branches are to be excluded, for the present, from the region considered. GEOMETEIC INTRODUCTION. 13 Analytic expressions, which involve x, y only in the combination t + iy, satisfy these relations. Cauchy used for such analytic ex- Dressions the term monogenic function. As we shall be concerned ilmost exclusively with functions of this kind it is convenient to Dmit the adjective. Functions such as x—iy, a? — y-, will not be •egarded as functions of z, inasmuch as they are non-mouogenic. The equations (i.) are equivalent to -i- = --5- (11-). 6X I 6y ind sufficient conditions in order that a function of x and y may possess a determinate differential quotient which is independent jf cly/dx, are that 8iv/Sx, Siv/8y are to be continuous and satisfy ;he equation (ii.) throughout a region which encloses z. [See Harnack's Diff. Calc, trans, by G. L. Cathcart, p. 141. J It wiU be seen subsequently that the initial assumptions as to the continuity Df u, V, and of their first derivatives, imply the existence and con- dnuity of the remaining derivatives. Hence, by ordinary differen- tiation, it follows from (i.) that u, v satisfy 8^ 8^_Q S^+S^ = (iii.) Sot 8i/^ ' Bx' Sy' ^ ^ The equations (iii.) show that u, v are solutions of Laplace's equa- tions for two dimensions. It is evident from (i.) that the solutions u and V are related. They are, in fact, the well-known conjugate functions of Physics. [See Clerk Maxwell's Electricity aud Magnet- ism, 3d ed., t. i., p. 285 ; Minchin's Uniplanar Kinematics, p. 226.] § 21. A theorem of great importance follows from the fact that iw/dz does not depend on dz* Let z, «„ z.2 be three near points in :he 0-plane and lo, w^, iv^ the corresponding points of a branch w ; then lim {iVi — io)/{Zi — z) = lim (joj — M)/(«i — z). Hence, if div/dz be neither zero nor infinite, lim (wj — w)/(wi — tv) = lira {z.^ — z)/{Zi — z). Now the absolute value of (Z2 — z) / {zi — z) is the ratio of the two sides of the triangle ZiZZ,, and the amplitude is the included mgle. Therefore the triangles Wiww^, ZiZz^, are ultimately similar, [n other words, corresponding figures in the two planes are similar in their infinitesimal parts, except at points which make dw/dz = 0, Dr 00. The scale dw/dz depends upon the part selected. * Riemann CWerke, p. 6) defined a function of z as a vuiiuble whose differential quotient A independent of dz. 14 GEOMETRIC INTRODUCTION. The significance of the equations dio/dz = 0, dw/dz = oo can be readily dis- covered in the case of the algebraic function. It will appear from the theorj- of series (Chapters III., IV.), which affords a precise basis for a knowledge of func- tions, that for a finite pair of values w, z the condition dv/dz = oo requires that two or more values of lo become equal at z, so that w cannot be regarded as one- valued. Similarly it will appear that dw/dz = requires that two or more values of z become equal at %o. Accordingly the theorem holds when the corre- spondence between ir, z is (1, 1) at and near the points considered. It is not implied that near points at which dw/dz is neither zero nor infinite, w must be one-valued. There may be values of lo and z which afford more than one value of dvfdz. (See Chapter IV.) If, for a given pair v:, z, n values of w and m values of z become equal, the theory of series will show that an angle in the w-plane is to an angle in the e-plane in the ratio m/n. As a simple example consider xifi = z'. Here three points z correspond to two points ic, and when any one of the three z's describes a circle, whose centre is 0, with angular velocity a, each w describes a circle whose centre is 0', with angular velocity 3u/2. Transformations which conserve angles are known as isogonal, or orthomorphic ; and the one figure is called the conform represen- tation of the other. If to = M + iu be a one-valued monogenic function of a; -f iy, the sj'stems of orthogonal straight lines x = a, y = h transform into systems of orthogonal curves in the ?«-plane. There may be excep- tion at points given by dw/dz = 0. We proceed to a brief discussion of some simple cases of ortho- morphic transformation. § 22. I. w = az + b. Let z describe any curve. The ?«-curve is obtained from the 2-curve by (1) changing the length of each vector from the origin in the ratio j a j : 1, (2) turning the curve about the origin through an angle = am (a), (3) givincr the curve a displacement b. The nature of the curve is evidently unaltered by these processes. §23. II. w=z\ Let zz=p (cos $ + i sin 6),iv = p' (cos 6'+i sin 6'). Then p' = p\ 6' = 2 0. When 2 describes the line p cos (5 — o)= k, w describes the curve p'cOS'(e'/2~a) = ^\ a parabola with focus at the origin 0'. When the line passes througli 0, K = ; ^ and ^ -f TT give congruent values of $' ; and the w curve ia GEOMETRIC INTRODUCTION. 15 a line from 0' to infinity, described twice. A semicircle with centre at gives a circle with centre at 0'. If 2 describe a circle, with centre a (which may without loss of generality be taken on the real axis) and radius c, then p'^ — 2ap cos e = c- — a', and the w-curre is p' — 2a Vp' cos 6'/ 2 = c^ — a^, a Cartesian. Since there is a node at 8' = tt, p' = c^ - a^, the curve must be a lima9on with a focus at the origin. [See Salmon, Higher Plane Curves, 3d edi- tion, p. 252.] a+o Fig. 8 To study the relations between the s-path and the w-path, we observe that, since both z and — z give the same point ic, the lima^on corresponds not only to the circle already considered, but equally to another circle, the two forming a figure symmetric with regard to the origin (Fig. 8). The points of intersection of these circles give the same to-point, say n'. If we start from n and describe the right-hand circle positively, when 2 is at — n, lo has returned to n', so that the arc n, — n of the circle gives a loop of the jo-path. The other arc —n,n gives another loop of the w-path. The reader will find it interesting to con- trast the manners in which the nodes arise in this paragraph and in § 47. 16 GEOMETEIC INTRODUCTION. § 24. III. w = z". Here p' = p", 6' = nO. The line p cos 6 = k gives the curve p'^'" cos 6' /n = K. We then get a well-known group of curves. A fundamental property of these curves follows from the principle of isogonality. Let ij/ be the angle which the 2-line makes with the stroke z ; then >p is also the angle which the Mi-curve makes with the stroke w. Since 6'= nO, and 0-^-^ = ir/2, therefore i// = ir/2 — 6'/n, the property in question. Fig- 9 The half-line 6' = ^ (mod 2 jr) represents the n equiangular half- lines 6= (P + 2mw)/n (m = 0, 1, •••, n — 1), while the whole line 26' = 2/3 represents the 2?t equiangular half -lines 6 = {/3 + rrnr) /n (m =0, 1, •", 2n — 1), which make up n whole lines. The circle p'^ — 2ap cos 6 + a' = c^, gives the curve p'V» — 2ap"/" cos ff'/n + a^ = c^. f/S. 70 GEOMETRIC INTRODUCTION. 17 This curve is easily sketched directly from the relation w = z". Let us, for instance, take n = 3 and suppose that the origin o is inside the circle and that the axis of x is a diameter. The points whose amplitudes are ± tt/S give the same value of w, i.e. give a node ; the points whose amplitudes are ± 2 ir/S give another node. As am (z) increases from to t, \z\ (Fig. 10) decreases, and therefore, as am(to) increases from to 3 tt, \w \ decreases. § 25. IV. w = az" + &«' where a, ft are real. Let z describe a unit circle with centre ; let the angular velocity be unity. The motion of w is the composition of two rotations : (1) a point w^ rotates with angular velocity p in a circle of radius a, centre 0' ; (2) ic rotates with angular velocity g in a circle of radius b, centre w,. The motion is thus epicyclic. The reduction to trochoidal motion can be effected in two ways. Fig. 11 (1) If we divide the stroke w, at I, and draw circles as in Fig. 11, choos- ing I so that the arcs Im, In are equal, then to is a point fired in the circle (w,) which rolls on the circle (0')- This happens if the radii of the circles be in the ratio (q —p)/p, so that the radius of the fixed circle is (3 —p)a/q, that of the moving circle pa/q. (2) We may write our original relation in the form w = bz^ + azr ; interchanging a, 6 and p, q, the same curve is produced by the rolling of a circle of radius qb/p on a fixed circle of radius (p — q)b/p, the tracing-point being at a distance a from the centre of the rolling circle. The w-path becomes cycloidal when the tracing-point is on the rolling circle, i.e. when pa = qb. A convenient pair of equations for the trochoid is obtained by writing io for the complex quantity conjugate to w. Then, since zz = l, to = az' + bzi, w = az »> -f 6z-«. 18 GEOMETKIC INTKODUCTION. §26. V. In III. let n = -l; then p'=l/p, 6'=-6. If w be represented on the 3-plane, the point w has polar co-ordinates 1/p and — e, and is therefore the reflexion in the real axis of the geometric inverse of z with respect to a circle whose radius is unity and centre 0. This combination of reflexion and inversion is called by Professor Cayley quasi-inversion. Since ti-z = 1, to each value of z, other than zero, corresponds one value w, but when 2 = 0, w = oo . This leads us to consider infinity as consisting of a single point, not of infinitely many. In ,the Theory of Functions, all points at an infinite distance from are supposed to coalesce in a single point z = cc . § 27. To make this supposition a natural one Neumann, follow- ing out one of Eiemann's ideas, chose as the field of the complex variable a sphere instead of a plane. Let 0, 0' be opposite points on a sphere of diameter 1. Take the tangent plane at o as 2-plane. Join each point z to 0', and let the joining line cut the sphere at p. Then the value z is to be attached to p, and we may treat the sphere as the field on which z assumes all its values. Each point at a Fig. 12 finite distance in the plane is replaced by a single point on the sphere, but all the infinitely distant points of the plane are replaced by the single point 0'. This process of replacing the plane by the sphere may be regarded as the converse of stereographic projection ; or, again, as the inversion of the z-plane with regard to the external point 0'. Let the tangent plane at 0' be the w-plane, with 0' as origin, and the line O'm in which xOO' meets this plane as real axis. Let angles be measured counter-clockwise for observers standing on the sphere GEOMETRIC INTRODUCTION. 19 at and 0', and let the line Op meet the ty-plane at w. Then the amplitude of to = — (the amplitude of z). Again, since the triangles OO'w, 2OO' are each similar to 0/)0', \z\\w\ = l. Hence toz = 1, or the curve traced out by w is the representa- tion, for wz = 1, of the z-curve. When z describes positively a curve which includes 0, w describes negatively a curve which includes 0' ; but otherwise a positive description of the z-path is accompanied by a positive description of the w-path. This transformation w = 1/z is of great use, for by its help the consideration of the behaviour of a function of m at w = is sub- stituted for that of a function of « at z = ao. § 28. The transformation wz = k gives | w | | 2 | = | fc |, and am w + am 2 = am A:. It is (1) an inversion with respect to a circle, centre and radius \k''-\, (2) a reflexion in a line which bisects the angle xOk. That is, it is a quasi-inversion. Now any circle remains a circle after inversion and also after reflexion. Hence, any circle in the z-plane becomes a circle in the to-plane. The following are the special cases : — (1) A circle through the origin in the 2-plane becomes a line in the to-plane {i.e. a circle of infinite radius) ; (2) A line in the 2-plane becomes a circle through 0'; (3) A line through becomes a line through 0'. "When to and z are represented in the same plane, all the jjairs of points which satisfy icz = k are said to form an involution. The points ± -^k, at which to and z coincide, are the double points ; the points of a pair are said to be conjugate ; and the point 2 = 0, whose conjugate is 2 = 00, is the centre of the involution.* § 29. The most general bilinear transformation is lu = (az -f b)/{cz -\- d). It contains four constants, but only three ratios. We may there- fore assume any relation between a, b, c, d. The most suitable one is ad — bc= 1. This transformation can be built up out of simpler ones, for writing w—a/c=w', z+d/c=z', so as to change both origins, we get w' = - 1/c-z', * GeDerally, if U and V be integral functions of z, of degree n, the points given by P'+AF=0, where A la arbitrary, form an n-ic involutioD. When n = 2, this gives a relation between two points of the form ZyZ^ — fi{Zi + z^) + v = 0; when «, = « , Zi= fi^ and taking fx aa a new origin we have a relation of the form in queetion. 20 GEOMETKIC INTRODUCTION. which is the quasi-inversion. Accordingly, circles change into circles with the following exceptions : — (1) A line through the 2'-origin, that is, a line through — d/c in the z-plane, becomes a line through ajc in the w-plane; (2) A circle through —d/c becomes a line through a/c; (3) A line in the «-plane becomes a circle through a/c. It is to be noticed that when z = —d/c,w = X) , and when « = CO , M = a/c. Hence if we regard a straight line as a circle of infinite radius, the bilinear transformation transforms circles into circles, without exception. We shall now show that the transfor- mation is orthomorphic throughout the plane. We have dw/dz = l/{cz + dY. The doubtful points are z = — d/c, w = oo , and 2 = oo , w = a/c. To study what happens in the neighbourhood of « = oo , write z = 1/z', and examine the behaviour of w in the region of the new origin z' = 0. We have w = (a+ hz')/{c + dz'), dw/dz' = {he — ad)/(c + dz'y. Hence when z' = Q, w = a/c, dw/dz' ——l/c; and the transforma- tion is orthomorphic at 2 = oo . If the equation w= (az-f6)/(c« -\-d) be solved for z and w be put = l/w', we find in the same way that dz/dw' is finite and not zero, when w' = 0. Thus the orthomorphism exists throughout the plane. Take four points z^, z^, z^, z^, and let the corresponding w's be M,'i, Wj, Wj, Wf. Then w, = (az, + h)/{cz, + d), w. = (az. + b)/{cz. + d), w, - IV. = (ad - be) {z, - z.)/(cz, + d) (cz, + d). Hence {rc,-w,){tc^-iCi)/{iv,-w3){w,-iv,) = {z,-Z2){z,-Zt)/{Zi-z,)(z2-Zi). Each of these is called an anharmonic ratio of the four points considered, and we have the theorem that an anharmonic ratio is unchanged by a bilinear transformation. [This theorem was given by Mobius : Die Theorie der Kreisverwandtschaft in rein geometrischer Darstellung. Abh. d. Kgl. Sachs. Ges. d. W., t. ii. ; or Ges. Werke, t. ii.J GEOMETRIC INTRODUCTION. 21 Since the relation between w and z involves three arbitrary ratios, any three points in the tu-plane can be made to correspond to any three points in the 2-plane. Mobius's theorem then gives the point in the w-plane, which corresponds to any fourth point in the z-plane. § 30. The anharmonic ratios of four iwints. Let Zj, Zj, Zj, 24 be any four points, and let \, /x, v stand for (zj - Z3) (Zi - zj), (Z3 - Zi) (z_, - zi), (Zj - Z2) (Z3 - Zi), so that X + jii + r = 0. Any one of the six ratios — ix/v, — v/A, — X//1, — v//x, — A./»'> — /^A is an anharmonic ratio of the four points. Let o- be any one of these, say — /a/i/. Then —\/v=l — %) ■ and, interchanging v and v"^, i ' ' ' (^^'^ ll_ = — {Z-Xi + v\Zi + VZ1Z2) /(2i + v\ + V23) _ The equations i., ii., iii., v., refer to A+. When z^ = h_, we must change the sign of ir/3 in the equations v. § 33. The Covariants of the Cubic. Let Zi, 2o, Z3 be the roots of 2^ + 3aiZ- + 3a^+as = 0. We have proved (§ 29) that a bilinear transformation does not change an anharmonic ratio. The points 7t^, 7t_ are deiined by special anhar- monic ratios and are therefore covariant. Hence if we form the quadratic whose zeros are 7i_,., h_ and notice that its coefficients involve z,, z,, Z3 symmetrically, we see that it is a covariant of order 2, i.e. it is the Hessian of the given cubic U. With regard to the cubic whose zeros are J], j.,, js, which is equally a covariant, "we know that the cubic, each of whose zeros forms with those of the given cubic a harmonic system, is the cubicovariant or Jacobian. [See Salmon's Higher Algebra, 4th ed. p. 183; Clifford's paper. On Jacobians and Polar Opposites, Collected Works, p. 27. J § 34. The canonical form of the cubic. The analytic condition that a triangle be equilateral is, if z/, z^', 23' be the points in positive order, (2i' -23')/(2^i' - ^J) = cos 7r/3 + i sin 7r/3 = - v\ or z,' + VZ2' + v\' = (vii.) But from (iii.) we have, when h^ is origin, l/2i + v/z, + „VZ3 = 0. Thus the quasi-inverse of z„ z.^, z^ as to h+ is positively equilateral, and therefore the inverse triangle itself is negatively equilateral. The triangle is now reduced to its canonical or equilateral form. The inversion has sent one Hessian point to infinity. If we now invert with regard to the other, the triangle must remain equilateral. Hence this other Hessian point must be the centre of the triangle. Taking this point as origin, we may write for the cubic U GEOIMETKIC INTRODUCTION. 27 In the canonical form, ji and z^ are harmonic with z^ and 23. Hence, jJJi is the counter-triangle, and the figure Zij-^z«jiZ^'2 is a regular hexagou. § 35. After the Jacobian and Hessian points we consider the polars or emanants of the triangle. "We shall consider, in the first place, the general case in which there are n points z^ given by an equation f{z) = 0, of the 9ith order in 2. Introduce the unit y by writing z/y for 2, and multiply by y" so that the equation takes the homogeneous form f{z, y)=0. Take any points z/y and z'/y', and let 2, = (2 + X^')/{1 + K) = (2 + X^')/(2/ + Ky') ; the n quantities A,, are roots of f(z + Xz', y + \y') = 0, which, if we write A t O , t O is either 8,8 f(z, y) + \A/(2, 2/) + 1^ A=/(2, 2/) + . . . + ^ A"/(2, y ) = or ^ A'»/(2', 2/') + ^^, A""->/(2', 2/') + • • • + A"/(2', 2/') = 0. n ! }i — 1 ! The coefficients of the powers of A are well-known covariants. We call the n — r points, given by A'/(2, y) — 0, the rth polar of the pole 2' with regard to the given n points. The geometric meaning is easily seen. "When A"/(2, y) = 0, the sum of the products of the A.'s, n — s at a time, is zero. But \, = (2-2,)/(2,-2'). Hence the sums of the products of {z, — z)/{z^ — 2'), n — s at a time, or of (z,. — z')/{z, — 2), s at a time, is zero. For the first polar 2(2,-2')/(2.-z) = or n/(2'- 2) =21/(2,-2), so that «' is the harmonic mean of the n points z, with regard to any first polar point of 2'. 28 GEOMETRIC INTRODUCTION. Comparing the two equations for X, we see that A'/(2, y) and A'"~'/(3', y') differ only by a numerical factor. Hence if 2 be an sth polar point of z', z' is an (u — s)th polar point of 2. § 36. By means of the polars of a group of points it is possible to frame a geometric definition of the Hessian and Jacobian. Let us consider the points whose first polars have two coincident points, or (say) a double point. We have 2'S/782 + 2/'8//83/ = 0, and if z be a double point, 2'S=//S2^ + 2/'Sy/8(/S2 = 0, 2'8y/8]/S2 + 2/'S!/78/ = 0. Hence such double points are given by the equation hz- SySz BySz Sj/- The expression H is the Hessian of the given function / Again, consider the points whose first polars with regard to / and H have a common point. We have z'8f/Sz+y'Bf/By = 0, z'BH/Sz + y'm/Sy = 0, whence such common points are given by the equation 7i\n-iyH = :0. n{n-2)J = ¥ 8/ hz % m m 82 ^y = 0. The expression J is the Jacobian of the given function. For a full discussion of the theory of the polars of a binary form, see Clebsch, Geometric, t. i., ch. 3. In the case of two points, say 2- = a, the polar of 2' is given by ez' = a, so that the polar point is the conjugate of z' with regard to the given points. For the triangle, we use the canonical form z^ = a. The polar pair of z' is given by GEOMETRIC INTRODUCTION. 29 Hence the polar pair of any point is harmonic with the Hessian points and oo ; and, in particular, the points which form the polar pair of a Hessian point coincide with the other Hessian point. § 37. We shall now transfer the figure from a plane to a sphere. This can be done by geometric inversion with regard to an external point s'. Let the constant of inversion be equal to the distance from s' to the plane, so that the plane and the sphere, derived from the plane by inversion, touch at a point t. The fundamental facts of ordinary geometric inversion hold good, — that a circle becomes a circle and that the magnitudes of an angle and of an anharmonic ratio are unaffected. Hence a harmonic quadrangle becomes a harmonic quadrangle ; an equianharnionic quadrangle becomes a tetrahedron in which the rectangles formed by opposite edges are equal, while the inverse of such a tetrahedron is another of the same kind. In particular, the inverse as to a ver- tex is an equilateral triangle and the point at infinity in its plane. "We wish to choose s' so that the fundamental triangle ZiZ.^^ shall become an equilateral triangle on the sphere. This will happen when the tetrahedron s'z^z^.^ is equianharmonic ; that is when, if ^1) ^2) ^3 denote the lengths of the sides, and 81, 82, 83 the distances from s^ to the vertices, 81X1 = 8A2 = 83X3. Each of these equations giving a sphere, we have for the locus of s' a circle in a plane perpendicular to the given plane, and meet- ing that plane at the Hessian points h^, h_. Choosing any point s' on this circle, let us denote the inverse of any point 2 in the plane by 2'; then the triangle ZjZ^^ becomes an equilateral triangle z^z^z^ on the sphere, and the points 7i^., A ., which are equianharmonic with Zi, z^, 23, become the ends A+', /i_' of the axis of the small circle z^z-Jz^. If we choose s' so that this circle is a great circle, we have a simple dihedral configuration. [See Klein's Ikosaeder.J Consider the set of great circles through h'^, h'-, and the orthogonal set of small circles. These become in the z-plane two sets of coaxial circles ; h^, h- are the common points of the first set and the limiting points of the second set. We have now to consider the plane z/z/zj' as the projection of the z-plane from the vertex s'. Let us denote the projection of z by f, of a by o, etc. It is evident that whenever a circle is projected into a circle, the symmedian point of an in-triangle becomes the symmedian point of the projected triangle. For the symmedian point was defined by means of conjugate chords of the circumcircle ; and conjugate chords project into conjugate chords. In an equilateral triangle the symmedian point is the centre j hence the projection of k is k, the centre of the circle z^iz.Jz^'. 30 GEOMETKIC INTKODUCTION. Next let c be the circumcentre of the original triangle. It is the pole of the line at infinity with regard to the circumcircle ; hence its projection 7 is the pole of the intersection of the f-plane and the tangent plane at s', with regard to the circle fif^fa. Hence the chords c's' and a/3 (Fig. 16) of the meridian circle Fig. 16 s'ft'+ft'- are conjugate, and therefore the chords c'k' and a/3 are parallel (see Fig. 15) ; this shows that the points c', i-'are harmonic with h'+, h'-. Hence c, k are harmonic with ft+, A_, and further the four points are coUinear. § 38. The three quadratic involutions determined by four points. Let Zi, Z2, 23, Zi be the zeros of a quartic. The points can be paired ofE in three ways : (23) (14); (31) (24); (12) (34). Now two pairs of points define a quadratic involution. Let e,, 62, 63 be the centres of the three involutions obtained from the three divisions of the four points into pairs. We have, to deter- mine Ci, (22 - Ci) (23 - 61) = (zi - e,) (2. - Ci), whence Cj = (22Z3 — ZjZt) / (22 + Z3 — Zj — Z4) . The focus of the parabola which touches the lines joining Zj, Z3 to 2i, Zj is fii. For let the line joining 2, and 2„ touch the para- GEOMETRIC INTRODUCTION. 31 bola at the point «,„, and let / be the focus. Then by the properties of the geometric mean /= = ±V(231 -/) («12 -/), /= ±V(2i2- -/)(2.4- -/), /= = ±V(% -/)(234- -/), Z4 -/= ± V(Z,4 -/) (Z34 -/)• The amplitudes of points on the parabola, when referred to the focus and axis, vary continuously from to 2ir. Let am ( + V2 — /) be e/2 ; then am ( + V(2,„-/)(2,„-/) ) = (6,™ + e,„) /2. But geometric considerations show that am {z,—f)= the same quantity. Hence the + sign must be chosen throughout, and (^=-/)(23-/) = (2.-/)(^4-/), whence / and ei coincide. Accordingly, to construct the point e„ we observe that it lies on the circumcircle of the triangle formed by any three of the four lines ^1^2) ^1^3) ^2^« ^3^i' [See E. Eussell on the Geometry of the Quartic, Lond. Math. Soc, t. xix., p. 56.] Let Ci, c.?, C3 be the constants of the three involutions, and let us eliminate the 2's from the six equations, (z, - ei) («3 - eO = (zi - ei) (2^ - e^) = Ci' ' (23 - e,) (zi - e.) = (2j - 62) (2i - fj) = Ca^ I (i.) (2l - 63) (Zj - 63) = (23 - 63) (24 - 63) = C3- We have 23 — ej = c// (2j — Cj) + Cj — e,, Z2 — ei = C3V(zi — 63) + 63 — ej. Hence Ci2 = {c2V(2i-e,) + e2-ei} {037(21-63) + 63 -fill . . (ii.) And, by proceeding similarly with the other pairs of quantities, we have precisely the same equation for 22, Z3, z^ Hence every coefficient of the quadratic (ii.) is zero. Now (ii.) is {C,' - (62 - Ci) (63 - Ci) I (Zi - fij) (Z, - 63) - C2V - Ci (63 - Cl) («i - 63) - 03^(62 -ej)(«i-e2) = 0. 32 GEOMETRIC INTEODTJCTION. Hence cf = ( e. — fij ) ( 63 — ej) , and similarly c/ = {e^ — e.^) (e^ — e^), If we regard the points e^, e,, e.j as given, the constants c/, c./, Cj^ are also given. We have only three independent equations to determine the four z's, say (2i-ei)(24-ei) = (e,-e,)(«3-ei) 1 {z,-e,)(z,-e.;) = {e,-e,){ei-e,} I . . . . (iii.) (23 - 63) (z, - 63) = (fi -63) {e, - e,) j From these follow three equations of the form {z, - eO («3 - e,) = (e, - e,) (e- - e^), a result which can be verified directly. We can state this in words as follows : Three involutions are determined by any four points ; the con- jugates of any point in two of these involutions are themselves conjugates in the third involution. Prom (iii.) we see that one of the points, z^, is arbitrary ; and when it is chosen, the rest are at once constructed from the preced- ing equations. Corresponding to the values 214 = 00, Ci, e^, 63 we have Zi = e,, oc , 63, 62 ; Zi = e.,, 63, co , gj ; 2:3= 63, e,. ei, 00 . § 39. The Jacobian of the quartic. The double points of the preceding involution are given by (z-e,y- = c^ (X=l, 2, 3). Let JK=e^+C),=e^+^/{e^—e^){e^—e,) (X, /x, v=l, 2, 3 in any order), j\ = Ca — Ca = e^ — V(eA — e^) (e^ — e^) . Now when we invert the points of an involution we obtain a new involution, and the old double points become the new double points. This is clear because any pair of points is harmonic with the double points. Accordingly the six points j are covariant. They are the zeros of a covariant of the sixth degree, the Jacobian of the quartic. § 40. Any two pairs of Jacobian points are harmonic. Tor ( j^ - e,) (i^' - e,) = (e^ - e. + c^) (e^ - e, - c^) GEOMETRIC INTRODUCTION. 33 \2 But c^' + cj' = {e^-e,y Hence {j^ - e,) {j^' - e,) = Cy% and J^, JiJ are harmonic with j^,, jj. § 41. The number of quartics which have a common Jacobian is singly infinite, for we have seen that one of the points, z^, is arbitrary. Each of tliese quartics is determined conveniently by means of its centroid m, as follows : We have 4m = x^ + x, + Z3 +«4, ei = (-^.?3 - 21^4) / (22 + Z3 - 2i - ^4) ; therefore 4(ei - m) = ^ (zj - z^)- - {z, - z^Yl/iz., + z.^-Zi- z^), and 16(60 — m) (63 — in) = (z.^ + 2:3 — 2:1— z^y. Xow (zo + 23 — Zi — «4)/4 is the stroke from m to the centroid of z, and Z3. Hence the centroids of the sides are determined, and the quartic is at once constructed. We shall denote the quartic of the system, whose centroid is m, by U„. Let us write the cubic whose zeros are e^, e.,, e^, so that the origin is the centroid of e„ e.,. 63, and therefore also of the Jacobian. Regarding this as a quartic with an infinite root, let us calculate the Hessian H. We have ^ _^ 24zy, 12z=-3^^= - ~ ' *^ 12 z^ - 3j7#, - Ggzy - 12g^' = -(z^ + i^2Z- + 2£73Z + T-V?/); thus the centroid of this quartic is at the origin, so that H^ = Uq. Any quartic U„ is now for in this last expression the coefficients of 2* and s? are — 1 and im, so that the centroid is m. It is proved in works on Covariants (see Burnside and Panton, 2nd ed., § 170 ; or Clebsch, Geometric, t. i., ch. iii.) that the Hessian oi kU+XIUs 8/c 8A 34 GEOMETRIC INTEODUCTION. ■where 4 n = 4 k^ — g^KX" — g^X^ ; so that if the Hessian be k' U+ X'H, the ratio of "' to X' is given by k'- — \- X' -— = ; and the geometric OK oX statement of this is that the centroid of the Hessian is the second polar of the centroid of the quartic with regard to the points e^, e,, e^. Thus the centroid of the Hessian is determined, and the Hessian can be constructed. If the Hessian be given, the first polar of its centroid, with regard to e^, e.,, e^, consists of two points, which are the centroids of the two quartics which have the given Hessian. § 42. Tlie anharmonic ratios of the four points. If the four points z,, 2,, z.^, z^ of § 30 be replaced by «], e^, e^, oo, we have ^/•'= (e3-ei)/(ej-e,), or X/(e2 - ea) = /^/(ej - e,) = v/(e, - e^) = {ii.- v)/{e, + e^ -2 e^). Now Ci + ej + 63 = 0, e„e3 + e.^e^ + e^e.. = — g,/4:, e^e.f^ = gf3/4, (62 - 63)^(^3 - e,)-(ei - e,y = (^/ - 27g/)/16 = A/16. Therefore /a — v = ke^, v — X = Tce^, X — /u, = ke^, (^-y)(y-X)iX-f.)=k'^g,/i, l/xv - 2X' = - k%/4: ; or, since 2X = 0, 3 2/iiv = — k-g.j/4: ; and since 3X= k{e.^ — e.2), 3« xyv' = fc" A/16. Eliminating k, _4(2Atv)^ _ (jit - vY (y - xy jx - ^y 27xvv The meaning of the vanishing of an invariant is evident from these equations. If g.^ = 0, then 2/xv = 0, and this, combined with 2X = 0, gives either X = vfj. = v^v, or X = v''iJ. = vv; that is, the points are equianharmonic. If g^ = 0, then two of the quantities X, ^, v are equal and the points are harmonic. GEOMETRIC INTRODUCTION. 35 § 43. The invariants ofmU^ + H^. In works on the theory of covariants it is shown that the invari- ants of this form are -3H'/i, J'/16, where H' is the Hessian and J' the Jacobian of the cubic 4 vf — g.m, — g^. Hence the quartic is equianharmonic when m is a Hessian point of the triangle efi^e^, and harmonic when m is a Jacobian point. In other words, we have in the quartic involution m U^ + H^ two equi- anharmonic and three harmonic quadrangles ; the polar points of Si, with regard to them, are respectively the Hessian points and Jacobian points of ZiZ^^. § 44. Canonical form of the quartic. Let one Jacobian point j^ be sent to infinity by inversion and let j\' be chosen as origin. Since two pairs of points are harmonic with j^, j/,', the quadrangle becomes a parallelogram, and the quartic takes the form {l-z^)(l-k~z-). The other pairs of Jacobian points, being harmonic with j),, j^' and with one another, form a square. § 45. Hie representation of the quartic on the sphere. Jjct the regular octahedron be constructed, of which the above- mentioned square contains four vertices, let one of the new vertices be selected as origin, and let the a-plane be inverted with regard to this origin. The six Jacobian points become the vertices of a regular octahedron in a sphere. Let |, i;, ^ be the co-ordinates of any point on the sphere, referred to the three diameters of the octahedron as rectangular axes. The other points which, with this point, make up a quartic with the given Jacobian, have co-ordinates For evidently any two of the four points are harmonic with regard to the points where an axis meets the sphere. The extremities of those four diameters of the sphere, which are perpendicular to the faces of the octahedron, are the vertices of a cube inscribed in the sphere ; the co-ordinates of these vertices are, if the radius of the sphere be 1, ±1/V3, ±1/V3, ±1/V3, forwehave e=r,- = S', and e + n' + ^ = '^- 36 GEOMETKIC INTRODUCTION. Choosing from the eight points those whose co-ordinates have an even number of minus signs, we obtain a reguhar tetrahedron. The other four, whose co-ordinates have an odd number of minus signs, form another regular tetrahedron, which we call the counter- tetrahedron of the former one. These tetrahedrons represent equi- anharnionic quartics of the involution. Again, draw the six diameters which are perpendicular to the edges of the octahedron, and, therefore, also to the edges of the cube. We thus get twelve points on the sphere whose co-ordinates are given by the scheme 0, ±1/V2, ±1/V2, ±1/V2, 0, ±1/V2, ±1/Vl'. ±1/V2, 0. The four which lie in any co-ordinate plane form a square, which from the form of the co-ordinates belongs to the involution. Hence the twelve points represent the three harmonic quartics of the involution. [For further information on these subjects the student is referred to Klein's Ikosaeder, to a memoir by Wedekind published in Math. Ann., t. ix., and to Beltrami's paper, Eicerche sulla geometria delle forme binarie cubiche, Accademia di Bologna, 1870.] § 46. Examples of many-valued functions. Example 1. If w be defined by the equation iv- = z, there are for a given z two values of w. li z = p(cos 6 + i sin 6), these values are Wi = Vp (cos (9/2 -f (• sin 0/2) | W2 = Vp(cos(e-f27r)/2-f-jsin(e + 2,r)/2)i " ' ' ' ^^■' We have always ii\ + u:, = 0, and when 2 = or oo the two roots Wi, tt'2 become equal. When z describes, in the 2-plane, a path which does not pass near the origin, there corresponds to a small change in z, a definite small change in either root; in this way it can be seen that the two points w^, w^ in the w-plane trace out con- tinuous paths as z moves continuously in the «-plane from one posi- tion to another. Moreover, assigning to one of the initial values of w the name w,, the final value of w^ is completely determinate. If 2 describe a closed curve round the origin, starting from a GEOMETRIC INTEODTJCTION. 37 point p, 6, the final values of p and 6 are p, 6 + 2tt, and the final values of lo^ tu.2 are If, = Vp J cos (e/2 + tt) + / sin (6/2 + tt) ; , zuj = Vp {cos (6/2 + 2 tt) + I sin (6/2 + 2 :r) j . Comparing tliese with equations (i.), we see that Wy, iVj have changed into iv.,, ivi. If z describe a contour which does not include the origin, the final values of p, are the same as the initial ones, and the final values of iv^, tCj are lu^, lu-i. Example 2. w- = {z — Oj) (z — a,) •■• {z — a„). Let C be a contour which contains %, ctj, ••• a„ but not a,+], a^+j, • •• o„. "When z, starting from z'. describes C, each of the ampli- tudes Oi, 6-i, •••, 6< of z — Ui, z — a,,, •••, z — a^ increases by 2 it, while those of 2 — o,+i, 2 — a,+2> •••> 2 — a„ return to their initial values. Let the initial value of the branch tfj be Vp,p2--- p„{ cos (6, + e,+ ■■■ + e„)/2 + isin (6, + 6,+ ••• + 6„)/2j, where 2 — aA = ^^(cos 6a + i sin 6a) . The final value is Vp,p2---p„Jcos(6i+6o+"-+6„+2K:r)/2+(siu(6, + 6,+ .-.+6„+2K,r)/2}. This is the same as the initial value of w, or Wj, according as k is even or odd. § 47. Example 3. a^w^ + 2 CTiIu + a^ = 2. By a linear transformation of w the equation becomes M= - 1 = 2. Let the branches be z«i and tfj. We have always lOi + w., = 0\ when 2 describes the circle 1 2 1 = k, the path of either w-point is given by |W-1||W + 1| = K, or p\P« = K) Pi and Pi being the distances of the point from ± 1. This is the equation of a Cassinian. (1) Let K be small. Then pi or p^ must be small, so that the curve consists of two small ovals round the points +1, — 1 ; iv^ must remain on the one oval, 110.2 ^^ the other. 38 GEOMETRIC HvTTKODUCTION. (2) Let K grow ; tlie ovals grow, but remain distinct until k = 1, when the circle passes through the point z = — 1, and the ovals join at «; = 0. If z, instead of passing through — 1, describe a small semicircle round it positively, then (Fig. 17) gj is consecutive to j)^ and q^ to Pi. But if 2 describe a semicircle negatively, q^ is consecutive to Pi, and jj to pi-i- The lines p,p., and q^q^ are of course at right angles. Fig. 17 (3) When k > 1, the curve is at first an indented oval, and after- wards an ordinary oval. As z describes its circle, w, and w^ together describe the whole oval, and the final position of either is the initial position of the other. § 48. TJie relation a^vf + 3 a,io- + 3 um + as = z. By linear transformation this reduces to w^ — Sw = z (!)• GEOMETEIC INTRODUCTION. 39 As in Cardan's solution of the cubic, put xc = t + \/t (2), tlien 2 = «3 + i/t3 (3). If z be given, we have six values of t, namely, t, vJ, vH, l/t, v-/t, v/t, and three values of lo, namely, Wj = J + l/t, IC^ = Uf + U2/J, 1^3 = vH + v/t. Let t describe the circle | j | = k ; the point z ^Yill describe an ellipse whose foci are 2 = ± 2 and -whose semi-axes are a = ifi + 1/k^, /3 = ] k^ — I/k^ j, and the three w's will describe an ellipse whose foci are ± 2 and semi-axes a^ = k + \/k, P^ = \k-\/k I, whence {a, + /3,)/2 = V(^+|3)/2, (a, - ^,)/2 = V(a-/3)/2. From either of these equations, and from ai" — §(■ = a- — (3^, the w-ellipse is determined when z is given. If S be the amplitude of t, the eccentric angle of 2 is 3 9, and those of w,, w^, w^ are 0, 8 + 2 t/3, 6 + i t/3 ; so that the points w form a maximum triangle in the Mj-ellipse. Therefore, when z is given, the points w are determined. In this way, if 2 describe any curve, say a circle with centre 2', it is easy to con- struct the id-curve. If !c,', ic.J, 1C3', be the points which correspond to 2', the equation of the ic-curve is I 10 — ^c^' \ \w — i€./ \ ' ic — ic/ \ = a constant, as is evident from the relations (io — ic/)(w — ic./){w — ic-/)= ic' — 3 10 — 2' = 2 — 2'. Two of the family of curves should be specially noticed. The points at which values of to become equal are given by mv" — 1 = 0, and are 2 = ± 2. Thus to the circles with centre 2' through 2 = ± 2 correspond tc-curves with nodes at !0 = ±1. Since for these values of z two values of w become equal, we have, approximately, near these points (,5±l)2 = c(2±2), so that the branches of the w-curve cut at right angles at these points. Fig. 18 shows these two special to-curves. Fig. 18 40 GEOHETEIC IXTEODUCTION. The orthogonal family of curves correspond to z-lines through z'. Their equations are of the form ^1 + ^2 + *j = 3, constant, where 9,. is the inclination, measured from the real axis, of the line from the fixed point u\' to u\ Since z' is any point on the j-line, the fixed jjoints iCr' can be chosen in infinitely many ways ; they may be conveniently taken to coiTe- spond to the point where the s-line meets the real axis. If the z-line pass through a point at which two branches become equal, the equation takes the form 2 9, + 9.^ = a constant, a nodal cubic, which can be very readily constructed. CHAPTER II. Real Fuxctioss of a Eeal Yakiable. § 49. One of the most noteworthy features in the history of mathematical activity during the nineteenth century is the careful revision to which the fundamental processes and concepts of analysis have been subjected. Cauchy, Gauss, and Abel were the first to insist upon the importance of basing the theories of their day on securer foundations, and they have been followed by a long line of brilliant successors. B. Bolzano of Prague (1781-1848) worked on the same lines, but his writings failed to attract the attention they deserved, until a comparatively recent date. The discovery that continuous functions exist which do not possess differential quotients and that a discontinuous function may possess an integral, the new light thrown upon discontinuous functions by Dirichlet's well-known memoir on Fourier's series and by Eiemann's masterly researches on the same subject, have all tended to reveal how little is known of the nature of functions in general, and to emphasize the need for rigour in the elements of Analysis. In the course of this revision it became necessary to place the number-concepts of Algebra upon a basis independent of, but con- sistent with, Geometry. If a zero-point be selected on a straight line and also a fixed length, measured on this line, be chosen as the unit of length, any real number a can be represented by a point on this line at a distance from the zero-point equal to a units of length. Conversely, each point on the line is at a distance from the origin equal to o units of length, where a is a real number. This theorem may seem evident, but a little reflexion will show that it cannot be true unless the word number is so defined as to make the number-system continuous instead of discrete. Definitions which will fulfil this requirement have been given by Weierstrass, G. Cantor, Heine, Dedekind, and others. Much of the theory of the higher Arithmetic is due to Weierstrass and was com- municated to the world in University lectures at Berlin. His start- 41 42 EEAL FUNCTIONS OF A REAL VARIABLE. ing-point is the positive integer, and the number-system is extended so as to include, successively, positive fractions, negative, irrational, and complex numbers. One noticeable element in his theory is the exclusion of geometric concepts. Certain laws of calculation can be stated, in general terms, only after the introduction of the new numbers, and it is for this reason, and not because of any geometric representation by extensive magnitudes, that these new numbers, namely the fractional, the negative, the irrational, and the complex numbers, are introduced. § 50. Hie irrational numbers. Three different definitions have been employed, due to Weier- strass, Dedekind, and G. Cantor. The following account of Cantor's method is based on a memoir by Heine (Crelle, t. Ixxiv.). Cantor's method. The starting-point is a sequence aj, a,, a^. •■• of rational numbers. This sequence is determinate when the law of construction of the general term a„ is known. A regular sequence is one for which a finite number /x can be found, such that for all values of n {> /j.) the absolute value of a„^j,—a„ {p = 1, 2, 3, •••) is less than an arbitrarily small positive rational number e. [Through- out this chapter and the next, c will be used, always, to denote an arbitrarily small positive number.] When the sequence a^, a,, 03, ••• is regular and when a rational number a exists which is such that a finite integer /x can be found which makes |a— a^^+p \ ^, I a„ +. - «„ l< V2, 1 &n+P -K\< V2 (P = 1, 2, ...) ; and from these two inequalities it is an immediate deduction that |(«„+,±&„ + p)-(«n±&n)| <£(P = 1, 2, ...). II. The sequence afii, ajb^, a^b^, -"is regular ; for a»+p K+p — aA = ««+? (^.+p — b^) + b„{a„+^ — a„), 4-i HEAL FUNCTIONS OF A KEAL VARIABLE. a quantity which is numerically less than (a„+p + &„)£, which, m turn, can be made less than any positive rational number, however small, by a proper choice of £. III. The sequence a,/6„ a.^jh.,, a,/%- is regular, if B be not 0. a fraction which can be made less than any positive rational num- ber, however small, by a proper choice of t. § 52. Definitions of the terms equality and inequality. If the symbols A, Bhe attached to the sequences a„ a,, a^, •■■, ., and if the elements of the sequence a^ — &„ a., — h-i, 63, •■•, from a definite term onwards, be always greater than some positive rational number, A is said to be greater than B. If the elements after that term be always less than some negative rational number, B is said to be greater than A. If a number /x, can be found such that ] o„ — 6„ | < e, {n > fx), A — B = Q, and A is said to be equal to B. This definition agrees with the facts in the case of rational numbers. For, if it can be proved that the difference of two rational numbers is less than e, the numbers must be equal. The two sequences aj, 02, aj, •••a^, cij+i, •••; Oj , a^,"-, a^, a,^.i, £114.2, ••■ define the same symbol, for Oi— a/, a., — a.^,---, flj+i — o,+i, ••• has for its symbol 0. It is to be noticed that the definitions of equality and inequality for A, B do not overlap. Let the sequence a^, a^, ••■, o,„ ••■ have the rational limit a; the symbol attached to Oj — a, 0.2 — a, a^ — a, •■■ is 0, since by hypothesis fi. can be so determined that | a — a„ | < c, when n > 11. Hence, {a^, a^, a^, •■■) — (a, a, a, •••) = ; i.e. the symbol attached to the sequence ai, a^, a^, ••• is a, or Za„. Thus, when a rational limit is known to exist, it must be the symbol attached to the sequence.* Example. If a„ = .333 .•-, where the figure 3 occurs n times, L (J — a„) = 0. Hence ^ is the symbol attached to .3, .33, .333, ••• . * When each number of the Beq\ience is ti decimal fraction, differing from the preceding number only by having one additional digit assigned, the necessary and sufHcient condition for a rational limit is thr.t there must be, sooner or later, a periodic recurrence of the digits. Bee Stolz, Allgemeine AriLhmetik, t. i., p. 100. REAL FUNCTIONS OF A REAL VARIABLE. 45 § 53. Definitions, (i.) The absolute value of (a) is (| a |) ; (ii.) the symbols Ay, A.,, A^. •••, A^ are said to decrease below every assignable value, when for every symbol E, except 0, a number fi. can be found such that, when n >/i, A„ < JS in absolute value. Consider the regular sequence cii — a^, a.^ — a,,, a^ — a„, ••• ; we have ^ — a„ = (tti — a„, a, — a„, ••■) = (a„^i — a,„ c(„+2 — a„, •••)■ When n increases beyond any assignable rational number, the elements of the sequence decrease below any assignable rational number, by the definition of a regular sequence. The absolute value of the sj-mbol attached to the second sequence decreases below any assignable absolute value ; therefore A — a„ decreases below any assignable value, as n increases. Hence we have the theorem that a limit exists for a„, namelj' the symbol A. When La„ is not a rational number, we call it an irrational number. When the limit of a sequence is a rational number, the symbol attached to the sequence can be regarded as that rational number. By the definitions of § 52, all symbols can be arranged in order of magnitude. Hence, when the limit is not a rational number, the corresponding symbol can be regarded as of like nature with the rational numbers, and is called an irrational number. From this point onwards the word symbol can be replaced by number. The symbols 1/3, 1 attached to the sequences .3, .33, .333, •••; 1, 1, 1, ••• are now the ordinary numbers 1/3, 1, and such an in- equality as (1, 1, 1,-)<(1, 1-4, 1.41, 1.414,. •■)< (1.5, 1.5, 1.5,...), where 1, 1.4, 1.41, 1.414, .•• are numbers which occur in the process of extracting the square root of 2, shows that the symbol (1, 1.4, 1.41, 1.414, . . . ) can be regarded as a number intermediate in value between 1 and 1.5. Cantor (Math. Ann., t. xxi., p. 568) calls attention to the impor- tance of perceiving that A = La, is a theorem and not a definition. The number A was not defined as the limit of a„ (n = x), for this would presuppose the existence of the limit. A different course was adopted. First, numbers were defined by regular sequences; next, a meaning was given to the elementary operations as applied to the new numbers; thirdly, a definition was given of equality and of greater or less inequality ; 46 REAL FUXCTIONS OF 'A EEAL VARIABLE. finally, there followed from these definitions and laws of construc- tion the theorem A = La„. To see that no new numbers are obtained by using a regular sequence of irrational numbers Oj, aj, asi "•. consider a regular sequence &i, b^, h, ■■■ of rational numbers defined in the following way : Let Oi and &i have the same integral part, a, and b^, agree to the first decimal place, a^ and 63 to the second decimal place, and so on. Form the sequence «i — b^, a. — b,, a. — h,--. The elements become ultimately < e, and the sequence defines 0. That is, the numbers ((ai, ao, 03, •■■), (&i, bn, h,--) are equal. Eational and irrational numbers constitute the system of real numbers. Deductions from the theorem La^ = (a„ a^, flj, •■■)• ^^y definition AB={a-fi^, ■■■ , a„bni---) = La,J}„. Hence La„ ■ Lb„ = La,fi„, similarly La„/Lb„ = ia„/6„, where Lb„ =#=0. We leave to the reader, as an instructive exercise, the proof that (Lan)" = Lan\ {La^y/" = La„y\ where it is a positive integer, and also the proof that (Xa,,)^''" =L(a„''''), when the a's and 6's form regular sequences. If 6 be irrational and a rational, a' is the number defined by the series n^, ah, aN, ••• . In this way it is possible to arrive at a definition of such expressions as a/^. The simplest mode of treatment is obtained from the exponential theorem. § 5J:. The unicursal, or one-one, correspondence between the totality of real numbers and the points of a straight line. I. To every point on a straight line corresponds a distance from the zero-point, whose ratio to the unit line is a real number. Let the line which extends from the zero-point to the point in question be named B, and let the unit line be A. To fix ideas, let the line B{— ob) be longer than the line A{=oa). Let lines ocj, C1C2, C2C3, ■■•be measured along the straight line so that oc^ is the last multiple of A contained within ob, c^c^ is the last multiple of ^4/10 contained within c^b, CoCjis the last multiple of A/ld^ contained within cjb, and so on. Either the point h is reached after a finite number of divisions, or it is not. In the former case the ratio of 5 to -4 is evidently a rational number. In the latter the points Cj, c^ C3, ••• approach h. The sura of the lines oci, CiC^, cfi^, ••• falls short of B, but REAL FUNCTIONS OF A EEAL VARIABLE. 47 the amount by which it falls short can be made less than an arbitra- rily assigned fractional portion of B. If the ratios of oci, CiCj, • • • to A, A/10, ••• = nil, *"'2; •■•< where m], m.,, ••• are positive integers of the set 0, 1, 2, •••, 9, the ratio of B to A = m^ + vi^/lO + wij/lO^ -\ = the number defined by the regular sequence wii, m, + m,/10, vii + m,/10 + m^/W, •••. II. To every real number corresponds a point of the line, as soon as a unit line has been selected. There is no difficulty in proving this for rational numbers. If the number be irrational, let it be defined by a regular sequence «!, Oi, «3, ••• of rational numbers, in ascencli7ig order of magnitude. Let 2^, ai, let b., be the first of these. Either b., is greater than all the succeeding a's or it is not. In the former case 62 is the upper limit, but, in the latter, some subsequent number 63 of the series is greater than b.,. A continuation of this process leads to the sequence of ascending numbers cti, b.,, 63, 6j, ••■; if the sequence contain only a finite number of terms, the last of these is the upper limit, whereas, if it contain an infinite number of terms, it is regular and defines a number < A. The number so defined is G. When no number A can be found with the assigned properties, the upper limit is said to be 00. There is alike theorem with respect to the existence of a loicer limit K. A variable x is said to be continuous in the interval a;,, ^ cf < Xj, when it passes through all the rational and irrational numbers of the interval (a'o and x-^ inclusive). Example. Let the values of x te 1-1/2, -1 + 1/.3, 1-1/4, -1+1/5,-.. •••, 1-1/2)1, - 1 + 1/(2)1 + l),.--(Thom£e). The upper and lower limits, which are nnattainahle, are + 1 and — 1. The variable is not continuous in the interval. § 56. Limiting numbers. In the neighbourhood of a limiting num- ber there is an infinite accumulation of points. To make the meaning more precise, let us consider an infinite sequence o,, a,, O3, ••• , a„, ••• whose members are finite and distinct. There must be in the neigh- bourhood of some member H, (A'< II ^G), an infinite accumulation of points. To prove this, divide the finite interval {Kto G) into j) parts, where p is some integer ; in at least one of these parts, say (Al to (?i), there must lie infinitely many points of the sequence. Divide (Ki to Gi) into p parts. In at least one of these parts, say (7u to G2) , there must lie infinitely many points. By continuing this process we arrive at the two sequences /li, luj A's- G„ G„ &,■ * Bolzano was the first mathematicinn to draw attention to the existence o( upper and lower limits of a variable. See Slolz, B. Bolzano's Bedeutung in der Geschichte der Infinitesimal, rfchnung, Math. Ann. t. jviii. Their Introduction into modern analysis is due to Weierstrass. REAL FUNCTIONS OF A REAL VARIABLE. 49 ■which are in ascending and descending order. Each member of the second sequence is greater than every member of the first, and G„ — K„ can be made less than any assignable number, if n be taken sufficiently great. The two sequences define one and the same number H. By the mode of construction of Hit follows that, how- ever small 8 may be, infinitely many values of x exist within the interval {H — 8 to .ff + 8) . This is what is meant by saying that there is an infinite accumulation of numbers at H, and that His a, limiting number. H need not be itself a member of the system a^, a^, cia, •••. A distinction must be drawn between "the upper limit" and a " limiting number" of a sequence. For example let a, ^a2lA (or <-l/£), lim t/= + oo (or -oo). ar=a-i-0 If, when x>h,\y — A\{y) be continuous at y = b, then <^\f{x)\ is continuous at xz=a. This theorem fol- lows immediately from the definition. § 62. Upper and lower limits of a function. TJieorem. Let f{x) be a function which remains finite for all values of x between x = Xf, and x = x^. The values taken by the function will have an upper limit Q and a lower limit K. For no value of X in the interval can the function be > G or < K; but further, however small e may be, there must be at least one value of X for which f{x) is greater than G — t, and at least one value of x for which /(a;) is less than K-^ £. The finiteness of the values of the function shows that there must be two finite integers A, B, between which all these values must lie. Divide the interval {A, B) into the parts {A,A + 1). (.1-f 1, ^-f.2), ..., {B-\,B). KEAL FUNCTIONS OF A REAL VARIABLE. 65 Let A' be the last of these numbers which is not greater than all the values of the function, and let ^1' + 1 = B'. Divide the interval {A', B') into 10 parts {A', A' + 1/10), {A' + 1/10, A' + 2/10), ..., {A' + 9/10, B'). Let A" be the last value which is not greater than all the functional values, and let B" = A" + 1/10. A continuation of this process shows that there can be found a value u4'"', such that (i.) jB<»>-^(»> 1 — e. When a function is said to be infinite for x = a, it is to be under- stood that this is only a short mode of expressing the fact that no finite upper limit exists. The value x= a must then be excluded from the list of possible values of x. When a function is continu- ous throughout an interval, it is implied that there is no place of this kind in the interval. Example 1. y = l/x isa. law which may define values of y for all values of X in the inten'al (— rtto«),a; = exclusive. As x approaches from either side, y tends to infinity ; but if a; = be included in the range of values, y is not defined by this law, but must be otherwise assigned. Example 2. The function y={x:-- or) / {x — a) has a determinate value, provided x^a. It is natural to assign to y the value 2 a when x = a; but that this is an arbitrary assumption has been pointed out by Darboux. Let (f>(x) be a function, for example a definite integral, which = 1 when x^a, and = A when x = a. The product G — t, and values < K+ t. Does it actually attain to G and /i"? Divide (x„, Xj) into two equal intervals. In one at least of these, the upper limit is G. In fact, if the two upper limits were G', G", quantities < G, the function would not take, in the interval (.r,,, x^), the values comprised between G and the greater of the two numbers G', G" Hence one, at least, of the limits G', G" = G. Divide that half interval, in which the upper limit is G, into two equal parts, and continue the process. The result is a sequence of sub-intervals, one comprised within the other, which tend toward zero as limit. For each of these inter- vals the upper limit is G. The further extremities of the sub-inter- vals are constant in position or else approach Xq ; the nearer extrem- ities of the sub-intervals are constant in position or else move away from Xo. Let a be the point defined by these sub-intervals. Within the neighbourhood {a — h to a + k) the upper limit of the function is G, hoicever small h may be. That is, there is a value of a;(= a T ^^0 comprised within this interval for which G —f{a =F 6h) < £. On the other hand we can take h small enough to make j/(a T Oh)—f{a) ] £, however often the operation of division may have been per- formed. The oscillations within some interval (i'o'"', a;/""'), which decreases indefinitely as m increases, are always ^ t/2. Now the sequences ,,) - <=> a; <« ...a; '"" ... .^2, Xj , Xj , Xj , • • • Xj , • • ■ T r (') r <-' T (■■" ... T (»■' ... X-O, Xy , Xo , .^0 J "^O J define a number X. Hence, since the function /(x) is continuous at the point X, it must be possible to find a number h, such that \f{XTeh)-f{X)\<./x, for all values of d from to 1, inclusive. Let X,, X, be any two of the values of ± 6h. We can deduce, at once, from the two in- equalities ^^^^ _ ^^^ _j.^^^ ^ ^ ^^^^ |/(X+X,)-/(X)|<./4, the further inequality |/(X+ X^) -/(X- X,) I < e/2, which implies that the oscillation within the interval (X— Xj to X+X.)<£/2, contrary to supposition. Hence a function of one variable, which is continuous at every point of an interval, is uni- formly continuous. Cantor and Heine were the first to enunciate the above theorem.* § 65. The continuous function plays a much more important part in analysis than the discontinuous, but the study of the latter throws light on the problems suggested by Fourier's series and simi- lar questions. Enough has been said to show the necessity for precise definitions and to emphasize our present ignorance of the discontinui- ties presented by functions of a highly transcendental character. The properties which are most commonly associated with a con- tinuous function are the possession of a differential quotient and an integral, and expansibility by Taylor's theorem. It is not our intention to examine these cases for the real variable, but we shall give an example of Weierstrass's which relates to the existence of a differential quotient. Wkierstrass^s example of a continuous function ivliich has nowhere a differential quotient. CO The f unction /(x) = 2 6" cos(a»a;7r), in which x is real, a an odd • Heine, Crelle., t. Ixxiv., p. 188; ThomsB, Theorle der analytisoheu Functionen. KEAL FTJKCTIONS OF A REAL VARIABLE. 59 positive integer, b a positive constant < 1, is a continuous func- tion, which has nowhere a determinate differential quotient if a& > 1 + 3 7r/2. We reproduce Weierstrass's proof : — Let Xf, be a definite value of x, and m an arbitrarily chosen posi- tive integer. There is a determinate integer a„, for which -l/2 ^" -Xo= (1- aj^+O/a" and x' < Xo < x"- The integer m can be chosen large enough to insure that x', x" shall differ from Xq by as small a quantity as we please. We have {/(^')-/(-o)^/(x'-Xo)=i ,(;,. eos(a"xV)-cos(a"Xo.) > n=0 (_ X — Xq ) =""2 ^ (abY ^°^ (a"-c'7r) — cos (g'^XpTr) ) „=o r' -^ a"(a;'-a:„) ) - I ^„+„ cos (a"+".-«'7r) - cos (a"+"a;„7r) > n=o 1 a;' — ajo ) Since sinf a" !^5r cos (a"a;V) — cosCa'Xoff) . / a;' + a;„ \ V 2 ^ — ^ ^ — ^—^ = — TT sm a" — is — 'T 1 • ^ a"(a;' — Xo) \, :i y ^„x' — Xq a"^-". i( a" — sinl- •'''' ') I 2 and since the value of — ^^-; — ^ always lies between — 1 and X — X(, + 1, the absolute value of the first part of the expression is less than m-i (ah')"' — 1 ab — 1 and therefore < 7r(a6)"'/(a6 — 1), if a6 > 1. Further, because a is an odd number, C0S(o"'+V7r)= C0S(a''(a„ — l)ir) = — (— l)"", cos (a"+»ai,5r) = COS (a»a„r + a'x„+iir) = ( - 1 ) ""• cos (a-x^+iff), 60 REAL FUNCTIONS OF A REAL VARIABLE. therefore 2 7^- 1 - COS (a'"+"a;V) - cos (a'°+"x,)7r) n = 1 -f" ^m+1 All terms of the sum 2 fc»{l + cos(a''x„^i7r)|/(l + a;„+,) n = are positive, and the first term < 2/3, since cos(x,+i7r) is not nega- tive and 1 + .T„+i lies between 1/2 and 3/2. Accordingly lf(^x') -/(a-o) l/{x' - a-o) = (- l)"'»(a6)'",7(2/3 + 7r,7,/(a6 - 1) ), where t; denotes a positive quantity >1, while rn. lies between — 1 and + 1. Similarly l/K)-/(^o)|/(^"-^o) = -(-l)°'"(«&)V(2/3 + W/(«&-l)). where t;' is positive and >1, while t;/ lies between — 1 and + 1. If a, b be so chosen as to make a6>l+3T/2, that is, 2/3>7r/(a6-l), the two expressions |/(.t') -f{x,)\/{x' - X,), \f{x") -f{xo)]/{x" - x„), have always opposite signs, and are both infinitely great when m increases without limit. Hence f(x) possesses neither a determi- nate finite nor a determinate infinite differential quotient. [Weier- strass, Abh. a. d. Functionenlehre, p. 97. j A real function is represented graphically by drawing ordinates equal to the values of the function and marking the terminal points. The greater the num- ber of these points, the more closely will the polygon through them resemble a continuous curve which admits tangents. This polygon will, in the limit, appear to the eye indistinguishable from the curve, but the polygon may have, in its smallest parts, infinitely many re-entrant angles. This is what actually happens in the case of functions which are continuous, without admitting differential quotients.* * For further Informntion on the reprcBentahllity of functions hy carves we refer the render to a paper by Klein, Ueber den nllgemeinen FunctionBheitriff, Math. Ann., t. xxii., p. 219, to two memoirs by Kopcke in the same jourual, tt. zzlx., xsziv., and to Tasch's Einleitung in die Differ- ential und Integralrecbnung. KEAL FUNCTIONS OF A REAL VAEIAULE. 61 § 66. Functions of two real variables. Let a?], i/i be the centre of a rectangle, whose sides are 2 h, 2 k. Any point within this rectangle has for its co-ordinates x^ =F 6^h, 2/i T &-J^, where 6„ 6-, are proper fractions. Allow 6^, 6.^ to take, inde- pendently of each other, all values from to 1. The one-valued function /(x, y) is said to be continuous at (x^, ?/,), when finite val- ues h, k exist, for which |/(x-, T ^A 2/i T ^2^)— /(xj, 2/1) |< e, for every combination {6^, 6.,). Erroneous statements are sometimes made with regard to the continuity of a function of two variables.* Let/(x, y) be a contin- uous function of y Avhen a; is put = a;,, and a continuous function of X when y is put = y^. It is an illegitimate inference that f{x, y) is continuous at {Xy,y^). For example, let /(a;, y)=xy/{x-+y-). "When a; = 0, this is a continuous function of y, and, when 2/ = 0, a contin- uous function of x ; but the function is not a continuous function of a;, y, conjointly, at (0, 0), as is easily seen by writing y = mx. Cauchy fell into this error in his Cours d' Analyse. As an example of the discontinuities which occur in the case of functions of two variables we may instance sin(tan" ^y/x), which is continuous at every point off the axis of x and discontinuous on that axis. That the discussion of such discontinuities is of more than purely theo- retic importance will be evident from these examples : — Example 1. Let f{x, y) be a function of x, y, which equals xsin (4 tan-\v/a;) when x=5tO, and equals when a; = 0, whatever be the value of y. When x^O, S-f/&xSy = S'f/Sy&x ; but when a; = 0, y = 0, S-f/SxSy = ± x, according to the sign of Ax, and 8y/8y8.c = 0, so that h-f/^xhy4^h-f/Zyhx at (0, 0). Example 2. Let f(x, y) = y- sin x/y when y^O, and = when x = 0, whatever be the value of y. Then, at (0, 0), Sy/8a,% = 1, 8=//8(/8.i- = 0. [See F. D'Arcais, Corso di Calcolo Infinitesimale, t. i. Padua, 1891. J Theorems strictly analogous to those already proved for functions of one variable exist with regard to upper and lower limits, limiting values, etc. * On this subject see a memoir by Schwarz, Abbandlungen, t. ii., p. 275. 62 EEAL FUNCTIONS OF A REAL VARIABLE. Dirichlet's definition can be at once extended to functions of two real variables x and y. The restriction that the function takes only real values may be removed. Such a definition therefore includes both monogenic and non-monogenic functions of a; + iy. The Theory of Functions, in the modern sense, discards Dirichlet's definition as too general and treats only of the monogenic function. References. Cantor's researches are to be found in memoirs publislied in the Mathematische Annalen and the Acta Mathematica. For those of Weierstrass, see Pincherle, Saggio di una introduzioue alia Teorica delle funzione analitiche secondo i principii del Prof. C. Weierstrass, Giornale di Matematiche, t. xviii., 1880 ; and Biermann's Analytische Functionen. An important memoir on Cantor's theory of irrational numbers was i^ublished by Heine in Crelle, t. Ixxiv., Die Elemente der Functionenlehre. The student who wishes to find a thorough discussion of many of the more difficult questions connected with the theory of the real variable should consult Dini's standard work, Fondamenti per la Teorica delle Funzioni di Variabili Reali. Pisa, 1878. Also Stolz, Allgemeine Arithmetik ; Tannery, Introduction h la Thcorie des Fonctions d'une Variable ; Fine, The Number-System of Algebra ; Thomae, Elementare Theorie der analy- tischen Functionen einer complexen Veranderlichen. Slany references to original memoirs are given at the end of the German translation of Dini's work, Leipzig, 1892. The student is also referred to the appendix in t. iii., of Jordan's Cours d' Analyse ; to Dedekind's pamphlets (i.) Stetigkeit imd irrationale Zahlen. Brunswick, 1872 (ii.), AVas sind und was sollen die Zahlen. Brunswick, 1888 ; to Cathcart's translation of Harnack's Introduction to the Calculus ; and to Du Bois-Reymond's Die Allgemeine Functionen-theorie. Erster TheD. Tubin- gen, 1882. CHAPTER HI. The Theoky or Ixfixite Series. § 67. We shall begin this chapter with a sketch, in outline, of the chief properties of infinite series of real terms, referring the student for a fuller discussion to treatises on Algebra. Series u-Uh Real Terms. QO An infinite series 2 a„ contains infinitely many terms, defined by 1 a law which permits the calculation of the general term. Let s„=Oi + a,,H 1-«„; if s„ tend to a finite limit s, the series is said to converge, and s is called its sum. In all other cases s„ either tends to oo, as in 1 -|- 1 -f 1 + ..., or oscillates, as in 1 — 1 + 1 — 1 + .--. Strictly speaking, an oscillating series is distinct from a divergent series, but it is usual to speak of non-convergent series as divergent. The necessary and sufficient condition for the convergence of the series 2 a„ is that a number p.* can be found such that |«a+i + ««+2H l-ci„+p| IX, 1^=1, 2, 3,.--). The condition is necessary, for L «„+, = Ls„ = s, and therefore L{^n+p - sJ = -^(a«+i + fl„+2 H l-«„+J = 0. To see that it is sufficient it will be enough to observe that when the condition holds good, a number /i can be found such that I «,.+,> - S" I and I s„+, - s„ | < t, when n > /tt, whatever positive integral values are assigned to p, q, and that these relations give L s„+j, = L s„+,. n=ij 71=30 Corollary. The condition | a„+i | < t (n > /a), is necessary, but not sufficient. * Here, aa elsewhere, e is given in advance, and ^ depends upon e, 63 64 THE THEOKY OF INFINITE SERIES. § 68. Many of the tests of convergence are derived from a comparison of the series in question with a few standard series. Such a series is /'a; > — 1^ \fc /x); that is, if Va~< 1. The cases require special treatment. A third test is to compare a series 2a„with another series 26„, where i&„/a„ = 1. When the latter series is convergent, the former is also convergent. Convergent series of positive terms have the properties of finite series, for they are subject to the associative, commutative, and dis- tributive laws. The same is always the case with series in which the negative terms have finite suffixes ; but series whose terms are ultimately both positive and negative present some new features. Theorem. If the series formed from a given series by the change of all the negative signs into positive hg convergent, the original series must be convergent. Let s„ be the sum to n terms of the original series and let s„', — tj be the sums of the positive and negative terms of s„. If i(s,/ -f f,/) be finite, LsJ and Lt„' must be finite, and therefore also L{sJ—t„'). A series is said to converge absolutely when it still converges after all negative signs have been changed into positive. A series which converges, but does not converge absolutely, is called semi- convergent. Theorem. If a,, Oj, a,, •.• be positive numbers arranged in de- scending order of magnitude and if Ln„ — 0, the series «■! — «2 + «3 — a4 + • ■ • converges. For a„+, - a„+2 + (r„+3 + ( _ i )p-i a„^^ = a»+i — (a„+2 - a„+3) — (a„+4 — a„+5) < a„+i ; hence | (- l)"a„+i | > | a„+, - a„^2 + ... + (- l)^-'a„J- THE THEOEY OF INFINITE SERIES. 65 Now a„^.i can be made less than any assigned positive number t, when n is taken sufficiently large ; hence the series «i — 02 + ttj — Oj + • ■ • is convergent. Example. 1 - 1/2 + 1/3 - 1/4 + ■•■ is semi-convergent. The series a, — a, + a, — ••■ will not converge, if Lon^O. It may diverge if a„ a.^, a,, ■■■ be not in descend- ing order, even when ifl„ = 0. § 69. It is natural to ask under what circumstances the compo- nent terms of a convergent series are subject to the associative and commutative laws. (i.) The terms of a convergent series can be united into groups, when this does not affect the arrangement of the terms of the origi- nal series. Let s„ be the sum of n terms of the original series, sj the sum of m terms of the series of groups. Whatever ?!. may be, m can always be taken large enough to insure that sJ shall contain all the terms of s„. AVhen this has been done, sJ - s„ = a„+i + a„+, -^ h a„+,, supposing that sJ contains n+jy terms. But, by the condition of convergence, |«»+i + «„+2H I-«„+p!<£ (">m); hence i(s„' — s„) = 0, or Ls^ = Ls„. (ii.) In a convergent series such as (1/2 - 2/3) + (2/3 - 3/4) + (3/4 - 4/5) + ••., it is not permissible to remove the brackets, without investigation. In the particular example selected, Ihe series oscillates after the removal of the brackets. In connexion with the commutative law it is convenient to define unconditional and conditional convergence. A convergent series which is subject to the commutative law is said to be unconditionally convergent ; otherwise it is said to be conditionally convergent. A semi-convergent series is conditionally convergent. To Kiemann is due the theorem that it is possible, by suitable derangements, to make a semi-convergent series have for its sum any assigned real number A. To fix ideas suppose A positive. 66 THE THEORY OF INFINITE SERIES. Let the positive terms, unchanged in order, be a^ + a^-l- a^ + •••, and let the negative terms, unchanged in order, be —(bi+b.,+h3-\ — ). If the first 11 terms of the original series consist of q positive and r negative terms, s„ = ,s,' — s/', where s,' = Oi + a, H f-a„ s/' = &i + &2 H h^r- When n becomes infinite, q and r also become infinite, and s,', s,", tend to the values s', s", where If the original series be semi-convergent, both the series 2a„ 26< must be divergent; for, if 2a,, 26« have finite sums, the series 2ci« + i&, has a finite sum, and the original series is absolutely- convergent; and if only one be divergent, the original series = s' — s" = X ) ^nd is divergent. The terms of the semi-convergent series can be rearranged in the following way: we write down the positive terms in order, and stop at the first term which makes the sum > A ; next we write down the negative terms, and stop at the first term which makes the sum A, and so on. The values of the sums thus obtained oscillate about A, and the range through which the oscillation takes place diminishes continually, tending to a limit 0. Hence Kiemann's theorem is proved. Since the value of A is arbi- trary, a semi-convergent series can be made divergent by suitable derangements of the terms. [See Dirichlet, Abh. d. Berl. Akad., 1837; Eiemann, Werke. p. 221.] Absolute convergence implies unconditional convergence. Let «i + a2+<^3+ ••• be an absolutely convergent series, and let aj'+ofj' + a^ + ••• be a series derived from it by derangements such that no term a„ with a finite suffix is displaced to infinity. If s„, s„' be the sums to n and m terms of these two series, it is always possible to choose such a value of m as will make sj contain all the terms a„ ttj, •••, o„, and others besides. If s^' be the first partial sum of this nature, the sole limitation on m is that it must be greater than /«.. Suppose that s„' — s„ = the sum of a certain number of terms •which lie between <7„ and a„+x+i. This sum is certainly less than I a„+i I -I- 1 a„+2 1 H — + 1 a„+x \, a quantity which can be made < « by choosing n sufficiently great. Hence LsJ = L.9„. THE THEORY OF INFINITE SERIES. 67 The preceding reasoning breaks down for a semi-convergent series 2a,, since ] a„+i 1 + | a„+2 1 + ••• + | a„^\ \ is not necessarily less than c. 1 Series ivith Complex Terms. § 70. Let tt„=a„+ij8„; then s„ = 2a„+ii^„. If s„ tend to a limit s when n is arbitrarily great, the series is said to be convergent and its sum is s. Clearly, the necessary and sufficient conditions oe CO are that 2a„ and 2;8„ both converge. In the limit both | a„ | and |y8„| = 0;' but I a„'| + I /3„ I > \a„ + il3„\, hence i | w„ | = 0. A series 2(a„+ i/3„) is said to converge absolutely, when 2|a„ + i'^„l converges ; this definition presupposes that the former series con- verges simultaneously with the latter, but this is evidently a legiti- mate assumption, for |a„| and |y8„| are separately less than |a„ + ij3^\. Unconditional convergence has the same meaning as before ; it is CO evident that it can exist only when 2a„ and 2/3„ are unconditionally 1 1 convergent. Absolute convergence implies unconditional convergence. Let (V -t- ty8„ = p„(cos d„ + i sin (9„). The terms p„ cos d„, p„ sin 6„ are less than p„ in absolute value, and therefore, if 2p„ be finite, the two real series 2p„cos^„, 2p„sin^„ are absolutely convergent. But absolute convergence, in the case of real series, has been shown to imply unconditional convergence. Therefore lp„(cose„-f »sin^„) is unconditionally convergent. ao Semi-convergence implies conditional convergence. Because 21m„| diverges, 2{ |a„| + I j3„|S must diverge. This shows that one, at least, of the series 1 1 a„ |, 2 1 18„ |, diverges. Let the former diverge ; then la„is semi-convergent, and therefore conditionally convergent. But, if the series 2M„be unconditionally convergent, 2a„ must be unconditionally convergent. Hence the series 2 m„ is conditionally convergent. Unconditional convergence implies absolute convergence. This is proved indirectly. We have seen that semi-convergence implies 68 THE THEORY OF INFI>riTE SERIES. conditional convergence ; hence a semi-convergent series can never be unconditionally convergent. Therefore an unconditionally con- vergent series can never be semi-convergent; that is, it must be absolutely convergent. The terms unconditional and absolute are, as applied to convergence, co-extensive. § 71. The sum, or difference, of two series %i.„, 2v„ is the series I(m„ + v„), or 2(«„-v„). 1 1 The product of two absolutely convergent series. If the absolutely convergent series 2?(„, 2v„ converge to sums s, s , the series' 1 1 UiVi + (Wil's -I- MoVi) -I- («iV3 -I- UiV, + Maf ,) H -f (?(iV„ + u,%\_i H 1- ?t„Vi) H will converge to ss'. Let ^„ = |i(i| + I ^(jI -I h|M„|, B„=\v,\+\v,\+:.+\v,\, then A„B^= | "i 1 1 Vi | -f h<2 1 | ^i i H h 1 u„ \\vi\ + |mi| I^jI + ImsI ^2^ 1-|m„| i'y2| + 1 Ml M «n ! + ! M2 1 K' J + • •• + 1 w„ I h„ I . If C„=\u,\\v,\ + l\v,\\v,\ + \u,\\v,\]+- + n«ilh'J + l«2lh„-,l+---+|w„lhil|, it is evident that A„B„ > C„, and that A„B„ < C/jn < C/jn+i.' Therefore, whether n be of the form 2 k -f 1, or of the form 2 «, the inequalities A,B,„|, an expression which is itself =A„B„ — C„. "When n increases indefinitely, -4„B„ — C„ tends to the limit ; hence L I s„s„' - f„ I = 0, or if„=ss'. Uniform Convergence. § 72. Hitherto the term w„ of the infinite series 2w, has been 1 regarded as dependent merely on n. We now allow u„ to be a func- tion of 2, as well as of n, and suppose that there is a region of the 2-plane at all points of which Mj, «2, •■- are one-valued and con- tinuous. The sum of a finite number of rational algebraic functions is itself a rational algebraic function; it is only when the number of these functions is infinite that new properties present themselves. In fact the passage from the algebraic to the transcendental function is effected by infinite series, whose terms are algebraic functions of z. Let/i(z),_/^(2),/3(2), •.. be algebraic or transcendental functions of 2, which are one-valued and continuous within a region A of the cc 2-plane, and let the series 2/,(^) ^^ convergent at every point of A. For a given value of z, the necessary and sufficient condition for convergence is that a number /i can be found such that, when n > ft, |/„+i(2)+/.+2(2)+ •••+/»+,(«) I <£ 0^=1,2,...). When this condition is satisfied, we call the sum of the series F{z). The complete system of points, for which the series converges, covers a region of the z-plane which is known as the region of con- vergence. This region may consist of one or more separate pieces, and the boundary of a piece may consist of discrete masses of points, or of linear masses of points. For such functions as we shall con- sider, each piece will consist of a continuum of points covering a doubly extended region. The series 2/„(2) is said to converge uniformly in a part of the region of convergence, when a positive integer /a can be. found such that when n > /n, |/„+iW + /„..(«)+-+/„.,WI = l, 2, ...), 70 THE THEORY OF INFINITE SERIES. whatever he the position of z in that part. Attention is specially- called to the fact that fi depends on the arbitrarily selected e, but can be made independent of z. If, when z approaches a point Zo of the region of convergence, the value of /a tend to cc, the series is said to be non-uniformly convergent in any region which contains z^ and the convergence is said to be infinitely slow in the neighbourhood of 2o- Example 1 (Du Bois-Beymond) . Let y _ nx (n — l)x Ux)-- (nx,+ l){nx-x + \) mx+1 (n - l)a; + 1 The sum of n terms of the series, when x is real, is -, and the sum to nx + 1 infinitv, when x^tO, is 1. The remainder after n terms is -• Let x move •" nx + 1 in an interval such that x> & positive number a ; the remainder < e, if nx + 1 > 1/e, and therefore, a fortiori, if na + 1 > 1/e. Let /i be a number which satisfies the inequality ^a + 1 > 1/e ; then, for all values of re > ^, \fn+i(x) + f„+2{x)+ - to 00 | log 1 /e -=- log l/x. The more closely x approximates to the value 1, the more nearly does the denominator approach and the greater becomes the value of ii. The series is non-uniformly convergent in the interval (x = to X = 1) , and the convergence is infinitely slow when 1 — x is infinitely small. Example 3 (Peano). Let l+(rex)2 e"^ + e--" When X = 0, L /„(x) = 0, n=3c and when x is greater or less than 0, L/„(x)=l, or-1. THE THEOEY OF INFINITE SERIES. 71 The series A(x) + {Mx) - /.(x)] + {f,{x)-f,(x)}+ ■■■ is convergent for all values of x, but the convergence in an interval which includes x = is non-uniform. § 73. Theorem. If a series formed of one-valued and continuous functions /i(2;),/2(i?), ••• converge uniformly in a region which con- tains 2o, then lim \f,{z) +f,{z) + ■■■] = lim/i(3) + lira f,{z) -f ... The right-hand side may be written 2/„(«o), for a continuous function attains its limit. Let 1 1 n+l = ^Mz) + p„iz). 1 00 Since the series 2/„,(2) is uniformly convergent, there exists a num- 1 ber /A such that, when n > jn, |p»(2)l < *, whatever be the position of z in the region under consideration. Let z = Zf, + h, and let the absolute value of h be made sufficiently small to insure that \Mz)-f.{zo)\<^/n{K = l, 2,-..n), where n is finite. Hence, if ^(2) = 2/„(«), we have when n> jjl, I F{z) -F{%) I = I 2 l/,(z) -/«(«„) } -f p„{z) -p„(Zo)| < 2 |/,(2) -/.(2o)H- 1 P„(«) I + I P„(«o) I 1 < n • e/n -f c -|- £, < 3 c. This proves that the limit of the sum oif^{z),f.,{z),f3{z),-" is equal to the sum of the limits of /i(2), f2{z), fai^), •••, when z approaches Z(i. This important theorem can be stated as follows : — If a series of one-valued and continuous functions converge uni- formly throughout a region T of the z-plane, the sum of the series is a continuous function of z throughout the region. It is evident from this theorem that a point of the region of convergence at which the sum of the series is discontinuous is a point in the neighbourhood of which the series converges non-uniformly. Du Bois-Reymond has proved that 72 THE THEOliY OF INFINITE SERIES. a series may converge infinitely slowly in the neighbourhood of a point at which the sum of the series is continuous. Hence a non-uniformly convergent series may be continuous ; but no uniformly convergent series can be discontinuous. Dini (Fondamenti per la teorica delle funzioni di variabili reali, p. 103), and Uarboux (.Me'moire sur les fonctions discontinues, Ann. de I'Eo. Xorm. Sup. Ser. 2, t. iv. , p. 77) adopt a different definition. A series is said by them to be uniformly convergent when a number /i can be found such that l/^ + .(2) + /M+2W+"-| fi and n>v, I Sm+p.n+, — S„^p \ fjL, n>v), by the condition of convergence ; hence Si + S2 + ...+ s,„= L ^S:„,„, and L {s, + 82 + •••+«,„!= L LS^,, = S. Similarly, if t^, t.,, ^3, "-be the sums of the first, second, third, ••• columns, L (fi + <2+--- + 0= L L^,„,„=5. 7l=ao ll=^x ni=x Also the series summed diagonallj^, namely «i, I + (Oi, 2 + «2, 1) + (fli, 3 + a.,, 2 + «3, 1) + ■ • • , has S for its sum. This is evident from the consideration that the sum of the latter series to k terms is intermediate in value between (Sj.j and iS,, „ where h is the greatest integer in k/2. Finally, since the series, summed diagonally, can be written as a single series Sa,, „ = 2a„, where (;-, s) take independently all values from 1 to 00, and since 2c"',, , = Sa,.,,., the double series is unconditionally con- vergent for derangements of the kind mentioned above. Conversely, simple series can be converted into double series. If the simple series 2a„ be converted into the double series Sa^, „ and thence into ai.i + «i.2+«i.3H + «2,l + «2,2 + «2,3H + 03,l + «3,2 + «3,3 + -" the sums of the simple and double series are equal. 76 THE THEORY OF INFINITE SERIES. Absolutely Convergent Double Series iviih Positive and Negative Terms. § 76. Let s„,„, -C„ be the sums of the positive and negative terms of S„,^. "iet >S'„',„ be the sum when all the negative signs in S„^„ are replaced by positive ; then and, since LS'^,„ is finite, the two positive quantities s„,» ^^^ U- must tend to finite limits, which we may call s and t. Hence LS„„ = s-t. -^m,n Absolutely Convergent Doable Series tvith Complex Terms. Let M,, , = a,, . + iPr, „ and let 2a,, , = p, 2/3,, , = o-. If 2|a,,, + J/3,,, I converge, 2a,,. and 2/3,,, must both converge; for I a,,, I < I a,,. + 1/3,,. I and !/?,,. I < I a,,. + i^,,. i. Hence the series 2 (a,,. + «/?,,.) converges and has for its sum p + ia: As before, if p, + io-« be the sum of the Kth row, %p = p and 2(7^ = a. 1 " 1 00 00 Hence 2«, , = 2/)« + i2(7, = p + ia: Similarly the sums by columns and diagonals = p + ia. As in the ease of simple series, if the series 2m'„,„ be absolutely convergent, and if i 1 m'„, „ j / 1 1«„, „ | = 1, when m, ji tend in any man- ner to oo, the series 2«„, „ is itself absolutely convergent. To come to the more general case ; we suppose the suffixes r, s to pass, independently of each other, through all values from — :» to + 00 ; the sum i + m-Mi. Let p be the abso- lute value of that point of the rim which is nearest to 0, and let p = np. For all values of n, p' will be finite and different from ; therefore the quantities p' must have a lower limit o- > 0. The series formed by the absolute values of the 8 n terms on the rim Hence the double series 1 22' X 8 " 1 tr^ 1 Jl* ' mjoji + wioojj This series is convergent when \ > 2. Then, when X > 2, the original doubly infinite series is absolutely convergent. § 77. A test of convergence. Cauchy's integral test for the con- vergence of simple series can be extended to double series. If the terms n„,„ of 2a„,„ be all positive and capable of being represented by a function {x, y), which diminishes continuously as x, y increase numerically and tends ultimately to the limit 0, then the double series converges or diverges according as ll{x, y)dxdy, taken over the part of the plane exterior to a bounding curve C, has or has not a meaning. We shall reproduce Picard's elegant proof that the • It will be shown in the chapter on elliptic functions thnt this nssumptlon is necessary In order to avoid the occurrence of Infiuilely small vulues in the eysiem m, o, + m,a>, i. 78 THE THEOEY OF INFINITE SERIES. necessary and sufficient condition for the absolute convergence of the double series 22' is X>2. Let (oj = a + ib, w, = c + id ; the series of absolute values is 22' = { (Miitt + m2c)-+ (mfi + vuiyi''/^ No denominator can vanish, for the combination mj = m.2 = is excluded ; moreover the ratio I {niiCi + m,c)-+ {mfi + m.d)-\^/^ is always finite. Hence the series in question will converge simul- taneously with 22' , but the latter series converges when rr dxdy^^^ has a meaning; i.e. when JY^^ is finite for in- finite values of p, or, in other words, when \ > 2.* Infinite Products. § 78. Let Ui, U2, «<3, ••• be a series of quantities, real or complex, which are determined uniquely by a known law of construction, audlet P„ = (l-|-Mi)(l + W2)-"(l + «n) =n(l+M.)- We shall suppose that all the factors, with finite suffixes, are different from zero. If, when n increases indefinitely, P„ tend to a finite limit P which is different from zero, the value of the coresponding infinite product is said to be P. When the value towards which the infinite product tends is either or oo, the product is called divergent, and, when there is no definite limit, oscillating. It is often convenient to regard the term divergent as covering all cases of non-convergence. f * Picard'a Tralt^ d' Analyse, t. i., p. 272. Simple proofs of Eisenslein's theorem that + 00 1 22...' -« im^' + — + mn')'^ converges when 2A>n, and of allied theorems, will be found In Jordan's Cours d'Analf se, t. i., p. 162. t Some writers regard the product as convergent when LPn = 0. The view adopted in the text is that of Pringsheira, THE THEORY OF INFINITE SERIES. 79 The necessary and sufficient condition for the convergence of the infinite product is that a number ^ can be found such that when w ^ ju., I (1 + «„+■) (1 + M„+2) (1 + w„+3) ••• (1 + M„+p) -1 1< £, where e is an arbitrarily small positive proper fraction, and p takes all integral values from 1 to oo. We shall first show that this condition is equivalent to the two conditions : — (i.) However great n may be, P„ is a finite quantity which is different from 0, (ii.) L P,,^^ = L /'„, where p is any positive integer. Let accents denote absolute values ; we have i P',,JP', - 1 1 ^ i P..,/P.- 1 1 (P = 1, 2, ...) and i Pu.+^/Pu. — 1 i < e; hence, if /u, +p = n, P'„/P'^ — 1 lies between — £ and + £. As p increases, n increases; therefore LP'„ lies be- tween {l—i)P'^ and (l+t)P'^. Thus LP„ is finite and different from 0. We are therefore at liberty to multiply both sides of the in- equality l^„.,/A-l| 1 -I- ia,+ a, + -+a„), II. (1 - aO (1 - a,)-{l - a„) > 1 -(a, -|- aj + .-.-f- a„). The proof of I. is immediate ; for (1 -f cti) (1 f Oj) = 1 + ai -+- aj -t- aiOj > 1 -f Oi -|- Oj, 80 THE THEORY OF INFINITE SERIES. therefore {l + a,){l + a.;){l + a,)>(l + a,+a,){l + a,)>l + ai+a,+a3, and so on. A similar proof can be given for II. ; for (1 - a,) (1 - a.) = 1 - (ai + a,) + a^a, >l-{a, + a,); therefore (1 - aO (1 - a,) (1 - as) > SI - (a, + a,) } (1- Og) > 1- {a. + a.+a^), and so on. These inequalities can be used to prove the following important theorem : — The necessary and sufficient condition for the convergence of the products n(l + a„), n(l — a„), where the quantities a are real I 1 so and positive, is the convergence of 2a„. Let us suppose, in the first place, that the a's are all less than 1. When %a„ is convergent, the second product is finite by the second inequality ; and, when 2a„ is divergent, the first product is infinite by the first inequality. Xow if P„ = (1 + a,) (1 + a.) •■•(! + «.), Q„=(l-a0(l-a2)---(l-a„), the expression Q„ < 1/P„. For 1/(1 + a«) > 1 - a, (k = 1, 2, — ,n). Hence when 2a„ is convergent, P^ is less than a finite quantity 1 CO 1/Qooi ^^'l when 2a„ is divergent, It is evident from their composition that P, Q2> Q3>Q4"-, and, therefore, that the seqiiences (Pj, Pji-Ps' "•)> (Qd Q21 Qs"') must define two numbers which are greater than P^ and less than Q, ; but we have just proved that these numbers are distinct from and 00 00 when, and only when, 2a„ is convergent. Hence the theorem has been established for the case in which the a's are all less than 1. This restriction can be removed; for the presence, in the finite part of the infinite product, of quantities a which are greater than 1 cannot affect the state of convergence or divergence of the infinite products. THE THEORY OF INFINITE SERIES. 81 CD Theorem. The valne of the convergent product 11(1 + a„), where the a's are all real and positive, is the same as the value of the sum 1 + ia„P„,i, 1 where Po=l, ^n = n(l + a<). Because P, = (1 + a,)P,_i = P,_i + a,P,.j, we have P„ = P„_, + a„P„_, = P„_, + o„_i P„_, + a„P„. i = -P„-3 + a.-iPn-i + «n-iP„-2 + a„Pn-i7 and so on ; n therefore P„ = 1 + 2a,P,_;. Therefore n (1 + o„) = 1 + ic(„P„_i. 1 1 If the product converge, the series must converge ; for the terms Pi, Pj, ••• are all finite and 2a„ is convergent. But further the series converges when the general term a„P„_i is separated into its component terms, by multiplying together the factors a„(l + Oi) ■•• (1 + o„_i), inasmuch as each component term is positive. By re-arranging the terms of the resulting series, we can write it in the form 1 + 2c[„ + 2a„ a„ -\ , where the signs of sum- j 1 J 1 2 mation indicate that ?i„ tij, •■• are to take, independently of one another, all integral values from 1 to oc. This proves that the value of the product is independent of the order of the factors ; for a product which is formed from 11(1 + a„) by a re-arrangement of the factors is equal to a series whose value is precisely that of the double series 1-f a, + o,+ --. + aia, -f ajajH + . . . . = l + 2a„^-|-2a„a„^ + ---. When a convergent product is subject to the commutative law, it is said to be unconditionally convergent. § 79. Let Ml, Mj, Ms, ••• be complex quantities; we shall prove that the infinite product n(l -f m„) converges when 2u'„ converges; that CO 1 ^ is, when n(l -|- m'„) converges, where accents denote absolute values. 82 THE THEORY OF INFINITE SERIES. Let ^P„=n''(l + M«), ,P'„=t/(l+M'«), n-t-1 n+1 and let 11(1 + w'„) be a convergent product. "We have 1 \pPn-l[=\ M,.+l + U„^2 H h M„+j, + W„+l«„+2 + ■" | ^ "'„+l + m'„+2 H 1- U'n+p + l«'»+l"'»+2 + ••• < P' —1 and therefore, |,P„-l|<£(n>^, ;)=1, 2, 3, .-.). This inequality establishes the convergence of n(l + «„). When n(l + w'„) converges, the infinite product n(l + ?(„) is said to be absolutely convergent. Thus (§ 78) the absolute conver- CD gence of ^u„ is a necessary and sufB.cient condition for the absolute convergence of n(l +w„). 1 Theorem. If 2«„ be absolutely convergent, n (1 + «„) = 1 + i«.P«-i, where Po = 1. The proof is similar to that of § 78. We have 1 and the series on the right-hand side is absolutely convergent, when n = 00 . For 11(1 -f «'„) is convergent, and therefore the series 1 + hi',P',_, 1 is also convergent. This means that the series 1 + i».P._i converges absolutely. Hence n can be made infinite in the equa- n n tion n(l + ?i,) = l + 2M«P,_i. The series on the right-hand side .1 1 remains absolutely convergent when the general term m,P,_i is separated into its components THE THEORY OF INFINITE SERIES. 83 for the absolute values of the terms of the resulting series form an absolutely convergent double series. Hence the terms of the series may be re-arranged, and n (1 + M„) = 1 + 2w„. + 2m„,m^ + • • • . This proves that II (1 + m„) is unconditionally convergent, and there- 1 fore that the absolute convergence of an infinite product implies unconditional convergence. To complete the proof that the terms unconditional and absolute are co-extensive as applied to infinite products, we must show that the unconditional convergence of an infinite product implies its absolute convergence. Pringsheim has established this fact by a purely algebraic method (Math. Ann., t. xxxiii., Ueber die Conver- genz unendlicher Producte). When the theory of infinite products is developed from first principles, any proof which depends upon the use of transcendental functions must obviously be avoided. On the other hand, by the employment of logarithms, the discussion of infinite products can be reduced readily to that of infinite sums. We shall illustrate the method by using it to settle the outstanding question. Let 1 -|- M„, = p„(cos ^„+ i sin ^„), be the nth factor of an uncon- n ditionally convergent product. In order that P„ = n(l -)- u,) may tend to a definite limit which is neither nor oo , it is necessary and suf&cient that the two series 21ogp.^ 2e„, where ip„=l, L6, = (i, 1 1 be convergent ; for P„ = Pip,-P„ Jcos (^1 + 6, + - + e„) + i sm{0, + e,+ -- + 6„)\. For the unconditional convergence of the infinite product it is CO necessary that pipzPs"- and 2^„ be unconditionally convergent; or, 1 CO oc what amounts to the same thing, that 2 log p„^ 2^„ be uncondition- 1 1 ally convergent. The two series just written down cannot be uncon- ditionally convergent without being also absolutely convergent. We have to show that the absolute convergence of the two series implies CO the absolute convergence of 2m„. Because L{p„^ — 1) /log pj = 1, the absolute convergence of 2 log p„' is accompanied by the absolute convergence of ^(pj' — 1). But, if «„ = a„ -f ip„, we have 84 THE THEORY OF INFINITE SERIES. Hence the series S(2a„ + a„- + j8„-) is absolutely convergent. But since | sin ^„ i < 1 6„ , , and since Lp„ = 1, the series 2y8„, = 2p„ sin 6„, is absolutely convergent. Theretore S^SJ is absolutely convergent. It follows that 2(2 +a„)a„ must be absolutely conver- gent, or, since L{2 + a„) = 'l, 2a„ must be absolutely convergent. Now, if both S2„ and 2/3„ be absolutely convergent, we know that « 1 1 oo 2((„ is absolutely convergent; a result which implies that 11(1 + i(„) 1 1 is absolutely convergent. Thus the unconditional convergence of an infinite product implies its absolute convergence. § 80. Uniform convergence of a product. If /i> ./i) /s! ■ • • be functions of z which are finite and one-valued within a region, throughout which the product 11(1 -f /"„) converges, the convergence is said to be uniform in that region when, to a positive number f which may be arbitrarily small, there corresponds a finite number ft such that, when n > fx, i (1 +/„+i)(l+/.^o)-(l +./;.+,)- 1 1 <£ (p = l, 2, 3,...), whatever be the position of z in the region. In conclusion we state two theorems, which foUow without diffi- culty from those already given. Theorem I. If 2»„ be absolutely convergent, n(l + «„2)=l+(2?^)2 + S2M„M„Jz=-f ... Theorem II. If n(l-t-«„) be absolutely convergent, the only values of z for which n(l-ftt„z) can vanish are 2 = -1/h„(« = 1,2,...). Coriolis seems to have been the first mathematician to state general rules for the convergence of infinite products. Cauchy gave the first detailed exposi- tion of the chief properties of infinite products in his Cours d' Analyse Alge'brique, Note 9; his proofs are simple and rigorous, but lie open to the objection that they depend upon logarithms. The proofs of § 78 are modifications of those used by Weierstrass in his memoir Ueber die Theorie der analytischen Facultaten, Abh. a. d. Functionenlehre. Dini gave the first proof of the theorem that unconditionally convergent products are, at the same time, absolutely conver- gent. We refer the student for further information to the valuable memoir by I'ringsheim, referred to on the preceding page. THE THEORY OF INFINITE SERIES. 85 § 81. Doubly infinite products. The discussion of double jjroducts is reducible to that of double series. The condition neces- sary and suificient in order that 11(1 + !r, ,) may be absolutely con- vergent is that 2»^,5 be absolutely convergent. lu conditionally convergent double products, the value depends upon the ultimate form of the contour which contains the terms, when r, s tend to oo. An example of this is the doubly infinite product ""[''+ ^ '' , , 1 . '»' '''' = 0, ± 1, ± 2, ..., L u -|- mai -I- m'u'J which depends upon the form of the bounding curve. See Cayley's Elliptic Functions, 1st ed., p. 303 ; also two papers by Cayley, Collected Works, t. i., 2-1, 25. The classical memoir on these products is that of Eisenstein, Crelle, t. XXXV., Genaue Untersuchung der unendlichen Doppelproducte, aus welchen die elliptischen Functionen als Qnotienten zusammengesetzt sind. The student who wishes further information on double series is referred to a memoir by Stolz, Ueber uneudliche Doppelreihen, Math. Ann., t, xxiv., and to Chrystal's Algebra, t. ii. Integral Series. § 82. TJie do7nain of a iioint. OS Let the series 2 fiX^), whose terms are one-valued and con- tinuous functions of z throughottt a region r, be uniformly conver- gent throughout F. If a positive number p can be found such that the series converges uniformly for all points within a circle whose centre is a and radius p, the series is said to converge uniformly in the neirjlihourliood of a, or near a. Let R be the greatest value of p for which this can be said ; the region bounded by a circle whose centre is a and radius It is called the domain of a, and E is the radius of the domain. In the domain of a let a point b be selected. If the domain of b lie partially outside that of a, the outside portion belongs to the region of convergence. In this outside portion select a point c and form its domain. A repetition of this process will lead to a continuum within which the convergence is uniform. This may be only one of several separate continua. Series in many variables. Suppose that 2;, z.,, ■■■, «„ fill continu- ous regions Tj, T,, ■••, r„ in their respective planes. These n regions may be called, for shortness, the region F, and each system of n values Zi, z.,, ••-,«„ is called a point z, or place z, of F. There is a theory of uniform convergence for series 2/^(^1. z.,, •■•, z„) as well as for series 2/,(z) ; the terms are supposed to be one-valued and 86 THE THEORY OF IXFINITE SERIES. continuous functions of «i, z-i, •••, z„. The series is said to be uni- formly convergent within the region V when a number ju. can be found such that for all points within r and for all values of n > /x, r2/,!a2*' • • • a/» | < fi., and therefore I "^'-^ ^»v-2^-^> I < '^(jT(^T-(^T' where | a< | = a,(K = 1, 2, •■•?i). Hence \ Sma„ a^, ... , a„«i*'«2*= ■ • • 2/- | < fi l-^l-ll-- Pi/ \ PJ Thus the given series is convergent within a region bounded by circles. The theorem may be stated for one variable in the follow- ing way : — Theorem. If the terms of the series P{z), for the value \z\ = R, be less in absolute value than a finite number /*, the series con- verges for every point z within the circle, centre 0, and radius R. This theorem was first stated by Abel (Remarques sur la serie Collected Works, ed. Sylow and Lie, t. i., p. 219; Crelle, t. i., p. 311). Tlieorem. If P(2) converge for a point z whose absolute value is R, it will converge for every point z whose absolute value is p, where p < R. y (1) 88 THE THEORY OF INFINITE SERIES. For if U; = I M. I (k = 0, 1, 2, •••), the terms U^R' have a finite upper limit 31. The terms of the series 2 C/,p" are less than those 30 of the convergent series Mlp'/R' ; therefore the integral series is absolutely convergent for all values of z which make \z\ < li. If P(Zi, 2o, • • • z„) converge for a poiut a, it will converge for every point z such that j z« | < | a, | (k = 1, 2, ■•• 9i). Let 1 z^/a^ 1 = a«. The series 1 + ai +a2-\ +a„ + ai2 + aia, +-"+a„^ + tti' + ai'aj + ai^os -\ \- a„^ + ■•• converges if the sum of the first p lines, S,, suppose, be finite for all values of 73. Let S ,,<■"'> be the sum of the first p terms of 1 + o-K + a<- + Ok' H — , then s,s;"-'---s;"\ Since a« < 1, S/''' is finite for all values of p ; therefore (1) is convergent. Multiplying the general term aj'^iao'^'-'-a/n by the finite quantity ] MAi\2...A„0i^»n/" •■•«/« \> we see that the series whose general term is | «a,Aj ... a„ • z/iZa'^j • • ■ z„*n | is convergent ; that is, P(zi, Zs) • • • 2„) is absolutely convergent. § 84. TJie circle of convergence. The theorems just proved estab- lish the fact that the region of convergence of P{z) lies within a circle of radius R, where R is the upper limit of the values of Z, = \z\, which make Uo, UiZ, U^Z'-,--- finite. For all points out- side this circle some of the absolute values are greater than any finite number, however large ; for such points the series is divergent. For points on the rim of the circle the question of convergence is left in doubt (see § 89). We can determine R, which is called the radius of convergence of the given integral series, when we know (§ 58) the upper limit of the mass derived from the mass I Ml I, I V^2|,..., I Vm"«| ..., this mass being supposed to lie in a finite interval. For if G be the upper limit, there is an integer fi such that, when n > ja, Therefore (§ 68) the series is convergent when {O + t)Z Cr — e, or | VuJ" > {G — c)Z, and therefore the series is divergent when {G — t)Z' 1/G ; and therefore R = 1/G (see Hada- niard, Liouville, Ser. 4, t. viii. ; Picard, Traite d' Analyse). A rule which suffices in many simple cases is the following : if ?f,/M«+i i lia-ve a limit when «= x, this limit is R. For the series is convergent or divergent according as i | m.^i2"+Ym,?« | < 1 ; that is, according as Z^L u„/u^^i \ . Examples. (i.) 1+z + z-/2\+z^/o\ +— , (ii.) l+z + z' + z' + -, (iii.) z-z-/2 + zy:i-..., (iv.) l + l!z + 2!22 + 3!«3+..., are integral series with radii of convergence cc, 1, 1, 0. The series (i.) is not convergent at 2=ac. Such series as (iv.) do not define functions, and are therefore excluded from the subject. In the case of n variables, let the absolute values of all terms of P(2i, Z.2, ••• 2„) be finite at a point a. The totality of circles | «, | = I a, | is called a circle of convergence. It is clear that we cannot always assign an upper limit of the radius of convergence in any one plane, until v,-e have assigned radii of convergence in the other planes. § 85. Theorem. The series P{z) is not only absolutely but also uniformly convergent within its circle of convergence. Let the radius of convergence be R, and let \z\= Z + Z»+V2Ji''+2 + -to oo] '^^ RrAl-Z/R^' -n+l 1 therefore | m„+i2"+' + - + w„+^"^' \<^R^^ l _ p/fi/ 90 THE THEORY OF INFINITE SERIES. where Z fx, this expression can be made < c. As this niimber /j, is finite and independent of z, the convergence is uniform within a circle of radius R^, and the only limitation placed upon R^ is Ri < R. Corollary. Since the integral series is uniformly convergent within the circle of convergence, it defines within that circle a con- tinuous function of z. In the series P(2i, z^, ■■•, z„) let z be within a circle of convergence A. Since the series is absolutely convergent, it may be re-arranged as an integral series in z^, which is convergent in A ; the coefficients being integral series in the remaining 2's, which are convergent in A. By the preceding theorem, the series in z« defines a continuous function. Therefore the series P{Zi, z^, •••, z„) defines, within^, a function which is continuous with regard to each variable. We may therefore use the term ' neighbourhood of ' as defined in § 82, for the region interior to a circle of convergence. Tlieorem. The series Ui + 2 u^ -J- 3 Usz' + 4 UiZ' +■■■ 2u^ + 3-2usZ + 4:-3uiZ'' + 5-4:UsZ'+-- have the same circle of convergence as the original series. For if M be the greatest term of the set Ui, U^Ri, U3R1, •••, and if 2 lie within the original circle of convergence, then U, + 2 U^Z + 3U,Z' -h - =U, + 2 U,R,- 4r + 3 UsR\-^ + - R\ R'l <3f\l + 2-^ + 3~ + .--l I Ri R\ i < i?i R\ M Z/R, 2' this proves that the first series is absolutely convergent within the circle of radius R. In a similar manner it can be shown that the other series have the same circle of convergence. These series are all uniformly convergent within the circle, and represent, accord- ingly, continuous functions of 2, which can be denoted be f{z), f'{z),f"{z), •••, where /(z) = iw,.z". To justify this notation we THE THEORY OF INFINITE SERIES. 91 proceed to a proof of Cauchy's extension of Taylor's Theorem to integral series, from wliicli the theorem lim/li±i^l-^) = «, + 2H.^ + 3w3^=+... follows as an immediate corollary. Cauchy's extension of Taylor's Theorem. Let f{z) = 5)(„2" have a radius of convergence B, and let (I ' I * I + . '* i < -^^ Then Cauchy's theorem is f{z + h)=f(z) + hf\z)+^^f'{z)+.... Because \z\+ h ' < E, the point z + h lies inside the circle of convergence, and f{z + h) is equal to the series Mo + "i(2 + /() + u,{z + h)- + ii,{z + ny + ... ; but \z\ + \h\ the absolute value of every term of /(«) on the 92 THE THEOKY OF INFINITE SERIES. circle with centre and radius r. We have j m„ | r" < C, and, if \z\ = Z p in. the to-plane and by circles of radius pi in the z-plane be called r'. As yet no limitation has been imposed upon p^ It must be possible to choose the finite quantities p, pi sufficiently small to secure that F{p, Zi, z,, ••• , z„) converges for all values of z,, Zj, •••, z„ which lie within r'. Inasmuch as Fi vanishes with the z's and F^ is independent of the z's, it must further be possible to choose p^ small enough to make ] 2^i | < j Fo | for all points of r'. In the region V determined as above, F F,-F, A=o2?'o^+i' and because ^=^"_§^, Sw 8iy 8w ^//F= f-yF, + i F,^ |?'72^o^+> - %Fr' l^'/Fo^ oiv 1 X8w * Welerstrass, Abtiandlungcn aus der Functionenlelire, p. 107. See also a Thesis by M. Dautlieville : fitude sur lea sdries entlirts i plusieurs variables. For the case in which F„ van. tshee Identically, sec Weiersiniss, p. 113. -Another proof of this important theorem will be found iu ricard's Trait^ d'.Analysc, t. ii., fascicule 1. THE THEORY OF INFINITE SERIES. 95 °° 1 The series 1 - (Fi/Fg)'' is absolutely convergent since, for all points in r', | J'l ] < , F^ . It may therefore be regarded as an integral series in ic, which is absolutely and uniformly convergent. We may therefore write f/F = f-yF,-f^{F,/F,y.... Sw biv dw 1 \ By hypothesis F„ = P^(it'); therefore the coefiB.cients in Pu(it;) being absolutely convergent integral series in 2], z.,, ■■■, z„, which all vanish at the origin, since there Fi = 0. Hence, uniting the various powers of u-, we get 2 \ {F,/F,y = i P(-) {z„ z,, ... z„)ic', 1 A. -» where again each coefficient on the right vanishes at the origin. Also ^/F, = ,j./w + P{ic). Therefore, throughout T', ^/F=,./w + P{ic)-^^ 8i« l/(zo-«'i) H 1- l/{tc-it\) = v/iv + sjw- + s,/it!^ + ••• , where s, = the sum of the xth powers of u\, ic,, ■■• , wv. Hence |?/J' = v/i« + P(ii') + 2 s. At.''+' • • ■ • (3). Sit) 1 Comparing (2) and (3), we see that /x = v; and also that Sj, s^ ^sr-- are equal to integral series in »„ z.,, ■■■,z^ which vanish at the origin. Let the equation whose roots are lUj, iC2, ■■■, w^ be M'^+/lM)''-l+.--+/^ = (4), then by Newton's theorem for the sums of the powers of the roots we have /i = - Sl, 2/2 = - S2 - S|/„ 3/3 = — S3 — S2/1 — S1J2, and therefore the coefficients/are integral series in z„ Zj, •••, z„, which certainly converge when these variables lie within the circles of radius pi, and which vauish at the origin. § 89. BeJiaviour of a series on the circle of convergence. We shall next examine the behaviour, when | 2 j = l, of the integral series 2a„2", in which the coefficients are all real, and the ratio fln/On-i can be expanded in a series where the coefficients are independent of n. We first suppose Cj = 0. Let Ci=c2= •••=c^_i = 0, c^^O. We shall show that a^ = a + h^/n''-^, where a is independent of n, and neither zero nor infinite, and &„ is finite for all values of 11. THE THEORY OF INFINITE SERIES. 97 Let (in/«,.-i = 1 + ^^./«^ so that Lk„ = c^. If we begin with some fixed term, say the pth, such that no sub- sequent term vanishes, we have Since k^^^ is always finite, the series Je+ Vi +...tooo be finite if ■ — | h • • ■ to oo P" {P + 1)" is finite, which is the case, since /x > 1. Therefore (§ 78) cfj,+„ lias a finite, non-zero, limit a when n = cc. Xow «„ — o„_i = A-„a„_i/»i''. If for n we write ?i -|- 1, •■• n -|- »• and add, we obtain "n+r — «n=/in.rO-n,r) where o-„,r=T — r^rrz + -, — t-^ttt + •■• + (rt-f-l)'' ()i-|-2)'' (n + r)'^ and ju,,^ , is not less than the least, and not greater than the greatest, of the quantities fc„+ia„, •••A;„+,.a„^.,_, ; 0. The case > Cj ^ — 1 remains to be considered. We have (1 - z)%aX = flo + 2(o„ - a„_i)2" - On^"""'- U 1 Now ia„2"+' = 0, and the series on the right hand is convergent when 71 = X, for a„ — a„_i a„_i aja^^ — l a„-i — a„_2 a„_2 a„-i/a„_2 — 1 Cl . ^ M - 1 "^ (n - 1) c, c, . \ n 1^ n-1 (n-l)2 n and Ci — 1 < — 1. Therefore (1 — «)2a„2'' is finite, and therefore 2a„2" is con- vergent unless z = 1. Summing up, the series in question, on the circle of convergence, is absolutely convergent when Ci < — 1, conditionally convergent when > Ci ^ — 1 and z^\, divergent when > Cj ^ — 1 and z = 1, and divergent when Ci ^ 0.* Example. Discuss the behaviour of the series '^z'/n", and of the expansion of (l + z)", on the circle of convergence; m being real. * Tbla proof is baaed on that of WeierBtrasB for the Integral eeries vrith complex coefficleDtB (Abbandlungen a. d. Functlooenlehre) and on Stolz's treatment of WeieVBtraae's tbcorcm (Altge- meine Arlthmetik). Pringsbeim has sbown that an integral aerlea can be conditioDally conver- gent at all pointa of the circle of convergence (Math. Ann., t. zxv.)> THE THEOKY OF INFINITE SERIES. 99 §90. The series f{z)= z/l-zy2 + z'/Z has a radius of convergence = 1. For the point z = 1, the convergence is condi- tional. Let 4>{z) = z/1 + 25/3 - zV2 + 2=75 + 277 - 274 .... when I 2 1 < 1, f{z) = (z) ; but, when 2 = 1, the values of /(I) and {z) must change its value abruptly as 2 passes along the real axis from a point inside the circle and very near 1, to the point 1 itself. The preceding considerations raise the question as to whether /( 1 ) is the limit of /(«), when x tends to 1. (See § 62.) The following important theorem of Abel's settles the point : — Let f{x) = tto + a^x -f a.^- + • ■ • be a series with real coefficients, which converges within a circle of convergence of radius 1 ; further, let aa + ai + a, + as+ ■■■ be a convergent series whose sum is s. Then lim/(l — ^)=fto + ai + ci2-\ , where ^ is a quantity which is real and positive. Let s„ = ao + ai + a2+ ••• +a„, then f(x) = So + (si — So)x + (s^ — Si)ar^ + (S3 - Sa)^^ + •••• If a; < 1, the series = {1 — x) (s,, + s^x + s.x' + s^a? -\ ). Thus /(I - ^)= ^ [s„ + s,(l - +S2(1 - 1)=^ + 83(1 - 1)' + - + s„(l - ^)"] + 1(1 - ^)"+' [s„^i +s„+2(i -i)+ s„+3(i - ^r + •••]• The series is divided into two parts. The number of the terms in the first part can be taken large enough to make s„+„ s„+2, s„+3, ••■ all > s — 8 and < s + c, when 8, £ are arbitrarily small. $ is still at our disposal. Choose it so as to make L n| = 0; e.g., make I = l/>i^ or l/?i'. Then because s„ + (l-^)Si + -+(l-f)X is a finite expression for finite values of n, ^[so + s,(l-0+-+«»(l-^)''l must vanish, when $ tends to 0. The remaining part lies between ^(1 - O'^'is _ 8) [1 + (1 - 1) + (1 - ^y + - to CO J and id - iY^Hs + [the same series] ; 100 THE THEORY OF INFINITE SERIES. i.e. between {l-^y+\s-S) and (1 - |)"+'(s + 0- ^o^ (1-^" lies between 1 and 1 — »|; hence lim (1 - ^)"+' = 1. Thus, ulti- mately, /(I — 0) lies between s — B and s + e, where 8 and e can be made arbitrarily small. This proves the theorem. (i.) The theorem holds when the coefficients are complex, for then the series can be separated into two parts, one real and the other purely imaginary. As the theorem holds for each part sepa- rately, it must hold for the two parts in combination. (ii.) Next the restriction that the variable is to be real can be removed. fl (1 — ^) (cos 4> + i sin <^) j = Oq + ai(cos (j> + i sin ) (1 — i) + a2(cos 2 ){l-iy-+ ■■■ can be separated into two parts, (tto-f Oi cos il-0 + a, cos 24,(1 - iy + •••), i{ai sin + ■••). To return to the example with which we started : — The series z/1 -z"/2 + ^/3 - z*/i + ■■■, and z/1 + 273 - 272 + 2^/5 + 277 - 274 + . • ., are equal when | 2 | < 1, but unequal when z = 1, in spite of the fact that both series converge for that value. The following is the explanation of the paradox. Write 2 = 1 — ^, and let $ be real and positive ; the two series are equal for all values of i, however small, but the convergence of the second series is infinitely slow in the neighbourhood of ^ = 0, and the infinitely slow convergence is accompanied by discontinuity. The result can be expressed in this way : when $ is real, lim{(l-0/l-(l-^)72 + (l-6V3 — •1=1-1/2 + 1/3-1/4+..., limS(l-^)/l+(l-a73-(l-072--| =1-1/2 + 1/3-1/4+... :^ 1 + 1/3-1/2 + 1/5+1/7-1/4 + .... § 91. The product of two integral series. Consider the two integral series iM„2", Sv^z", with a radius of convergence =1. At a point within this circle the product is cc 2h'„«" = Vo + (mo«i + u,Vo)z H h (moV„ + u,i\_i H 1- u„Vo)z" +■-. THE THEORY OF INFINITE SERIES. 101 If 2k„, 2i'™ converge absolutely, we know that the expression just written down converges when z=l. In virtue of Abel's theorem we can extend this theorem ; for ec (c 00 lim (Sw^x") lira (Ic^af) = lira {lw„ar) gives 2?/„ • 2v„ = 2!f„, when 2u„, 2t'„, 2iy„ are convergent. The theorem 2u„ x 2i'„ = 2(f„. when 2«„ and 2v„ are uncondi- tionally convergent, was stated by Cauchy in his Cours d' Analyse Algebrique. Mertens proved that the result still holds, when one of the two series is conditionally convergent (Crelle, t. Ixxix., 1S7J). The theorem which we have just proved is due to Abel (Ueber die Binomialreihe, Crelle, t. i.). It will be noticed that Abel's theorem gives no information as to when 2[o„ is convergent. For informa- tion on the product of tiLo conditv mally convergent series we refer the student to a memoir by Priugsheim (!Math. Ann., t. xxi., pp. 327-378). Weierstrass^s Tlieory of the Analytic Function. § 92. Series derived from an integral series. Let the integral series Wo + «i2 + U.X- + n^ H have a radius of convergence R. Within the circle (i2) it has properties analogous to those of integral polynomials, but without the circle it is divergent. If /(z) = Mo + «i2 + 'w' + Ma^" + ••■ = P(2), /(z) is also equal to an inte- gral series in z — Zq for all points inside the circle A whose centre is Zq and radius i? — j 2o ,(§ 87). This series in z — z^ is denoted by F(z i z„), and is said to be derived di- rectly from P{z), with respect to the point z^. For aU points of ^, P(2|2o)=P(2). Let z, be a point interior to A. With centre Zj, describe a Fig. 20 102 THE THEORY OF INFINITE SERIES. circle B -which touches A and lies entirely inside it. A series can be derived from P{z) with respect to Zj, namely, P{z , e^, and its circle of convergence will include the whole of the circle B ; but a series in z—z-^ can be derived from P{z \ «„) with respect to Zi, namely, P{z «„ Zy), and this latter series is convergent in B. The two series are identically equal in the circle B, for the former = P{z), and the latter = P{z z,,), throughout B. There is an obvious extension of this theorem, namely P{z\z:) = P{z\z,\z,\.:\z,), when 2„, Zi, z,, ••• lie inside D ; Zi, Zo, Zsi •■• inside A ; z^, 23, • • • inside B ; and so on. The importance of the result consists in the circumstance that a series directly derived from P{z), with respect to a point inside the circle of convergence, has been shown to be the same integral series as that obtained by the indirect process described above. The function defined by the integral series P{z) has a meaning for all points within the circle of convergence. A method will now be described by which the integral series can be continued beyond this circle. It depends on the fact that the circles A, B, of Fig. 20, are not, in general, the full circles of convergence of the correspond- ing integral series P{z \ Zf,), P(z | «]). Let the circle of convergence of P{z 1 3„) be not A, but A'. ■\Vithin A, P{z) = P{z \ 2„). The question is whether the two expres- sions are equal in the shaded region. Let a point z^ be taken within A, and let the circle B' be drawn to touch the nearer of the two circles D,A'; and the circle B to touch A. Since B' lies within the circles of convergence of P(z) and P{z I Zo), the integral series P{z 1 Zi) must be identically equal to P{z), and P{z j z^ \ Zj) to P(z]zo), throughout B'. But we know that P{z \ Zq) is identically equal to P{z) throughout B, a part of B'. Hence P(j I zi) and P(2 I Zo ! z,) are identically equal throughout B. These two integral series, which proceed according to powers of z - z^, must be identically equal for all points for which either has a meaning (§86). But we know that Fig. 21 THE THEORY OF INFINITE SERIES. 103 the circle of convergence of P (z ] 2o | Zi) extends at least as far as B'. Hence P{z \ z,) = P{z | z„ | Zj) throughout B'. Therefore P{z) and P{z I 2„) must be identically equal throughout B'. This proves the equality of the two series for that portion of the shaded region which is contained within B'. By adding on this portion to the circular region bounded by A, and by selecting suitably a new point Zj within the region so extended, the equality can be affirmed for a further portion of the shaded region, and the process can be continued until the equality of the two integral series has been established for all points of the shaded region. § 93. Theorems on derived series. Theorem I. Let P(2 | Zo), P{z | z„') be two integral series which are derived directly from the common element P(z), where Zq, Zo' are points in- terior to D, the circle of convergence of P{z); and let A, B be the circles of convergence of 7'(z|zo), P{z \ Zo'). The two series are identically equal with- in that region, which is common to A, B, D. We have to prove that they are identically equal within the shaded region. Let us take a point c, near D (Fig. 22). Within the circle (not drawn in Fig. 22) which has centre c, and touches D, P(z 1 Zo I c) = P(z I Zo' I c) identically. As these two series proceed according to powers of z—c, and as P{z ; z„ I c) converges for all points within the circle C of Fig. 22, the series P(z|zo|c), P(z|z„'ic), must be identically equal throughout G. Hence P(2lz„)=P(z|Zo') for that portion of the shaded region which lies within C. Add this portion to the unshaded region common to A and B. By a suitable choice of a new point c, within the region so extended, a further extension can be effected ; and the process can be continued until Fig. 22 104 THE THEORY OF INFINITE SERIES. the equality of the series has been established for all the shaded region. Theorem II. If the series P{z \ z^ | 2i) be derived directly from P{z I z„), the series P{z | 2„) can be derived from P{z | Zq \ ^i). If the series P{z\Zo\ z^) have the point z^ within its circle of convergence, we have immediately, P{z\z,\z,\zo)=P{z\zo), and the theorem is proved by a direct derivation. This will always be possible if j 2j — «o | < p/2, where p is the radius of convergence of P{z I Zo). If Zo be without the circle of convergence of P{z \ z^ \ Zi), it is possible to interpolate, between Zq and z^, a series of points a^, a,, ••• , a„, where n is finite, such that ttj lies inside the circle of convergence of P{z I z„ \ Zj) ; a2 inside the circle of convergence of P(z | Zq I ^i | fli) ; ffs inside the circle of convergence of P{z ^ ZolZi'aia.,) •■■ ; and finally, z„ inside the circle of convergence of P(2 { Zo | Zi | %■ ••• ]a„). From this last series can be deduced P{z ] 2,, | 2i | aj j cto | ••• j a„ | Zq), an integral series in 2 — z^- As the latter integral series is identi- cally equal to P(z \ z^) for points within a neighbourhood of Zg which is not infinitely small, the two series must be identically equal throughout the original circle of convergence. Therefore P{z ' Zq) can always be derived indirectly from P{z \ z„ ! 2,), when P{z | 2,, | 2i) is directly derived from P{z , z^) (Pincherle, Giornale di Matematica, t. xviii., p. 348). Theorem III. If c be a point of the region common to the two regions of convergence, A and B, of P{z — 20), P{z — z^), and if the two series derived directly frolu these, with regard to c, be identically equal : — (i.) The given series can be derived, indirectly, the one from the other. (ii.) The series derived from P(2|Zo), P(2|2i), with regard to any other point common to A, B, are identically equal. (i.) By supposition, P(2 l2„ ] c)= P(2 |zi| c). But P(2 j 2o) and P{z \ z^) can be derived indirectly from P(2 | Zo | c) and P(2 1 2i 1 c) ; hence P{z \ Zq) can be derived, indirectly, from P(2|zO. (ii.) Let Ci be a point in the maximum circle which can be drawn from the centre c, without cutting A or B. With regard to Cj, we can derive the two series P(z I 2o I c I Cj) and P(z | Zj | c | Cj) ; THE THEORY OF INFINITE SERIES. 105 and these series are identically equal to P{z | ^o | c), P{z \Zi\c) through- out the maximum circle. But P{z \ z^ \ c), P(z | «i | c) are themselves identically equal throughout this circle ; therefore P{z | Zo | c | Cj) must be identically the same series as P{z | Zi | c | Ci) . The same must be the case for the series directly derived with respect to Ci, namely P{z ; z^ , q), P{z \ z^ | Cj). We can repeat the same reasoning for a point C2, inside the maximum circle, described from Ci as centre without cutting A, B, and prove that P{z,Zo\c,) = P{z\z,\c,). By a continuation of the process, the theorem can be established for every point of the region common to A and B (Pincherle, loc. cit., p. 349). § 94. The analytic function. If P(z) be convergent within a circle D, and if from this integral series be derived, directly, the integral series P{z Zq), where Zj is a point of D, the circle of con- vergence A of P{z I Zo) will, in general, cut the circle D. A function which is equal to P{z) in the circle D ; to P{z ; Zg) in the circle A, Zq being any point in D; to P{z Zq \ z^) in its circle of convergence, Zi being any point in A, and so on, is termed by Weierstrass an analytic function. The analytic function is defined by an integral series, whose radius of convergence is not zero, together with all possible continuations of that series. Each of the series is called an element of the function ; thus P{z) is an element of the function at 0, P{z I Zo) an element at Zo, and so on. The value of any one of the totality of integral series at a point 2 of its region of convergence is one of the values of the analytic function at z. In defining the analytic function the choice of the primitive element, from amongst the totality of integral series which serve to define the function, is perfectly arbitrary; for if P(z | Zj) be derived from P(z|zo), the latter series can be derived from the former. Hence each series of the totality is an element of the function. In other words, any series selected from the totality will give rise to the other members of the totality, and will, in conjunction with them, completely define the analytic function. § 95. It must not be inferred from the preceding work that the analytic function is necessarily one-valued. Let P{z \ c) be an inte- gral series. It may be continued beyond its circle of convergence until it passes into P{z \ z'). Suppose this effected by the interpola- tion of n points. We have then a chain of n -f 2 points, and the 106 THE THEORY OF INFINITE SERIES. element at each (the initial point c excepted) is to be directly derived from the preceding element. "We may regard the n points as lying on a path L from c to 2', the path being subjected to the restrictions that it is not to cut itself, and that each of the n + 1 arcs into which it is divided by the n points is to lie within the circle of convergence of the element at the initial point of that arc. By what has been already proved (§ 92) the same final element is obtained when additional points are interpolated on L. And further the system of n points can be replaced by another system of m points on L, pro- vided that each of the m + 1 arcs so formed lies within the circle of convergence at its initial point. For the compound system of n + m intermediate points leads to the final element P{z j «'), and we can regard the n points as interpolated among the new m points, and suj^press them without affecting the final element. When an analytic function is said to be continued along a path L, it is understood that L lies wholly within the chain of circles of convergence. What is proved is that the final element then depends uniquely on the initial element and the path L. It readily follows that the reverse path — L restores the initial element. On the other hand another path M from c to 2' leads from the same initial element to a final element Q(z [ 2'), which may be dif- ferent from P{z I z') . When the difference exists, the contour formed by — iy from z' to c, and M from c to 2', leads from P{z j 2') to Q{z\z'). This is due to the presence, within the contour, of a irancJi-point, at which there is no element of the function. A point at which there is no element of an analytic function will be called a singular point of that function ; and the branch-points of a function are included among its singular points. We shall return to the singular points, in connection with the present theory, in § 97. They are recommended to the reader's attention, inasmuch as the character of a function is betrayed by its behaviour at these exceptional points. § 96. Tlie coefficients of an integral series. Consider the series f(z) = iu^z", with a radius of convergence R, and let a be a primitive 2)th root of unity. Let ^z\=pm, When p = 00, the right-hand side reduces to u„. To prove this, we must show that , lim {v^^y + n„^.y^ + u^^^^p'p -f- • • • ) = 0. THE THEOKY OF INFINITE SERIES. 107 00 The proof follows from the fact that 2i(„2" is an absolutely conver- gent series for j z | = p, for we have I OWpP^ + w.+^y" +•••)?"• I < I «»+, ; p"^" + 1 w.+p+i : p""-"^' + ■ • ■ to cc ; and this expression tends to the limit 0, when p increases indefinitely. Hence «„ = limU"•> 108 THE THEORY OF INFINITE SERIES. where O is the upper limit of the values of | f{z) \ on the circle p. Further, if two series 2it„z", 2y„2", have the same region of conver- gence, and if they be equal for all points of a neighbourhood, how- ever small, of a point z„ of this common region of convergence, then the theorem of undetermined coefficients holds good and m„ = v,^ § 97. Discussion of the singular points. Suppose that we know that a given integral series P{z) converges for all points \z\ pi ; therefore, by the theorem of the preceding paragraph, ^\r{a)\ R + p, the lower limit of the quantities R^ cannot be p. In the proof of this theorem of Weierstrass's we have followed Stolz, Allgemeine Arithmetik, t. ii., p. 180. That the true circle of convergence must contain at least one point such that in any neighbourhood of the point, however small, p is infinitely small, is evident from the following considerations. The lower limit of the quantities R, is zero, hence there must be at least one point, inside or on the rim, in every neighbourhood of which, however small, the lower limit is ; and this point must lie on the rim, since the radius of convergence at an interior point is not less than the distance from the point to the rim. 110 THE THEORY OF INFINITE SERIES. The points on the rim can be divided into two classes according as they are or are not singular : — A non-singular point Zi lies inside the domain of some point a, interior to the circle of convergence of P{z). The value of P{z) at 2i is P(zi \a\zi) ; for since P{z) and P{z \a\zi) have a common region of convergence which includes the point a, the two series are equal to P(z I a) throughout the region common to their circles of convergence, and in particular at the point z^. Thus for non-singular points the limit to which P{z) tends, when z tends to 2,, is the value P{zi\a\zi). The function defined by P{z) can be continued over the rim at Zj. A singular point is one within every neighbourhood of which, however small, p is infinitely small. These points do not lie within the circle of convergence of any point a, interior to the circle of convergence of P(z). As no elements of the function cor- respond to them, they are the singular points which we have already defined. By what we have already proved, no singular point of an analytic function can lie inside a circle of convergence, but there must be at least one on the rim. As an example, consider the function . Throughout the 1 — z interior of the circle, centre 0, and radius 1, 1 — Z but the point 1 is a singular point. If a be any point within the circle, the derived series with respect to a is /(a) + (z - a)f{a) + ^^fp^ /"(«)+ - , where /(z) stands for the series. But 1 f{a) = l + a + a'+- = /'(a)=H-2a-)-3a2-f ...= 1-a' 1 (1-ay /"(a) = 2 + 2.3a + 3.4a^+... = — 2i— , (1-a)' and BO on. THE THEORY OF INFINITE SERIES. Ill The derived series f{z a) is, therefore, ■ + ■ ; + {z - ay- (z - a) 1-a' {1-ay ' (1-0)3 (1- a) and the radius of convergence is \l — a\. -.+ ■ But 1 1 (z - a) ( 2 - g)" _ 1-z ({l-a)-{z-a)) 1 - a (1 - a)- {1 - a)" ' throughout the circle, centre a, and radius 1 1 — a | . Thus the inte- gral series l+« + 2- + z-'+ •••, together with its continuations, defines the analytic function . In this example, if the suc- cessive series be derived with respect to points a^, a2, a3---a„ 0, each point being within the circle of convergence of the point immediately preceding it, the final series is the same as the initial series. The figure is intended to show the way in which a function can be continued in the neighbourhood of an isolated singular point. Fig. 23 § 98. A one-valued analytic function f(z) is said to be regular at a point c, if it can be developed, in the neighbourhood of c, in a convergent integral series P{z — c). 112 THE THEORY OF INFINITE SEPJES. If /(z) be infinite at c, while (z — c)'"/(z) is regular at c, m being a finite positive integer, c is said to be &pole of the f-unction, and m is said to be the order of multiplicity, or simply the order, of the pole. If there be no such finite integer m, c is an essential singularity* It is evidently a matter of some importance to determine, from the integral series which serves to define a function, the positions of the singular points on the circle of convergence. This problem is discussed by Hadamard in Liouville, Ser. 4, t. viii. In particular it appears that if the ratio u^/u^^i have a limit a when k = cc, a is in general one of the singular points on the circle of convergence; that a is on the circle is evident from § 84. The ideg, of determin- ing a singular point in this simple way is M. Lecornu's. Reference must also be made to Darboux, Sur I'approximation des fouctions de tres-grands nombres, Liouville, Ser. 3, t. iv. § 99. When the circle of convergence of P{z) covers the whole of the z-plane, with the single exception of the point z = cc, the function defined by the series is nowhere infinite in the finite part of the plane, but has an essential singularity at z = oo. The algebraic function Ua + n^z + u^^ + ■■• +u^z' is finite throughout the finite part of the plane, but has a polar discontinuitj- at oo. The resem- blance in properties caused Weierstrass to name the integral series P{z), with an infinite radius of convergence, a transcendental integral function. As examples of transcendental integral functions we may instance those defined by the well-known integral series, l+z + zy2\ + z'fZ\ + ..., 1 _ zY2 ! -1- zY4! - ««/6! -f •••, 2 _ 23/3 ! -I- 25/5 ! _ 2Y7 ! + .... Such functions are necessarily one-valued, since there can be no branch-point at a finite distance from the origin. A transcendental function is a function which is not algebraic. The question of ascer- taining whether a given integral series defines an algebraic or a transcendental function was considered by Eisenstein, who proved that when an integral series Pi(z), with rational coefficients, defines an algebraic function, the coefficients can be rendered integers by the substitution of kz for z, where k is a suitably chosen integer. » The discuBslon of the singular points of » function resolves Itself into that of branoh-pointo, Infinities, and essential singularities. These will be considered In the following chapters. The pole is a case of an InHnlty. THE THEORY OF INFINITE SERIES. 113 [Eisenstein, Monatsberichte der Akademie der Wiss. zu Berlin, 1852 ; Heine, Kugelfunctionen, Second Edition, t. i., p. 50. For a remark- ably simple proof by Hermite, see Proc. London Math. Soc., t. vii., or Hermite's Cours d' Analyse, Third Edition, p. 174.J § 100. The sum of an infinite number of integral series. Let P<-''>(z) = Mo,« + Mi_,2i-|-M2,«2'+ •••(«= 1, 2,---,cc), be the xth term of the series ^k PM (^z) . We assume that the component terms, 1 as well as the series itself, converge absolutely and uniformly within the same circle (J?). (i.) The infinite series M„,l + M„,2 + M„,3+--- converges to a definite value m„, whatever be the positive integer n. (ii.) For every value of z, within the circle (B), 2"„2" converges absolutely and =2«PW(^)- 1 Let p be a positive quantity < R. When \z\ = p, there exists, by reason of the suppositions made above, a number /j. such that I P"'+"(z) 4- P'"-'^'(«) H 1- P^'+^Hz) i < « (" > m)j p being any positive integer. The expression P("+"(2)H [-/""""^'(z) can be written as an integral series, and in this series the absolute value of the coefficient of z^ ^ e/p^. Thus I Ma, n+l + Wa, „+2 + ••■ 4- "a, n+P I ^ ^/P''- As this is true, however large p may be, Ma,1 + Ma,2H to 00 must be a convergent series. Suppose that its sum is Ma- We have, ii. z^ = Z.+2 + ■■• to °<^) ^^^ 'i is arbi- trarily small. But we know that this inequality exists ; for we have seen that also |p(»+i)(2)+p("+2)(2)4....] is, by the definition of uniform convergence, < c, when »i > a suit- ably chosen number fx.. Thus 'S,p^ „z'^ and SPt"' {z) are, simultane- ' n+l ously, arbitrarily small; and part ii. of the theorem has been established. An application of the theorem. Let P(z), Q{z) be integral series, with radii of convergence E, R', and let R" be a positive number ! UiZ + U^- + UiZ' -\ I, and therefore, if u^z + u^- + u^z' -j- ■•• = v, 1 1 1 ( ;i-^+4-4 -fo(2) Wo + « "O = P{Z). Hence ^^ ' , being the product of two integral series, is itself ■•oi^) an integral series. To determine this integral series, let and, after multiplying both sides by Fo{z), equate coefficients. The resulting equations determine uniquely the values of Wq, Wi, ^2, •••■ Corollary. The quotient of an integral series Po(«) by an inte- gral series P,(2) is z-'-Qo{z). Theorem. If w — w' = P^{z — z'), then (Mj - w') V« = («.) V«P, (3 _ z') . For, if W-W' = U,{Z - Z'Y + M,^i(2 - z'Y+^ + •", then {w-w')y'' = {z-z')\u, + u.+^{z-z')->ru,^^{z-z'y+-\y- "l+iQ(z-«') + |=^lQ(^-Or+" = {u>,y'{z-z') 2U2t = (m.)V«Pi(2-Z'). The K values are furnished by the xth roots of «,. 116 THE THEORY OF INFINITE SERIES. § 102. The reversion of an integral series. Let y = Oo + aj^z + a._^- + (1), be an integral series, whose radius of convergence is 1, and whose coefficient a^^O. By a simple transformation any integral series can be changed into one with radius 1. When {y — rto)/«i = Vu ttie equation becomes y,= z-A.a--A,z'-A,z*- (2). The problem is to find an integral series for z, which has the form z = yi + b^i+biyi''+ (3). If (3) be, in reality, a solution of (2), the relation 2/1 = (2/1 + 6.2/f + h,y,' +■■■) -A,(y, + b,_y,' + b.j),' +■■■)' - A,{y, + h,y;' + h^,'+-Y must be satisfied identically. Equating coefficients of the various powers of y^, we find that h,-A, = 0, 63-2X62-^3 = 0, hi - Ao{h' + 2 63) -^3 . 3 62 - ^^ = 0, (4); equations which determine uniquely 62, 63, ••• as integral functions of the A's. To determine whether these values of 62, 63, •■• make the right- hand side of (3) convergent within a domain whose radius is not infinitely small, let us replace, in the relations (4), the quantities A by their absolute values \A\. The (k — l)th of the relations (4) gives, in conjunction with the preceding (k — 2) equations, a value for 6, in terms of the ^'s : 6, = an integral expression in A^, A3, •■■, A^, whose signs are all positive. Suppose this expression is (A^,A3,---,A,). The change of the A's into | ^ | 's cannot diminish the absolute value of the expression. A fortiori the change of all the ^'s into G, the maximum value of their absolute values, cannot diminish the value of I 1 ; while both functions exist for all values of 2. But there are functions which have no existence in regions of CD the 2-plane. For example, consider the integral series 26"2''", where a is an integer > 1, and b is positive and < 1. The series a/m« or 6"/"" has the upper limit 1, and therefore the radius of con- vergence is 1. On the circle of convergence there is certainly one singular point 2,,; if we write z = z' (cos 2/u,7r/a'' + i sin 2/xjr/a''), where /x and v are integers, all but the first v -j- 1 terms are unaltered, and therefore the series in 2' has the same circle of convergence and the same singular point 2o. Therefore the original series has the singular point 2o(cos 2 fin/a'' + i sin 2/i7r/a'') . By a proper choice of H and V this new singular point may be brought as near as we please to any assigned point on the circle. The circle is therefore a singular line ; no circle of convergence of an integral series derived from the series in 2 can extend beyond this line, and the function defined by the series has no existence except for points within the circle. The rest of the plane is said to be a lacnnary space of the function.* * The above example is due to WeieratraBB (Abb. a. d. Functionenlelire, p. 90) ; this proof is Hadamard's (Liouville. Fourth Sei-iea, t. viii., p. 115). Instances of Buch functions, defined by means of integrals, are given by Ilermite (.\cta Societatis Fennicse, t. xii.) ; other important sources are Mittag-Leffler (Darboux's Bulletin, Second Series, t. v.) and Poincar^ (American Journal, t. xiv.). Such functions occur naturally in the theory of modular functions. Bee Klein- Fricl:e, Modulfunctionen, 1. 1., p. 110. 120 THE THEORY OF INFINITE SERIES. § 105. The Exponential Function. The generalized exponential and circular functions exp z, sin z, cos z are defined by the integral series l + z + «V2!+2^/3! + .-, 2 _ z3/3 ! + 2^/5 ! - z'/l !+••■, l-zV2! + z'(»-3-i+f^- (ii.) exp (z + 2 -ri) = exp z- exp(2 -rri) = exp z (cos 2 tt+i sin 2 ir) = exp z. (iii.) cos z + i sin z = exp (iz), cos z — i sin z = exp (— iz), hence cos^ z + sin- z = 1. (iv.) The addition theorems for sines and cosines can be proved by replacing the sines and cosines by their exponential values. Enough has been said to show how the formulae of Analytic Trigonometry can be established.* • The reader will find a full treatment of the oubjcct in Hobson'e Trigonometry and Chryelal'e Algebra, t, ii. THE THEORY OF INFINITE SERIES. 121 § 106. The Generalized Logarithm. The generalized logarithm log z is defined as the inverse func- tion of exp 2. Let z = p(cos 6 + i sin $) ; the equations w = log z, z= expw, express the same relation between w and z. Hence, if w = m + iv, p(cos 6 + i sin 6) = e"+" = e"(cos v + i sin v), and, therefore, p = e", 6 + 2mr = v. If the real solution of e" = p be u = log p, we have log 2 = log p + i{6 + 2 rnr). At a given point 2o, one value of log z is log p + i9, but there are infinitely many others. If 6 be regarded as capable of unlimited variation, these infinitely many val- ues must not be regarded as distinct functions of z, but as branches of one and the same function, for after a de- scription of the curve with initial point 2,1, z returns to Zq, but log p + id changes into log p+i{0 + 2ir). If 6 be limited in its range to an angle 2 ir, as for instance if - TT < e < ir, Fig. 24 it is possible to separate the infinitely many branches. The chief branch is log p + iO, the nth branch log p + i{e + 2 rnr). If log z. Jog z denote respectively the chief branch and the 7ith branch, we have log 2 = log p + 15 {—■ir<6<-ir), „logZ = logp-l-l"(6 + 2n7r) (-ir 3, each point lies within the circle of convergence of the preceding point. Tlierefore if logXj = Pi(zi — 1), we have log 2, = log 2i + p/^= ~ ^ log «3= log 22 +py = 3P,(2^-l), and finally log (— 1) = })iPi(2i — 1). By taking the integer m large enough, the absolute value of m(zi — 1), or the perimeter of the half polygon, is as nearly equal as we please to the length of the semicircle; while the amplitude of m(2i — 1) is, as nearly as we please, a right angle. That is, lim m(Zj — l) = Tri. The limit of the ratio of any other term of the series mPi(2i — 1) to the first term is evidently zero; and we obtain + log(-l)=.n-i. Similarly log ( - 1) = — ,ri, and therefore the constant difference of log x — log x, where x is any negative real number, is 2 iri. It follows that if we allow z to cross the barrier and return to the point 1 after a positive description of the contour C + C, the value of the function at 1 is 2 tti, and the value near 1 is 2 Tri; + Pi(2 — 1). THE THEORY OF INFINITE SERIES. 125 To remove the restriction that C + C must be symmetric with regard to the real axis, we shall show that two curves L, M, which l^ass from 1 to z', without in- _, eluding the point 0, lead to the same series at z'. For let the dotted curve be at a finite but arbitrarily small distance from L. Let Ci, c/ lie within the circle whose centre is 1 ; c,', c/, c, within the circle (c,) ; and so on. Also let the arcs c._,c',_„ e',_,c'„ c'„c^, c,Q_i lie within the circle (c\), ("Mz) + 7«"-'/i(z) -t- w"-%{z) + ... +f,{z) = 0, in which the functions fo{z),fi{z), ■■■, f„{z) are integral polynomials in z of degrees not higher than ?>i. To a given value of z correspond in all cases n branches of to, namely, w,, ic^, ••-, w„, which are, in general, finite and distinct. Those points of the z-plane at which two or more branches are equal, or at which one or more branches are infinite, are named critical points. The finite points z, for which two or more values of w are equal and finite, are found by eliminating w from F=0, SF/Sw^O. The points at which a branch is oo are given by /o(2) = 0. This eqiiation is to be regarded as of the »»ith degree ; if it appear to be of the degree mi, it has m — mi infinite roots. A value of z which satisfies fo(z) = 0, but not /](2) = 0, is a pole ; a value which makes not only ^(2), but also f\{z), f.y(z), • • •,/k(2) is an infinity at which K 4- 1 branches are equal. The nature of the point z = 00 can be determined by arranging the equation i^= in powers of z and equating to zero the coefiicient 127 128 ALGEBRAIC FUNCTIONS. of 2". We thus obtain an equation in iv which is to be regarded as of degree n, having n — % infinite roots, wlieu the highest power of w therein is M"t. If this equation have equal roots, or infinite roots, « = oo is a critical point. Evidently 2 = 00 is a critical point when the polynomial F does not contain the term w"z"'. It is worth remarking that the view taken of the equation F = is different from that to which we are accustomed in Cartesian Geometry. Here we ask how many ro's there are when z is assigned, and how many z's when w is assigned ; in a word, it is the correspond- ence of w and z which primarily interests us. For example, take the equation w-z- + m' + z- — 3 = 0. In Cartesian Geometry it would be said that this is a curve which is met by any line in four points, and therefore, when z = l, the values of iv are ± 1, 00, 05. But the present point of view is that we have a (2, 2) correspondence always, and that when z = 1 the values of w are ± 1. § 109. Continuity of the branches. An immediate deduction from Weierstrass's theorem of § 88 can be made by writing m = 1 ; in this way we see that if /«. roots of F{io, z)=Ohe equal to b when z=a, the equation has fi roots nearly equal to b when z is nearly equal to a. For write z= a +z', w = b + 'w', and suppress accents ; when 2 is put equal to 0, F(w,0)=w''-B(w), ■where R{w) is an integral polynomial, which may reduce to its constant term. Hence = ii]''B{w) + (a polynomial in z, to, which vanishes when 2=0). By the reasoning of § 88 this equation is satisfied, in the immediate vicinity of 2 = 0, by /i. values of w which are nearly 0. Expansions of the branches at ordinary points. In the terminat- ing or absolutely convergent infinite double series F{W, 2) = Cio2 + CoiW + — (C2022 + 2 Cii2Z0 -I- Co2M)2) +•.; assume Coi i= 0. It is evident that F{w, 0) is a series which starts with CoiW. Hence /«. = 1, and the equation W' +fiW>'-'+f^w^-^ + . . . +/^ = ALGEBRAIC FUNCTIONS. 129 reduces to to+/i = 0, and, therefore, one and only one of the branches of w is equal to an integral series in z which has no term independent of z ; for fi = — Sj, and Sj contains no term independent of z. This deduction is of great importance, for it establishes the existence, when Cd =?t 0, of a single integral series of the form w = P^{z) (7->0). Hence if a single branch lo of the algebraic function given by F{iv% z") = be equal to b when z = a, there is only one expansion of the form iv — b = Pr{z — a), (r > 0). If all the n roots b^, b^, ■••, b„ be distinct when z = a, there are ?i integral series w-b,= PJ''\z-a) (r>0, K = l, 2,--;n). § 110. Discussion of the integral series Pr{^)- (i.) Let )■= 1, and let the series be written IV = a^z + a.yS- + a^z' -\ — . In the immediate neighbourhood of « = 0, iv =0i2 approximately; and the z-plane near 2 = is conform with the w-plane near tv = 0. (ii. ) Let r > 1. iv = a^ + a^^^z'-^'^ -] — ; when 2 describes a small circle round the 2-origin, the approximate equation w = a^' shows that tv makes r turns round the w-origin ; in both cases (i. and ii.) the final and initial values of the series are the same. § 111. The behaviour of a fractional series in the neighbourhood of a branch-point. Let IV be a function, of which r values at a point z near z = are defined by the fractional series W,+i = C,(a'2V')'. + C2(a'z'/-)'' + CsCaV/')'' + -, where q^, q^, ••■ , r are integers, 9i < 92 < 93 < •••> a = e^^'", K = 0, 1, 2, -, r - 1, z'/' is the same for each term, and one at least of the exponents g,/r, q^/r, •■■ is in its lowest terms. When z describes a small circle round the origin, z^'' changes into az^'% and the series for w,, w^, •••, w, change into the series for Wj, Wj, • • •, tOi. Thus the r values tv^, Wj, • • •, w. 130 ALGEBEAIC FUNCTIONS. permute cyclically round 2 = 0. When the description of a circle round a critical point permutes some of the branches, we shall call that critical point a hranch-point. When a branch-point permutes only two branches it will be called a simple branch-point. § 112. The theory of algebraic expansions is purely analytic, depending merely on the nature of F{w, z) = ; but the understand- ing of what follows will be facilitated by the use of the language of Cartesian Geometry. By speaking of z and w as co-ordinates, it becomes possible to demonstrate the parallelism between the ex2:)an- sion-problem in the Theory of Functions and the problem of the resolution of higher singvilarities in the Theory of Higher Plane Curves. It is important, however, that the reader should bear in mind that oo is appoint in the Theory of Functions, whereas in pro- jective geometry all points at an infinite distance are regarded as situateiupon a line. The investigation resolves itself into an examination of the points of intersection of the curve F{ic'', z") = with a line z = a. When this line intersects the curve in n distinct points, which lie within the finite part of the plane, the n expansions for the n branches are all integral. When the line touches the curve or passes through a multiple point, there are still n expansions which may be integral or fractional according to circumstances. A finite pair of values which satisfies both F=0 and hF/hio = is in the Cartesian theory a point such that the line drawn through it parallel to the axis of w touches the curve or passes through a multiple point. In the parabola ic- = z, the critical point z = is a branch-point, for the tangent at (m = 0, z = 0) has simple contact. The expan- sions z'''-, — z'^^ are terminating fractional series. In the curve vf = z, the inflexional tangent at the origin yields a two-fold branch-point with a cyclic sys- tem (wiWjMJ,,). This two-fold branch-point can be resolved into two coincident branch- points (see Fig. 27). More generally the tangent to w''+> = z at the origin is vertical and yields a (c-fold branch-point, which can be resolved into k simple branch-points. To examine the behaviour of the curve F— at the point (z= a, w= b), it is customarv to transform the origin to the z—axis Fig. 27 ALGEBRAIC FUNCTIONS. 131 point {a, b) by writing z = z' + a, iv = w' + b, and thus change F {w. z) = into F'{io', z') = 0, aneqiiation free from a constant term. The problem of tracing the curve F'(io', z') = at the new origin is equivalent to the examination of those values of iv' which become infinitely small with z'. Omitting accents, it is evident that there is no loss of generality in considering the point (0, 0) on the new curve instead of (a, b) on the original curve. The passage back to the original form is effected by writing z — a, iv — b, for z, iv. § 113. When the form of an algebraic curve near the origin is to be determined, the usual plan is to arrange the equation in the form (?(,', 2)1+ (W, Z)r,+ (lO, 21)3+ ••• =0, where the number of terms on the left-hand side is finite, and (u; z)^ denotes the terms of the rth order in w, z combined. Let the equa- tion be Cjo2 + Cojty + — (C20Z- + 2 CiiZic + Costf^) + IT] (<"30^ + 3 <^2lZ^J« + 3 CijZZC^ -f Co^lV^) -I- • • • , so that c,„ is the value of ^"'' ^ , when z = iv = (i. "When the lowest terms are of the order fc {k ^ 1), the origin is called a fc-ele- ment. Preliminary to the discussion of the most general case, we shall examine the expansions when the origin is a 1-element, or a 2-element with no tangent parallel to the axes. The 1-element. (i.) Suppose C]o, Coi ^ 0. There is, by Weierstrass's theorem, one and only one expansion of the form w= Pi(z). (ii.) Next let c,„=Coo= ••■ =c,_i,„=0, c^, Cm^O; then there is still a unique solution in the form of an integral series without a constant term ; as the lowest power of z is z"", this series is to = Pr{z), which gives on reversion 2 = Pi{v}^''). (iii.) Finally, let Cm ^t 0, Co, = Cos = ••• = Co,s-i = 0, Co» :#= ; then, by reasoning similar to that just used, «=P,(w) and w = Pi(2'/*). The geometric interpretations of these three cases are imme- 13: ALGEBRAIC FU^s'CTIONS. In (i.) the origin is an ordinary point with a tangent oblique to the axes ; whereas in (ii.), (iii.)) the tangent meets the curve in r or /(. consecutive points, and is parallel to the z-axis or lu-axis. The origin yields in case (iii.) an (/i— l)-fold branch-point, which is equiv- alent to 7i — 1 simple branch-points. In ease (ii.) the origin is an (?■ — l)-fold branch-point when z is regarded as a function of w. The 2-element. Here Coi = c^ = 0. The equation of the curve is = ^77 (^20^^ + - '^11^^" + ^ ^)3 + ■ • •• The nature of the singularity is dependent upon the factors of the (i.) Let CjoZ- -f 2Ci]2lC + Co2J(.'- = C(i2(2W — aZ)(K' — ^z), where Co2, a, (3 ^0. The origin is a node with tangents oblique to the axes of z and w. When iv — az is put = il\z and the equation, after division by z-, is re-arranged according to ascending powers of ««i and z, the origin becomes a 1-element with a term W]. Hence, if ic' be the series connected with the factor tc — az, w, = P,{z){r^l), and zo' = az -(- P,+i(z). Similarly the factor w — /3z contributes an integral series iv" with the first term fiz. Analytically the two series iv', iv" at the node are unaffected by the description of a small circle round the z-origin. Thus the node, with tangents oblique to the axes, is a critical point at which two roots become equal, but not a branch-point at which two roots permute. Geometrically it may be regarded as equivalent to a point at which two simple branch-points unite and destroy each other.* Fig. 28 (ii.) The simple cusp, with a tangent inclined to the axes. Let /J = a i^fc ; and let the factors of {w, z), be A{w — \z){io — f).z){w-vz) where X-a, /a — a, v — a^Q. • The node may equally be legaided as the unioa of two branch-poinu with regard to w. Fig. 29 ALGEBRAIC FUNCTIONS. 133 The transformation to — 02 = w^z leads, after division by z-, to an equation in %i\, z, \?itli a 1-element ; the lowest power of 2 is the first, aud the lowest power of w^ the second ; hence it'i = Pi («'/-) and io-az = I\{z^'-). Since z^'- is a two-valued function, two branches of w are represented by this series, and these branches permute, when z describes a small circle round the origin. Thus an ordinary cusp yields two fractional series in z, which are integral in z^'-. Analytically the corresponding critical point behaves as a combination of a node and a branch-point. Geometrically the simple cusp, with a tangent oblique to the axes, has for its penultimate form a vanishing loop, and the branch-point corresponds to the vertical tangent to the loop. Without stopping to examine the more com- plicated cases which may arise in the case of the 2-element, we pass on to some geometric considerations which will be of help in the treatment of the prob- lem of expansions. Tlie Cartesian Problem. § 114. The resolution of a higher singularity of F{w, 2)= into simple singularities, namely, v nodes, k cusps, r double tangents, I inflexions, where v, k, t, t are integers which satisfy Pliicker's equations, was first effected by Professor Cayley (Quarterly Journal of Mathematics, t. vii.), but the proof, which rested upon the nature of the expansions and the properties of the discriminant, was merely outlined. Afterwards Nother, Stolz, H. J. S. Smith, Halphen, Brill, and others completed the theory. The determination of these numbers suggests the existence of a penultimate form in which these 8 nodes, «• cusps, etc., are to be found separate from, but in infinitely close proximity to, one another, the actual form being derivable from the penultimate by a continuous deformation. Proc- esses by which this can be effected have been described by Halphen and Brill (Math. Ann., t. xvi.); see the memoirs by Halphen, Memoire sur les points singuliers des courbes algebriques planes, (t. xxvi. of the Meraoires presentes a I'Academie des Sciences); and Neither (Math. Ann., t. xxiii.). These we shall not discuss, as all that we require is the expansions at the point, as determined from the given equation. We shall have occasion to discuss two distinct methods for the 134 ALGEBRAIC FUNCTIONS. determination of the expansions at a multiple point : one of these was used by Puiseux, the other by Hamburger and Nother. The principle which underlies ]S^6ther's process is the determination of the nature of a higher singularity on a given curve, by the employment of a series of Cremona transformations. His inference was that any higher singularity can be regarded as composed of one or more ordinary multiple points at which the tangents are all separate, combined usually with branch-points. The transformation used was the special quadric inversion Avith coincidence of two of the principal points, namely ic = u\z. This arises from by writing ^ = z, ri = iv, t, = 1, e = 2, r,' = u;, V = l- jSTo loss of generality results from the special form of the trans- formation, for every Cremona transformation is reducible to a suc- cession of quadric inversions. It is not possible to replace, in this waj", a curve with higher singularities by another with nodes only, but the penultimate form of a higher singularity with k separate tangents, not parallel to the axes, is easily conceivable. The singu- larity in question results from the union of ^^■(A- — 1) nodes. The principle that an equation F{tv, z) = 0, with higher singu- larities at which the tangents are not separated, can be replaced, by means of Cremona transformations, by a new equation which has only ordinary multiple points with separated tangents, is useful for many purposes of the theory of algebraic functions. The theory of such transformations is given in Salmon, Higher Plane Curves, and in Clebsch, Vorlesungen fiber Geometrie. Nother s Melliod. § 115. We resume the discussion of the expansions in the neighbourhood of the non-critical and critical points of the algebraic function defined by the irreducible equation i^io", a'")=0. It has been explained already that no generality is lost by supposing the point (ft, b), at which the expansions are required to be the origin; at the end iv, z can be replaced by w — b, z — a, it being understood that z — x, w — CO are equivalent to 1/z, 1/w. It has been proved that, when Cqi^O, a root iv, which vanishes when z = 0, can be expressed as an integral series M- = iv'z + w"z-/2 : + ic"'zy3 l + --, ALGEBRAIC FUNCTIONS. 135 the series having a finite region of convergence. The next question is to determine the coefficients ic', iv", w'", •••. These coefficients are the values of the derived functions dw/dz, dhv/dz-, ... at (0, 0); and the problem is merely that of expressing the derived functions of IV, when given by the implicit equation F{w, z) = 0, in terms of the partial derived functions. We may substitute for iv the infinite series ic'z + iv"z-/2 I -\ , and equate the coefficients of the powers of z; or we may proceed as follows : — Let IV, z be functions of t, and let F=F{iv, z) = 0. If accents denote differentiations with regard to t, F' = iv' ^+z'^=D,F suppose. dw bz To find y (DiF) we observe that, while 8F/Sw and BF/8z are functions of w and z, w' and z' are functions of t, and that now dt ho &z &t ' St Let ^ = w"^ + z"^=D„ 8t Sw Bz ' -~ = £>3, etc. ht " Then F''=(d, + ^^D,F= (A' + D,)F, F"'=fD, + IjF" = D,F" + (2 AA + D,)F, = {D,' + ZD,D, + D,)F. F"" = D,F"' + (3 A'A + 3 A' + 3 AA + A)-?; = (Z)/ + 6 D,W, + 4 A A + 3 A' + A) F* Fut t = z; then3' = l, «"' = (k>1), and A = - + «-•'—> A = w" —• etc. Sz Bw oiv * The law of these exprcssioDs is that id F*''' we first find all positive integral solutions of ffi*! + ^2*3 + •■• = ^1 that is, all the unrestricted partitions f or r ; and then for each solution we form r! D "^D "'.. F a,!a,!...(s,!)''>(s,!)''^.. 'i 'i and add the results. 136 ALGEBRAIC FUNCTIONS. In applying these formulae of differentiation to our case, it is convenient to write w = v^z, and divide by z, so that F—0 becomes = Cjo + CoiVi + — 2 (Cso + 2 Cii^l + c^v^) + ■■•, 2" Z^ or 0=(^i + Z(^2 + ^<^3H \--^4>g+i! g being an integer not greater than m + n — 1, and ^ being a polynomial in v^, of degree > r. The operator Di is now v'l — + ^ ; ol'i 02: 8 it may be replaced by v'l \- E, where E is an operator which changes (t>r into <^^+i (where if r > g", <^,+i must be zero), for = EF. We have now only operations in which z does not occur, and we may put z = before effecting the differentiations. Let z = 0, and let the values of Vi, v'l, ■■• then become v, v', •■•. The equations FW = become <^i = 0, (A' + A)<#'i=0, (D,' + 3D,D, + D,), = 0, (A* + 6 A' A + 4 A A + 3 A + A) <^i = o, In these equations E must operate before — . The two are not always commutative ; for instance, E~i>, = 0, while ^E <^, = ^. Further, A'^.= <^., + .v|= + ^-(^.'^s^' + .. Il ^,Mr^^-. \\r — s ! hv' ALGEBRAIC FUNCTIONS, where s>r + l — s, or 2s>r + l. We have, accordingly, 137 ^•i 6y by- 6y 6y It will be useful to write down the analogous results when the origin is a fc-element; the equation F=0 becomes, after the trans- formation w = vz, = <^4 + 2'/>4+l + ^ <^t+2 + the suffix denoting in each case the degree in v. The equations are <^* = 0, II. <^».3 + 3 v'^ + 3 1;'^^ + 1;''^* In the system of equations I., — or % ^ 0. Hence the equa- Sv tions give in succession v, v', v", •••as rational fractional functions of the c's. The value of Vi, when z is not zero, is z- Vi = v+zv' + --v" + — , 138 ALGEBRAIC FUNCTIONS. this series being legitimate near z = 0, since we know that there is an integral series Pa{z) for w or v,z. Finally, since w = zvi = zv + z-v' + —v" +•■■, the coefficients in the series for the branch of w are determined.* § 116. To return to the expansions at a fc-element, we have now to discuss in detail the behaviour of the branches at such a point. If k = l, and if c,o ^ 0, Co^ ^0, c,^ = c^, = - = Co, ;,-i = 0, we have proved that there is an expansion z= A (?(-■) and have shown how to find the coefficients i (^p\, -J— -^^, etc. On reversion, hl\chc^J h + llduf'^'^ this series gives a cyclic system of h series, ?o = Piia'z^'"), where a = e-"'^" and r = 0, 1, 2, ••• , /i — 1. For instance, if iv^ = z + zw, the roots in the neighbourhood of the origin are iv = o^y' + -z'''-—^^'-, 3 81 a being any cube root of unity. When k > 1, all the partial derived functions of the orders 1, 2, • • • , fc — 1 vanish, and the terms of the lowest order are — {c,^,z'' + c,_-^^''-hu -\ h Coiio*), or (iv, z)^. With each factor of {ic, z)^ are associated one or more expansions for w in terms of z. Each factor must be examined in turn. The simple factors of {ic, z)^. (i.) Let lu — PiZ be a simple factor such that P^ + O. Write w — 13^ = WiZ in the equation to the curve and, after removal of the factor z", arrange the resulting equation according to ascending powers of i«i and z. The origin of the new curve is a 1-element, in which to, occurs to the exponent 1. Hence there exists a single integral series for «', of the form Pr{z), {r > 0) ; i.e. 10 = p,z + w,z = /3,z + P,^,{z) = P,{z). (ii.) Let w be itself a simple factor; i.e. let Cjo = 0, Cj_i, i =;t 0. The only difference from the preceding case arises from the circum- stance that ^1 = 0. Thus iv = P^{z) , (r > 1). *See PlUcker, Tbeorie der algebraischen Curven, p. 156. See also Stolz'e memoir, Ueber die singularen Punkte der algebraischen Fiinctionen und Curven, Matb. Ann., t. viii., p. 415. The quadric transformation io = t\z was first used by Cramer in connexion with the resolutioQ of higher singularities (Analyse des Lignes CourbesJ. ALGEBKAIC FUNCTION'S. 139 (iii.) Let 3 be a simple factor of (jc, z)^ ; i.e. let Cot = 0, c,, i._i ^t 0. The transformation to be used in this ease is 2 = Zyic. called by Xothei' a transfovmatioa of the second kind, the one previously used being a transformation of the first kind. After division by lu', the resulting equation in Zi and lo has a 1-elemeut at the origin, and {z^, ?f), reduces to Zi + aw, where a may be zero. If a^O, z = z^iv = Po{iv), and reversion gives w = P,(z'/-). In this case z is not a factor of {iv, 2)j.+i. But if z be a factor in each of the groups {ic, z),^„ {w, 2),^„, -.., [lo, 2),+,_„ and not in {w, 2),+,, then z^ = I\(w) and z = z^iv = P^_^i{iv) . On reversion this gives u'= Piiz^"-'^'-'}, a cyclic system of 7 + 1 expansions. It will be noticed that the first power of w, which is not multiplied by 2, is w'+« and that the difference between the exponent k + q and the order k of the element is precisely q ; also that from the point of view of fhe Theory of Functions there is a (/-fold branch-point asso- ciated with the cyclic system oi q + 1 expansions. Combining the preceding results, we see that to each simple factor lo — p^z of {iv, 2)4 there corresponds one and only one integral series, and that to a simple factor z correspond q + 1 series, where q is the difference between the exponent of the lowest separate power of w and the order of the element. Tlie Multiple Factors of (iv, z)^. § 117. We have now to examine the expansions connected with the multiple factors of (iv, 2)4. Let (iv — ^^z)' be a multiple factor of these terms, and for simplicity assume /3i ^ 0. Apply the trans- formation of the first kind w — fS^z = il\z, and let the transformed equation be divided by 2* and arranged according to ascending powers of Wi and 2. The lowest power of w^ yielded by (w, z)^ is Wj', and the new equation begins with a fcj-element, where ki ^ I. The order of the new element depends upon the manner in which the factor w —fiiZ is involved in the groups {iv, 2)j+„ {w, z)t+2i •••' («") ^)t+ii- It is necessary that it should occur in these groups raised to expo- nents not less than the numbers of the sequence A'l — 1, fci — 2, •••, 1, and that, in one case at least, the exponent should be exactly equal to the corresponding number of the sequence. By Xother's reasoning the resolution has been partially effected: the ft-element and the A;i-element contribute ordinary fc-tuple and A:i-tuple points, while the absence of the terms w"^, «/*i+\ •••, m'i"^"~*i~'', 140 ALGEBRAIC FUNCTIONS. shows that there are I — Jci branch-points. To account for the presence of the branch-points, all that we need do is to show that at an ordinary fci-tuple point with separate tangents oblique to the axis of ?0i, there are only k^ values of u\ which vanish when z = 0, whereas in the present case there are I values of ici. Let the new equation Fi{iVi, z) = be written (tt'l, Z)i-, + (?"l, 2;)4, + l + ('(-'H 2)t,+2 + ••• = 0. (i.) If (it'i, z)^ — contain a simple factor Wj — fi.z, the equation must be transformed by writing ?Ci — /JjZ = icx. Divide by z'l and arrange according to ascending powers of w.j, z. The new origin is a 1-element, and the reasoning of § 113 shows that ii.:,= l\(z) , ()->0), or w = li^z + p.z- + P,{z), (i->2). The expansion associated with the pair of factors iv — ySjZ, «'i — ftz has been found. (ii.) The necessary modifications when /Si = or ^2 = can be easily worked out. (iii.) If (ly,, 2)j. = contain a multiple factor {iv^ — /Sj^)'', where /Ju is finite and different from 0, use the same transformation as before, namely «'i — p.z = ir.^z. After division by z'l, the equation in u-2, z begins with terms of order k.,{^li), say {ic.,, 2)1^ To each simjile factor Wj — P^z corresponds an integral expansion M = /?,2 + p^z- + P.z' -f P,(2), (r > 3) and the process terminates. The existence of the ^velement shows that the resolution of the higher singularity can be carried a stage further by combining the fc-tuple pioint, the fcj-tuple point, and the I — ki simple branch-points, with an additional fcj-tuple point and li — Aij simple branch-points. The further resolution of the Avelement is accomplished by selecting a multiple factor {ii.'2 — l3jZ)'2 and using the process already described. Assuming the truth of a theorem, — which will be proved below in order not to interfere with the continuity of the argument, — the process must terminate. That is to say, it is impossible to continue the expansion indefinitely by means of multiple factors {iv — PiZy, («'i — /JjZ)'", (ecj — /?3Z)'2, •■•. Sooner or later this process would lead to an equation F^{it\, z) =0, in which the origin is a fc^-element, with simple factors itv — /iv+iZ, and possibly multiple factors in z alone. All the expansions connected with the linear factors ii\ — l3,,+iZ are integral, and the process stops, so far as they are concerned, at this stage. It remains for us to examine the case in which one or more of the expressions (w, 2)4, (ii'„ z)jj, (wj, z)^,--- contain powers of z as factors. When, at any stage of the process, a power of z ALGEBRAIC FUNCTIONS. 141 makes its appearance as a factor, and when the expansions are to be continued by means of this factor, Mother's second transformation z=Ziiv must be used. The eifect of this transformation will be evident from what has been already said ; z takes the place of w and w of z. § 118. The following is the general process : After m, trans- formations of the first kind suppose that a factor z is selected, and that the expansion associated with this factor is to be found. The part of this expansion due to the mi transformations is l3iZ + l3^' + l3^ + .-+ p„,z'", + k'z'"^. In the equation which connects w' with z, the terms of the lowest degree contain a factor z, and the expansion connected with this factor requires the transformation z = Zyw', to be followed by Zi — y^iu' = z.,w\ Zj — y^w' = z^w', and so on. Thus Z = y,iu'- + 73?«'^ + 74l'j" +•■•+•/„ to""2 + iv"iv''"t, the series stopping when the lowest terms in z„^ and w' contain a factor «.•', and when the expansion is to be continued with regard to this factor. The next series of transformations is w' = iL\'ic", w'/ = Sjw" + iv.Jiu", ic.J = S-iW" + lu^'w", and so on. The process terminates as soon as the factor to be considered is linear. For if to<'+'> — Xtu"' be a linear factor of the lowest terms in «o<'+", io'-'\ we know that zu('+» = PX««*")> ('•>0). The system of expansions, when w — ^^z is the factor selected from (iv, z)j, is the following: IV = ftz +p.x'' + — + /S^.z". + WiZ"!, z = y^iv'- + y,w'^ + • • • + y^y^' + io"w"'r, w' = h^iv"- + Ssw"^ + • • • + Ky'"^ + «;'"(«""•», I.^ In these expansions some of the constants may = 0. Because ?«"+" is expansible as an integral series in iv,{=t), it follows that w"~^', w"~^', ••• can be expressed as integral series in t. ::.} • The two series are termed by Weierstrass a " functionenpaar." See Biermann, Analytische Functionen, ch. iv., sect. 1. It is understood that t lies within the region of convergence of both series. 142 ALGEBRAIC FUNCTIONS. when \, IX are positive integers and /x ^ X. By reversion of series and therefore w = B^''^'' + Az'"""'^^ +"• • If (?c, z), contain a factor in z, and the corresponding expansion be required, the system I. must be replaced by II. 2 =J3,IC'- +ftil-''+...+/3„„2f"i +«-'ic»., w =y,ic'- + y,li:"+--- + y^jv'"'-- + ic'w'""^, As before, zy<'+" is an integral series in lo"', and but now the positive number ft. ^ the positive number \. The series for IV in terms of z is again m; = 5^"/^ + Az"'^"'" + • • • • Eeturning to the point {w = b, z = a) the corresponding expan- sion is iu-b=B,{z-aY/^+B,{z-ay''+'^/^ + --, where ju, is a positive integer. If Z; = a: when z = a we write for w — x, 1/to ; hence 1/ic = 5„(z - a)"/^ + A(2 - a)'""""^ +••■, which gives B,w = iz-u) -■"/* s 1 + Cj (z - a) '/^ + C. (z - a) =/^ + • • • | . When a = X, z — a must be replaced by 1/z. § 119. To complete the proof that at (a, b) vi — b can be ex- pressed as an integral or fractional series, we have to show (i.) That after a finite number of Nother's transformations of the first kind there must result an equation Fp{Wy, 2) = in which the factors are either simple or powers of z. (ii.) That t is in all cases finite. If it be possible to continue the transformations of the first kind indefinitely, there must exist two or more branches of iv which coincide, near z = 0, to any order of approximation. To see that this is impossible, form the equation {W, z) = 0, whose roots are the squares of the differences of the n roots of Fl^w", 2") = 0. ALGEBRAIC FUNCTIONS. 143 Let it be AJV^"-'^ + ... + Li,W+ 3f^ = 0, where A, B, ■■•, L, M, are integral functions of z, of finite degrees a, ■••, X, ju., and M is the discriminant, save as to a factor. If two series for iv, near z — a, coincide as far as a term (z — a)' exclusive, one value of TFis of the form P.2,{z — a) ; and the equation shows that M^ = Pp{z-a), where p is certainly as great as 2q. Hence 2q'^iJL, and cannot be infinite. This shows that v is finite. It must next be proved that t is finite. We know that after a finite number g of transformations, the process either terminates or must be continued by a transformation of the second kind. Hence nil, ^f/, is finite. That the order of the element in the (««', z) equa- tion > fc — 1, can be shown by the consideration that, after each transformation of the first kind, the order of the element is not increased, while the transformation of the second kind must dimin- ish the order. For instance, if 2 be a factor of the fci-element of § 117, Jci must be less than I, and I is at most = k. Thus the order of the element in the {u'\ z) equation > fc — 1. Since, after each change in the character of the transformation, the order of the ele- ment is diminished by at least 1, the number t must be finite. § 120. Rhumi of the preceding results. The equation F{iv", z") == gives n expansions of w for each value of z. If the n branches for z = a be finite, distinct, and equal to 61, 62, ••-, 6„, there are n integral expansions IV— 6a = P{z — «) ; but if two or more of the values be equal, say 6i=&o='-' = 6^( = &), the corresponding expansions may be all fractional, some fractional and some integral, or all integral. To a given value z = a, there may belong several separate systems of equal roots, consisting of /I], iM, ■•■ members. Geometrically, these cases of equal roots occur when 2 = a is a vertical line, which is a tangent to the curve at one or more points, or passes through one or more multiple points. In the most general case, the /x. expansions of a system (/ix^P{{z — ay^''>C) ■■■ r^s^ expansions. 144 ALGEBRAIC FUNCTIONS. where /«, = TiSi + r^s^ + ••• +''aSa. The 9\s, expansions (w -b) = {z- ay''KP{ {z - ay/^K, form a set, which can be subdivided into s, sub-sets, each forming a cyclic system of r, expansions. The typical member of such a sub- set is where a = e"""' ; v = 0, 1, 2 ■ ■ • r - 1, and one at least of the fractions q^/r, q.Jr, • • • is in its lowest terms. In the special cases o=cc,& = cc, z — a and w — & are replaced by 1/2, l/2«- The preceding treatment of higher singularities is based upon the memoirs of Hamburger (Zeitschrift fiir Mathematik und Physik, t. xvi.), Xother (Math. Ann., t. ix.), and Stolz (Math. Ann., t. viii.). § 121. Resolution of the sinr/ularity. AVhen the branches of an algebraic function at a singular point are expanded in integral and fractional series in the way now explained, we are able to determine the number of nodes and of simple branch-points into which the singularity may be supposed to be resolved. It is convenient to define the relative order of two branches w„ w^ as the least exponent of z in the difference of the two expansions of these branches. Geometrically speaking, if w, and w^ be equal to integral series, the relative order = the order of M, — W), = the number of intersections of the branches at the origin. The least exponent of z in the product of all the differences of the branches, or the sum of all the relative orders, may be called the total relative order. Now a simple branch-point, at which there are two expansions w, = P,{zy'),w, = P,{-z'''), contributes 1/2 to the total relative order; and a node, at which w, = a^z + Pi{z), Wa. = o-kZ -{■ Q,i{z) , a, ^ ax, Contributes 1. Therefore, if the origin can be resolved into 8 nodes and /3 simple branch-points, we have where I is the total relative order. But fi is determinable at once from the expansions ; for if r be the least common denominator of the exponents which enter into a cyclic system of expansions, that cyclic system contributes ?• — 1 simple branch-points. Therefore ^ = 2(7' — 1), and then the preceding equation gives 8. ALGEBRAIC FUNCTIONS. 145 For example, suppose we have the single cyclic system Here /? = 5, Z = 3 x 5/2 + 12 x 4/3 = 231, and therefore 8 = 21. The 5 branch-points indicate that there are in the penultimate form 5 small loops, and therefore the singularity is equivalent to 5 cusps and 16 nodes. The number of branch-points at a multiple point ceases to be equal to the number of cusj^s if the ?f-axis be parallel to a tangent. To take the simplest case, while lu = z-'^ gives two branch-points and one node, vi — £^'^ gives one branch-point and one node. The expla- nation is given in Fig. 29. If the case occur, we may change the direction of the tu-axis. It is sufficient to write, in the cyclic system in question, Zj -|- aw for 2, where a is arbitrary. §122. Example i. Let {k — azf + z' + {a- - azYz- =^ Q . The origin is a 3-element ; we wish to determine the expansions of w. Let w — az= vi\z, and remove the factor z^. The new equation is and the origin is a 2-element. Since z- is a factor of the terms of lowest degree, use a transformation of the second kind, z = z^Wi. Removing the factor ic^, n\ -f z^ + z^ic^^ = 0. The origin is now a 1-element, and u\ — P^i^i)- Therefore, reversing the series, «, = AO«i"-), and z = z,ii', = P,{ic,'''). Reversing again, U' V2 = p, (2V3) ^ „,j = p^ (^1/3) , „• _ az = w,Z = P, (zV«) , and finally, w = oz + PsCz'^'). The singularity at the origin is determined by /3 = 2, 8 = 4; therefore the penultimate form consists of four nodes and two vanishing loops, and the singularity is equivalent to two nodes and two cusps. Since the equation is of degree 4 in w, and since only three expansions have been found, there is one root which becomes infinite when z = 0. To obtain its expansion put w = 1/u. Then (1 - aZuYu + 2-(l - aZllY + Z'u* = 0. The origin is a 1-element, and u = Piiz). 146 ALGEBEAIC FUNCTIONS. Therefore 1/w = PjCz), and iv = -PoC^)- The following two examples have been suj^plied by Professor Charlotte A. Scott of Bryn Mawr College. For explanation of the method by which the penultimate forms are inferred, refer- ence should be made to Professor Scott's paper in the American Journal, t. xiv. Fig. 31 Example ii. Instead of w, z use y, x, and let 2/^ — Zx*y + 2a;" — x'^y — 23? + ^x'y = 0; we seek the expansions at the origin, which is a 3-element. Write y = 2/iXi, X = Xi. The new equation is 2/i' - 3 xcy^ + 2 aji^ _ x^^y^ _ 2 x/ + 9 x/2/1 = :0. The origin is a 3-element. First consider the simple factor 2/1 + 2 Xj. Write 2/i-t-2xi = X22/2, Xj^^x.^. Then yi = X2{y2 — 2), and the new equation is 2/2(3 - y^y - x„% - 9x/(2 - 2/,) = 0, or 92/2 -18 x/- :0. Therefore y = y^x^ = x/(2/2 — 2) = x}(^—2-\-2x}-\- higher integral powers of x^ . Next consider the repeated factor y-^ — x^. Let 2/1 — Xi = X02/2, Xi = Xj ; then 2/1 = X2(l + 2/2)- I'lie resulting equation ALGEBRAIC FUNCTIONS. 147 has for its lowest terms o{y.f — x.r), and the expansions associated ■with the two linear factors y.^ — x.,, y., + x., are integral in x^. Thus )/=?/i.Ti = .r/(l + ?/,) = x.rd ± a\, + higher integral powers). Therefore the expansions are ?/ = ar + .r'+ ••-, 1/ = ar — af + ... , y = —2ar + 2:if -\ . Here /3 = 0, 8 = 7, and the origin can be resolved into seven nodes. Fig. 31 shows the ultimate and penultimate forms. Example iii. Show that the branches of y, given by y^-3x*y + 2af + dx'y = 0, are y = xr + 3iaf '-•••, y = x^-3ix"'—, y = —23r+2x^'-: The ultimate and penultimate forms are given ia Fig. 32. Netftoii's Parallelogram. § 123. We shall next describe briefly a method of determining the expansions at a point which is historically interesting as emanating from Newton. In the hands of Fuiseux it proved to be a potent instrument for the advancement of Cauchy's Theory of Functions. [For Puiseux's important memoirs, see Liouville, ser. 1, tt. xv., xvi., or Briot and Bouquet, Fonctions Elliptiques, 2d edition.] Let us take, as a typical expansion, IV -b = Ci{z - a)'.'-- + C2(z - «)'="• + Ci{z - a)h'' + in which one, at least, of the integers gi, q^, •■• is prime to r, and 9i< 5'2<5'3"'- I'St qi/r, q^Jr, etc., when reduced to their lowest 148 ALGEBRAIC FUNCTIONS. terms, be ju,i, jni + /i2, ^i + /X2 + /H3, •••. The expansion for w — b is obtainable from the series of equations tv-b ={z-a)>'i, whose Cartesian co-ordinates are /, g. Let the points nearest to the origin on og and o/be fc and I. Assume w = CjZ"!, and insert this value of ?« in the equation "^nf^z'tc^ = 0. The order of the term Of^zhif in 2 is y^-^g +/ Now that line through p, which makes with og an angle tan"' /lii, intercepts on of a length t^i9+f- Therefore all terms of the same order in z lie on a line. In any approximation we keep the terms of lowest order only, and of course there must be at least two such terms. Hence iv = CiZ'h will not be a first approximation for tv unless it makes at least two terms give the same minimum intercept on of. To find the admissible values of ^1, suppose that pegs are placed at all the points p, and that a stiing fog is pulled in the directions of, og, until it is stopped by some of the pegs. The string has, between I and k, the form of a polygon, convex with regard to 0, each side of which begins and ends at a peg, and may be in con- tact with intermediate pegs. (1) The number of these sides is the number of values which /x, can take. (2) The value of /i, for each side is the tangent of the angle made by that side with og. (3) The number of terms to be retained in the approximations which correspond to one of the sides is the number of pegs in contact with the side. 9>. Fig. 33 ALGEBRAIC FUNCTIONS. 149 (4) The polygon begins at I and ends at k. Let the pegs in contact with a side be p^, o- = 1, 2, •••, p. Let u- = CiZ'i for this side. We must have and therefore Ml =(/!-/;)/( 0, the terms of lowest order that survive are Hence 2 + 2/^0 = 5/2, /u,2=l/4, 4c,c/=l; and to the second approximation :C,8 ,1/2 + -i;=^'^ (c.^=l). 2v'ci The four branches are approximately XV,= 2^I'^\Z^I\ XK., = -^-l'-\i^l\ 11-3=2-/^-123/4, io,= -^-i''^\ii^i'; the exact values are integral series in g'/^ Example 2. To see how equal roots of the Ci-equation introduce a new radical, take the general equation of the form just considered, «04W* + ai.jzw- + a.jjZ- + a^^zhv + a4o«* = 0, and first substitute u- = Ciz'/-. This gives, if {c{) + a,,z'r- + a^z'^0. ALGEBRAIC FUNCTIONS. 151 Xow substitute, for Ci, c, + cz'^'^. Then <^(Ci)+0, defines uniquely the value of w at z'", so long as z'" lies within the circle of convergence, and a fortiori so long as | z'" — z'"' |< p*"*, where p'"> is the distance from z'°' to the nearest critical point. This value of w is the value of the selected branch at z'". In the same way the value at z'^* is deter- mined uniquely so long as \ z'^' — z'" | < p<", where p'" is the distance from z<" to the critical point nearest to it. No ambiguity arises so long as the path of z lies wholly within the chain of circles whose radii are p"", p<", •••. See § 95. The preceding argument does not apply to the case in which z passes through a pole or a branch-point, when the pole or branch- point affects the branch which we are considering. At a pole the indetermination can be readily removed by the use of the series in 1/w. But at a branch-point a there is real indetermination. If /i 152 ALGEBRAIC FUNCTIONS. branches permute at a, when z passes through a, any one of these branches can, without breach of continuity, move in any one of /x directions. To avoid this case, the 2-path must be so chosen as not to pass through a brancli-poiut. Wlien the initial value of a branch is selected, the final value is now fully determined, for the particular path. § 127. Theorem. If a contour C include no critical point of the function, and if, starting from a point «'"* with the value «,■«'"' of it\, z describe the contour, the final value of to„ when z returns to z*% is w,'"*. Since z'"' is not a critical point, there exists an integral series w,-w,«" = P,(z-3('')), (r>0), which within its circle of convergence defines w, as a one-valued continuous function of 2. Let a circle smaller than but concentric with the circle of convergence meet C at the points 2*", «'^*. The branch w, is one-valued along the contour z'^z'^a'-'g'"* ; therefore the theorem holds for this con- tour. Let a circle, with centre 2''* and less than the circle of convergence whose centre is 2"', meet the con- tour 2'"^3(-> at 2<3' and z'« ; then the theorem is true for the contour 2<"2^'z<""2<'*. f^'g-36 It is therefore true for 2(0)^(1)^,4,^,1)2(3.2(412(2)^(0)^ that is, for the contour z'^^V^^z^^iz^zOK Proceeding in this way with circles whose centres are z<-', 2<% •■• , we prove the theorem for the contour C. Corollary. The paths 2""2")^ and 2""2'2'^ lead to the same final value of M,, the initial values being the same ; that is, a path may be deformed in any manner provided it does not pass over a critical point. It has appeared in § 46 and § 48 that a closed path which includes a branch-point may lead to a final value of w different from the selected initial value ; therefore also two paths from z'"' to ^ may lead to different values of w. As infinitely many paths lead from z*"' to ^, it is a matter of importance to reduce these to combinations of certain standard paths. This can be effected by means of the preceding theorem. ALGEBEAIC FUNCTIONS. 153 Let a small circle (c) be described with a critical point c as centre. Let i be a line from any point z'"', which is not a critical point, to the circle. The path from zC' along L to the circle, round A loop is sup- It is positive or the circle, and back to 2*°' along L, is called a loop posed neither to cross itself nor any other loop, negative according as (c) is described positively or negar tively ; but it will be supposed to be positive unless the con- trary is stated. If a contour C contain several critical points Ci,C2, •■•,ca, the path C which begins and ends at a point 2,„, of the con- tour can be contracted, without passing over any critical point, into X loops from z"" to the points Ci, c,, •••,ca; therefore by the preceding theorem the path C has the same effect on the branches of w as the successive X loops. A loop to an infinity which is not a branch-point will reproduce the initial value. For near such a point, say c, we have ^=P,(z-c), r>0, or w^ = Pa{z — c)/z% so that, when 2 describes the circle (c), w, does not change its value. Accordingly, in the present discussion, the loops to infinities which are not branch-points may be disregarded. The nature of the changes produced by a loop to a branch-point a is to be decided from the expansions for w near that point. The selected value of lu at z'"' leads by a given path to a definite value near a. If this value belong to one of the cyclic systems near a, it is changed by the description of the circle (a) into another, of the same system, which leads to a definite new value at z'"'. If r be the least common denominator of the exponents which enter into the system of expansions, r descriptions of the loop will restore the initial value at z*"'. The sphere-representation (§ 27) shows that the effect of the description of a contour C may be determined as well from the external branch-points as from those inside ; if the contour be flescrihed nositivp.1v. the Innn s to the external branch -points must 154 ALGEBKAIC FUNCTIO'S. be negative. In particular, a contour which encloses all the branch- points in the finite part of the jjlane will reproduce the initial value of a branch, or not, according as the point z= x is not, or is, a branch-point. § 128. The prepared equation. It has been stated that the theory of quadratic transformations enables us to replace any given algebraic equation by another in which the multiple points have no coincident tangents. Further, if we project to infinity a line i = which is not a tangent, by means of the linear transformation ic' = ic/t, z' = z/t, we have an equation which has no contact with the line sc . Lastly, if n be the order of this equation, a suitable change of axes yields an equation in which the term it" occurs, and in which there are, parallel to the lu-axis, only ordinary tangents with finite points of contact. The branch- points are now all simple and at a finite distance. "When an equation is brought to such a form it will be said to be prepared. The great advantage of such a form is that at a branch- point for which z=a, the expansions are all integral in z — a, except two which are integral in (z — a)'/-. Since there are no cusps, and since from the point at oo on the to-axis the tangents are all vertical, p, the number of branch-points, = the number of vertical tangents = the class of the curve. From Pliicker's equations (Salmon, Higher Plane Curves, p. 65), if 8 = the number of nodes and p = the deficiency, we have y8 = H(»i-l)-2S, 2; = l(ft-l)(n-2)-8, and therefore l3=2p-\-2{n — l). § 129. Deformation of loops. Let F(%v, 2) = be an irreducible prepared equation, and let /J loops be drawn from z'"' to all the branch-points. Since all the branch-points are simple, if a loop lead from an initial value w, to a definite final value w,-, the initial value uv leads to the final value 10, ; and any other initial value is not aifected by the loop. The negative description of a loop has the same eifect as the positive description. Beginning with an arbitrary loop, we assign to the branch-points the names ai, a„, ■■•,a^ in the order in which their loops are met by a small circle described positively round z(">. We denote the corre-. sponding loops by A^, A^, ■■■, A^, and say that A is in advance of A^ when X < ^. ALGEBRAIC FU^'CTIOXS. 155 A loop A^ to a branch-point a^ can evidently be deformed so long as it does not pass over a branch-point. On the other hand, let two loops A,^, A^' to the same branch-point a^ enclose a single branch-point a^+j. In Fig. 38, the loop .-1^' can be deformed into the broken path without crossing a branch-point, and the broken path is equivalent to the loops ^-l^+„ ^1^, A^^^i in succession.* Let A^ and ^^^i permute the branches u\, u\' and ic„ tv^: (1) If I, i' be diiferent from k, k' the new loop A^' still permutes iL\ and M\'. For, starting with u\, A^^^ leaves it unaltered, A/, changes it to )(•.', A/^+i leaves il\' unaltered. (2) If I, t' be the same as /' -h a.i{z — 0)"''' + a3(« - c)'3/'- + ..., 160 INTEGRATION. 161 where q^, q2,---,r are finite integers and r is positive, the branch-point is called algebraic. The point is an infinity of the r branches when gi is negative. In the expansions at oc , z — c must be replaced by 1/z. Definitions. If a function be one-valued and continuous, and possess a derivative, at all points within a region r, it is said to be holomorphic within V. Such a function can have no singular points, of either kind, within r. If a function be holomorphic within r, except at certain poles, it is said to be meromorpMc within r. For example, sin z is holomorphic throughout the finite part of the plane ; tan z is raeromorphic throughout the finite part of the plane, the poles being the zeros of cos z. The words "holomorphic" and "meromorphic,'' introduced by Briot and Bouquet, are intended to suggest that the functions behave respectively like the integral polynomial and the rational fraction. Halphen proposed that the terms be replaced by " integral " and "fractional." Cauchy's term "synectic" is synonymous with "holomorphic." § 132. Curvilinear integrals. The properties of integrals of a real variable, taken over a curved path, are required as preliminary to the study of the inte- gral of a function of z. For our immediate purpose they may be stated very briefly : — Let U(x, y) be a continuous function of. two real variables, and let 2/ be a path defined by the equations x= .2{t). We suppose that (^i(O) "^^(O ^""^ continuous functions of the real variable t, such that t increases constantly from to to T, as the point {x, y) describes the path L from «„ to Z. For instance, t may be the length of the arc of the curve. While it is supposed that L is an unbroken path from ;„ to Z, it is not assumed that the various portions of the path are represented by arcs of one and the same curve. That is to say, while i{t), «{t) are to be continuous along L, they need not be represented throughout the path by one and the same pair of arithmetic expressions. We may assume, accordingly, that <^i'(Oi (t>2{t) are subject to a finite number of discontinuities of finite amount. Suppose (a-o, .Vo) and (X, T) to be the terminal points Zq, Z, and let {xi, yi), (xj, y,), -, {x^i, Vn-i) be intermediate points on the 162 INTEGRATIOK. path L, while t^, U, ••-, t„_i are the corresponding values of t. Let us construct the sum Ua,, r,,) {X, - x„) + U{i,, 7,,) (a;, -x,)+ ... + U{t„ r,„) (X-x„_,), ■where (li, r/i), •••, (4„ 77,,) are points on the n arcs which connect the points (,r„, y„), {x^, y{), ■••, (X, Y), and tj, r,, •••, t„ are the corresponding values of t. When tlie number 71 is increased, and each arc diminished indefinitely, this sum tends to a definite limit, whatever be the mode of interpolation of the points (x„ y,) and (^«, 77,<). This limit is the same as that of or that of ?7J0i(ri), ^■2{r,)\\^,\t,) + t\{U ~t,)+ ..., where | e | vanishes with | <; — f,, j . This limit, by the properties of ordinary definite integrals of a single real variable t, = rV(0i, ,)^dt This is the meaning of the curvilinear integral | Udx. Similarly, if F(x, y) be a continuous function of x, y, the meaning to be attached to | Vdy is | F(i, .,).,'dt; and generally by lUdVis XT f^Y U — dt, where U, V are expressed as functions of t by means of the equations x= i(t), y= 4>-2{t). The conception of the definite integral of a one-valued function of a real variable, as the limit of the sum of infinitely many ele- ments, is capable of extension to functions of 2. Let f(z) be a function which is one-valued and continuous along a path L, of finite length I, which leads from 2:^ to Z; and let Zi, z.^,..., 2„_i be intermediate points on L. Further, let ^1, ^o, ■••, ^„ be points situated on the arcs (2o, «i), (^i, 2..), (^2, 83), •••, (2»-i, Z). We shall prove that the sum 2/(^«) (««+!-««) =/(^i) (2i-2o) +/(^.) (22 - 2i) + •••+/(« (Z-z„_,) tends towards a perfectly determinate limit, when the number n increases indefinitely and each interval z^_^^ — 2^ tends to zero. This limit will be called the integral of f{z) with regard to the path L from 2o to Z, and will be written | /(2)rf2. That the absolute value INTEGEATIOK. 163 of the integral is finite is evident from the consideration that for all values of n, and all jiositions of z^, z.j, ••■, z„_,, ^i, ^n, •■•, ^„, 1 2/(^«) (2«+i - z«) ^M\z,-z,+'z,-z,\ + --- + 'Z- z„_, I , where M is the greatest absolute value of f{z) upon the path L. AYhen n tends to x, the coefl&cient of 3/ becomes the length I of the path L. This proves that the absolute value of the definite inte- gral is finite. To show that the sum tends to a litnit, let us write u + IV for /(z) and x + ii/ for z. Thus lini 2/(^,) {z «^i — z,) = lim 2 [hC^„ r),) + i'r(^„ 7;«)](a;,+i — x\ + i'!/,+i — y,) = I (iidx — w?^y) + i 1 {udy + vdx), an expression whose value does not depend upon the mode of interpolation of Z;, z.j, •••, z„_„ ^,, •••, ^„. It is very important to notice that the limit has been proved to exist, for a given path between Zq and Z, on the two conditions that /(z) is one-valued and continuous along that path. If /(z) be a many-valued function, one of its values must be selected at z^, and then the succession of values along the path from Zq to Z is com- pletely determined by the principle of continuity, provided the path does not traverse a singular point at which the selected value ^becomes discontinuous or equal to another value. It follows from the definition that, for one and the same path, £f{zY\z = -^y(z)dz, since the element, (z,+i- z,)/(^k), of the first integral is replaced by (z, _ 2^^j)/(^;t) in the second. As the integral | /(z)c?z is reducible to real curvilinear integrals, the ordinary processes such as inte- gration by parts, change of the independent variable, and so on, are still applicable. When z is changed to a new variable z', the path L in the z-idane must, of course, be replaced by the corresponding path V in the z'-plane ; z' must be so chosen as to be one-valued and continuous along the path L'. § 133. The problem which arises is to find the effect of a change of path from z^ to Z upon the value of J f{z)dz. Cauchy discovered, in this connexion, a theorem which is of vital impor- tance in the Theory of Functions. 164 IXTEGRATION. Cauchy's Theorem. If /(z) and its derivative f'(z) be one-valued and continuous on tlie boundary of and within a region T, the integral Cf{z)dz taken in the positive direction over the complete boundary C is equal to 0. For simplicity we shall postpone, for the present, the examina- tion of the case in which the bounding contour is complex. Many proofs have been given of this fundamental theorem in the Theory of Functions. The one which we shall give is due to Goursat (Demonstration du Theorfeme de Cauchy. Acta Math., t. iv., p. 197) . This proof is instructive inasmuch as it makes direct use of the property of monogenic functions that the value of the derivative of f{z) at Zi is independent of the path along which z approaches Zj.* Let the region T, with a simple contour, be divided by two systems of equidistant lines, parallel to two rectangular axes, into a network of small regions. The interior regions are squares of area Zi^, while exterior regions are portions of squares. The integral Fig. 39 C-f(z)dz is equal to' the sum of the integrals taken positively over the rim of each region, for the figure shows that each line other than C is traversed twice, in opposite directions, with the same value of /(z). We shall first examine the value of an integral taken round a complete square, such as pqrs, whose contour may be denoted for shortness by C If Zi be any point inside Ci, we have, for each point of a, "' z~z^ " where | e J varies along C„ acquiring a maximum value e^. * "We reserve for Chapter VI. a proof by Riemann of the generalized theorem which relates to multiply connected surfaces. Rieraann's proof has justly become classical, but it is open to the objection that it employe double integrals in the proof of an elementary theorem. rSTEGUATIOX. 165 Hence I f{z)dz= | e{z — z,)dz, because i dz is zero from the definition of an integral, and if z- = t, Czdz = i Cdt = 0. Thus the absolute value of j f{z)dz 'jvJU^^rJi »^ *- *' ^ r \£(2 - z,)dz < £. • h^2 ■ 4 h t/Ci < 4-Y'2ec(area of square) (A). "We next examine the value of j f{z)dz, where D, is the contour of the incomplete square klmn. As before, let z, be a point inside Dt, then I f{z)dz = j c(2 — 2c)c?2, but I Ciiz - z.yiz I < £./iV2 f i rf2 I < £i7iV2 1 perimeter of complete square + arc A:Zj < £,V-;{-4 • area of complete square { + c./iVii • (length of arc W) (B). The result of adding up all the right-hand sides of the set of inequalities A and B is an arbitrarily small quantity, for the sum is < fj,V2\4: sum of all the complete squares C„ D, + (the length of C)h\, when ju, is the greatest of the quantities c,. But, if ^ become arbi- trarily small with h, this expression vanishes in the limit. Hence ■£f{z)dz=0. To complete the proof, it must be shown that a quantity h can always be found such that ;ix 1, | dz/{z — c)" = 0, though lim (z - c)f{z) ^ 0. This instance shows that the condition of the preceding paragraph, though sufficient for the vanishing of j/(z)dz, is not necessary. More generally | y'(z)dz = 0, where C is any contour lying in a ' /" dz region within which g{z) is holomorphic. If t taken from J z — c any initial point o along a given path to a point b, be equal to I, we can, by making z turn n times round c, give to the integral at b the value I+2nTri; that is, the integral has at b infinitely many values all congruent to the modulus 27rt. If, however, we draw any line from c to oo and re- gard this as a barrier which z must not cross, the value (^ 1 2_ ""of the integral is fully deter- ^^" mined when the value at any point is given. For instance let c = 1, and let the barrier lie along the real axis. Let the value of the integral at 2^. be ; then the = log I 2 — 1 1 + i^o, log denoting the 2^.3 — 1 real logarithm, and 6o the acute angle 21:. Compare § 106. § 138. An example of Cauchy's treatment of the integral of a many-valued function. Though the subject of the integral of a many-valued function will be treated in detail later by the help of Riemann surfaces, it seems advisable to give an illustration of the way in which, when by means of loops such a function has been rendered one- valued, Cauchy's theorem may be used. INTEGRATION. 171 Let us consider a branch of the two-valued function defined by w-= 1/(1 — 2-). The points ± 1 are both branch-points and infini- ties. Let iL'i be that branch which = 1, when z = 0. As 2 moves along Oj^ in the direction of the arrow, lu^ remains posi- tive. After the description of the small circle ^j^c, its value at ?• is — 1/ Vl — 2-j, and along ?-0 the value is different from that along Op. "Within the region bounded by C and JW the branch il\ is not holomorphic, and we are not Justified in saying that Ju\dz= I it\dz. "Within the region bounded by the simple rim ddbOrqpO, u\ is holo- morphic ; for Or must now be regarded as distinct from Qp, and a complete circuit round 1 is prevented by the barrier L. Hence 7(0, a, 6, 0) = 7(0,;;, g, r, 0) flz Jo Vl-2- Jo Vl — 2- since and lim Vl = 7r + = '-« clv s. vr "<:>' dz Vl - 2- in the limit, owing to the fact that lim (2 - 1) . along pqr = Vl-; = 0. More generally, if Cj, c,, •••, c„ be those branch-points of f(z) which are situated within a closed curve C, and if there be no crit- ical point in the region T contained between C and the small circles round c,, Ca, •••, c„, a branch of the function f(z) need not be one- valued in the region T, but must be one-valued within the region bounded by the single rim, composed of C and the n loops from a 172 INTEGKATIOK. point of C to the branch-points. But, if m poles d^, d^, •••,(?„ be contained within T, ( f{z)dz — [the sum of the integrals taken posi- tively over the n loops to the branch-points] = 2 I f(z)dz. It is interesting to observe the way in which Cauchy's theorem accounts for all the values of the integral ( '^^ ■ Any path from to z can be contracted, without passing over any singular point of 1/Vl — Z-, into a path formed by single or repeated descriptions of loops from to 1, —1 and of a definitely chosen line from to z, as for instance the join of the two points. Let u, — u be the values of the integral along this latter line when the initial values are w'l, — ztj. A single positive description of the loop to 1 changes il\ into — Wj, and the corresponding value of the integral j u^dz is ir — m. A de- scription of the loop to 1, followed by a description of the loop to —1, is equivalent to a description of the first loop with the value il\ and of the second with the value — w^, the final value being Wj, and there- fore the corresponding path from to « with initial value Wi leads to the value 2ir + ?* of the integral. By a consideration of all possible combinations of the loops, it is easy to see that I c?z/Vl — z- has infinitely many values, which are all included in the formulae u + 2 n-ir, TT — u + 2 mr, where n is any integer, positive or negative. The resulting theorems in Trigonometry are ( sin (it -|- 2 nw) = sin u, i sin {it — u + 2 »i7r) = sin u. A similar process of reasoning would establish the double perio- dicity of the function z = sum, defined by the equation n dz_ -'» Vl-z-'-l- VI but we shall consider the elliptic functions (chapter vii.) when the general theory of periodicity has been given in the chapter on Eiemann surfaces. We refer the reader who wishes for further information on Cauchy's Theory of Periodicity to the great trea- tise of Briot and Bouquet, Theorie des Functions Elliptiques ; to Clebsch and Gordan, Abelsche Functionen, ch. iv. ; and to Jordan's Cours d' Analyse, t. ii. INTEGKATION. 173 § 139. A second theorem of Cauchy's. Let g(z) be holomorphic within and on the boundary of a region r, bounded by a simple contour C, and let t be any point within the region. The function g{z)/{z — t) becomes infinite when z = t. The function g{z)/{z — t) is holomorphic in the region bounded by C and (0. Hence rg_(z)dz _ f m^z rg(z)dz ^ r g{z)c Jc Z — t J^t) Z—l Now, since g{z) has a derivative within r, lgiz) — g{t)\/{z — t) has a finite and determinate limit when z = t, and therefore is a holomorphic function throughout r. Cauchy's theorem gives r g(^)-.9(0cfe=o, Jit) z — t Cauchy's integral for g(t) is I ^-^ — . 'JiriJc z — t From this result can be deduced expressions for the successive derivatives. By subtraction, and after division by At, there results from the two formulae ^^ !iTdJcz-t ^ 2iriJcz-t-M the third formula g'{t)=^C fi^,. 2iTtJc{z — t)- But, everywhere on C, z — t differs from 0; therefore g(t) has a single first derivative, and this derivative is holomorphic throughout r. g'{t) has, in turn, a holomorphic first derivative g"{t), and so on. Hence when a function is holomorphic in the region T, all its derivatives are holomorphic in the same region. § 140. Taylor's theorem. Let g{z) be holomorphic in and on a circle C; and let c be the centre of the circle, c + t any inside point. We have -2^X'«*{.-^ + -^+- + t_ c {z-cy ' ' («-c)"+i + (2-c)»+'(«-C-0 ) ' = ^(c) + <\z\), can be made as great as we please. That is, the absolute value of a constant coefficient is less than a quantity which can be made arbitrarily small. Such INTEGRATION. 175 a constant coefficient must be zero. In this way it can be proved that if G be finite, all the terms, except the first, in Maclaurin's expansion for g{z) vanish. This important theorem (clue to Liou- ville), shows that if a function (j{z) be holomorphic throughout the finite part of the plane, and also at cc, it must be constant. A function g{z), which is holomorphic over the finite part of the plane, is of course no exception. Its property is that it has no infinity at a finite distance from 0. § 141. Laurent's Theorem. Jjet g(z) be holomorphic in the ring bounded bj" two concentric circles C, C", with centre c; and let ^^ c + t be any point in the ring. Laurent's theorem states thatg{c+t) can be expanded in a convergent +^ series of the form Sa^t"*. Describe — » a small circle C" with centre c + ^ Then g(c + t) = ^ C -SMi^, ^^ ' 2niJc-z-c-t or, since g(z) is holomorphic in the . shaded region, g^C + t)^J- C ^(^^--1- CJI(^ ^^ ' 2.TriJcz — c — t 2TTiJc"Z — c — where the integration is performed in the sense of the arrows. On the circle C',\z — c\>t, 1 ■ + + ■ Z — C — t Z~C (2— C)- (2— C) ;+' r (z-c)"+'(2-C-0 and — C iMl^- = ao + a,t + a.f+-+ a„r + R„, Z-n-iJc- Z — C — t where 1 r gj^yi^ p ^ 1 f r^'!7(2)rfz 2 7rtJ(7 (Z-C)'+'' " 2TrVJc(z-c)»+H2-C-f) Now it can be proved precisely as in § 140, that the limit of iJ„, when n = so, is zero. 176 INTEGRATION. Again, on the circle C", \t\>\z — c\, 1 ^1 z-c {z-cy (z-c)" (z-c)"^' c + t-z t f e r+i r+'(c + (-z)' and -L f X(?)^ = ''-'+%2 + ... + ^' + J?„, where a_, = — I (« — c)" 'a(z)(7«, ii„ = I --r-— ^ ^, 27riJc-- ^ J\ J ' 2TriJc- r+'(e + f-z) but |iJ„|{z„ z,)dz,dz, Sci».Sc2»> (2 TTl) -Jc, Jc, {z, - c,)"i+' (Zj - C2) ">+' ■ and This expression for ' ""^ shows that the order of differentia- 8ci"i8c2"2 tion is indifferent, and also that each partial derivative of (^(z,, z.^) is INTEGRATION. 179 holoniorphic throughout the regions F,, r,. By an easy extension of the method of proof used in finding Taylor's expansion, we have {2 TTt)-Jc\Jc^ («i — Ci — tj) I {Z.2 — C, — fj) § 144. Some general theorems on holomorphic and meromorphic functions. A function g[z) which is holomorphic in a region T and con- stant along a curve of finite length, lying in the region, is constant throughout r, however small the length may be. This follows at once from § 86 and § 140. Hence it results that two functions which are holomorphic in T and are equal for infinitely many points of T, are equal throughout T. This theorem marks off Cauchy's monogenic function (Weierstrass's analytic function) from tlie function used by Dirichlet. In the historical account of the growth of the present views on the nature of functions, we showed that, adopt- ing Dirichlet's definition, no connexion need exist between the values of a function for different values of the variable. The monogenic function u + iv of z is not so general, for it is fettered by the differential equations 5i( _ , 5p Su _ _5v , Sx ~ dy' dy Sx ' but the reader must not suppose that these restrictions interfere with the use- fulness of the Theory of Functions. Without them u -|- iv would cease to possess a definite differential quotient, and fundamental results, such as the expansion by Taylor's theorem, Cauchy's theorem, etc., would fall to the ground. By § 87, a holomorphic function g{z), which vanishes when z = c and is not identically zero, is, near c, of the form P,„(z — c) ; that is, it is equal to (« — c)'"x (a holomorphic function which does not vanish at c). The integer m is called the order of multiplicity, or simply the order, of the zero c. Differentiation shows that g'iz)=P„^,iz-c), g"iz)=P„ ^{z-c), and so on. Since a region containing c can be assigned whose area is not infinitely small, within which there is no zero of g(z)/{z — c)"", it follows that the number of zeros of g{z), which lie in T, is finite if r lie in the finite part of the plane. 180 INTEGKATION. Next let us consider a function h(z), meromorphic in r. Let c be a pole ; then by definition there is a positive integer m, such that {z — c)'"/i(z) is holomorphic near c; therefore h{z) = {z-c)-~Po{z-c), and l/h{z) = {z-c)'"Po'{z — c), and c is a zero of l/h{z), of the order m. Since there is a region containing c, of area not infinitely small, within which Po{z — c) is not zero and is not oo, it follows that the number of poles of h{z) which lie in F is finite, if T lie in the finite part of the plane, and that a zero and a pole cannot be infinitely near. At a pole C of order m. the function Ji{z) = {z-c)-''PJz-c)= "^' + ^ — ,+- + ^^ + P(z-c). The importance of this expression lies in the division of /t (z) into a purely algebraic part (z-c) which accounts for the polar discontinuity, and into a function holo- morphic at c. § 145. Theorem. A function 7^(z) which has, within a region r in the finite part of the plane, a finite number of poles, can be ex- pressed as the sum of a rational fraction and a function holomorphic throughout F. For let ^(z) have, within F, v poles Ci, C2, •••,c^ of orders wii, m2, ••• , «i„. In the neighbourhood of Cj, where Ai(«) is not infinite at Cj, but is infinite at Cj, C3, •••, c„ with orders mj, wij, ••• . As before, it can be proved that (2-0,)"= (Z-C2)'"2-1 ^2_C2 ^^ " where 7^2 (z) is not infinite at c„ Cj, but is infinite at Cg, C4,---, c^. Proceeding in this way, we arrive finally at a function h^(J) which is holomorphic throughout F. Hence INTEGKATION. 181 h(z) = ^ + ^ 4-... + .^". (2 — Cj)"! (2 — Ci)""'"' Z — Ci (2 - C^) "-2 ^ (2 _ c) "'2-' 2 - e, + a function which is holomorpliic throughout r. Theorem. If g{z) be holomorphic throughout the finite part of the plane, with a jjole at oo of order m, it must be an integral polynomial of degree m in z. For the one-valued function g(—] is holomorphic everywhere except at the origin, where it has a pole of order m. It must be of the form ^ + 9j!i^ + ... + ^ + ^(z) ; Z'" 2""^ 2 but <^(2) has no singularity in the plane, and reduces to a constant etc (§ 140) ; hence g{z) = ao + a^z + a.^^ H \-a„z~. Theorem. If a function h{z) have no infinities in the plane other than a finite number of poles, it must be a rational fraction. For h{z) = a rational fraction + a function which is holomorphic throughout the plane = a rational fraction 4- a constant. Theorem. Two meromorphic functions h(z) and hi{z), which have the same zeros and the same infinities, with the same or- ders of multiplicity, are in a constant ratio to each other. For hi{z)/h(z) can be infinite at no places other than the infinities of the numerator and the zeros of the denominator ; but, by hypothesis, each infinity of the numerator is neutralized by an infinity of the denominator, and each zero of the denominator by a zero of the numerator. Hence hi{z)/h{z) is holomorphic throughout the plane, and must be a constant. Residues. § 146. Let c be an isolated singular point of a one-valued function f(z) ; the function can, as has been said, be expanded in positive and negative powers of 2 — c, when z is near c, by means of Laurent's theorem. If we integrate round c, the integral of each term is zero, 182 INTEGRATION. excepting the term in Let the coefficient of this term be a ; 2 — c therefore \f{z)dz = 2 ■ma. The coefficient a is called (after Cauchy) the residue of /(«) at the point c. The integral being taken in the positive direction with regard to c, we have in the case when c is oc f{z)dz = -2TTia', where a' is the coefficient of - in the expansion at oo. There is z accordingly a residue at x when it is a regular point, while at other regular points of the function the residue is zero ; the reason being that the residue belongs to the integral rather than to the function. If lim {z — c)f(z) be finite, this limit is the residue at c, and if lim zf{z) be finite, this limit, with sign changed, is the residue at CO. If we cannot apply Laurent's theorem at a point c, as may happen when c belongs to a cluster of essential singularities, we may draw a contour C including the cluster, and define the residue as iirlJo It is evident, by differentiation, that the only infinities of h'^z), in r, are poles of orders ?Hi + 1, •■-, m„ + 1, at Cj, •••, Cy. Cauchy's theorem (§ 135) may now be stated as follows : If a function /(z) be holoniorphic in a region r, bounded by a simple contour C, except at isolated singular points Ci, c^, •■■, c^ and if the residues at these points be a^, a.,, ■■■, a„, then X f{z)dz = 2 TiSa,. C 1 For (f{z)dz =2 f f(z)dz, and j f{z)dz = 2Tria,. The theorem is true for both the regions into which C divides the plane, if the conditions apply to both. Therefore, for a func- INTEGEATIOX. 183 tion holomorphic over the whole plane, except at isolated singular points, the sum of the residues is zero ; the residue at infinity being included. Example. Verify that the sum of the residues is zero in the cases : — (1) ?^ , (2) e'"- ^ ^ (z-a)(2-6)(2-c) ^ ^ When h{z) has a pole of order m at c, g{z)D log h (z) has a residue — mg{c) ; g{z) being any function holomorphic in r, which does not vanish at c, and D standing for — dz For h{z) = {z-c)-''P„{z-c), g(z) = g{c) + P,{z-c), and therefore g{z)Dlogh{z) = - -^^^ +P(z-c). z — c Similarly at a zero c' of order m' the residue of g {z)D log h(z) is m.'g{c'). Therefore, if C be any simple contour lying in T, we have, by the preceding theorem, Cg{z)D log h{z) = 2 Trilm'g{c') - 2 ■7ri:img{c) , the summations being for all zeros and poles of h (z) which lie in C. The theorem is equally true if g{z) vanish at any of the points c or c'. Corollary. If /x' be the number of zeros, and /x the number of poles, within a region V bounded by a contour C, J- C D log h{z) = ^'-f., l-KlJc provided a zero of order m! is counted as m' zeros and a pole of order m as m poles. Example. Show from this formula that the number of zeros of the integral polynomial 2" + Oi?""' + a^"'^ -\ + a„ is n. Essential Singularities.* § 147. If ao + ttiZ + a.^- + ■■• be a series whose circle of conver- gence is infinitely great, the series ao + ai/z + a.,/z'' + a^jz" -\ * See § 99. 184 INTEGEATION. will represent a function which is holomorphic throughout the plane, except at the point 0. The singularity at this exceptional point is of an entirely diiierent nature from that at a pole, or, to state the matter in another way, the behaviour of Oo + tti/z + 02/2- + a^jz"' + ■ • • to 00, at z = 0, is entirely different from that of ao+ai/2 4 l-a„/2", at the same point. Consider the function e^i'. This function is holomorjjhic throughout the plane, the point z = excepted, and = 1 + 1/z + 1/2 ! z^^ + 1/3 ! z' + •• ■. If z be near the origin, we may put z = pe'', where p is small. The function e^'' becomes e'^'"^*/c x e'*''"*/p, and the absolute value of giA_gcos«/^_ The value of e'^"'*/'' depends upon the direction Q in which z approaches the origin. If cos ^ be negative, e'=°'^/'' = when p = 0; if cos 5 be positive, e™'*/i'=», when p = 0; if cos 6 be zero, «'='"'/.' = 1. ' There is also indetermiuation as regards the amplitude — sin 6/p. It is easy to show that e"'' takes an arbitrarily prescribed value a + ifi at infinitely many points within a circle, whose centre is at the origin, no matter how small the radius may be ; for the equation &'' = a+ij3 = ce"' gives 1/z = log c + i{v + 2 imr), where m is any integer ; or x+ iy = 1/{A + iB), where A = log c, B = v -\- 2m,Tr. Thus x = A/(A'+B'),y=-B/{A' + B'); that is, the absolute values of x and y can, by varying m, be made to take infinitely many values less than assigned positive quantities, however small. The argument fails when A=x; that is, when c = or c = CO. Next let us consider the function 1/sin (1/z). At a point z = l/wiTT, where m is an integer, the function has a polar infinity, for the reciprocal function sin 1/z vanishes. This is true however large the integer m may be ; therefore, however small be the radius of a circle, centre 0, it is always possible to find an integer p, such that for 7)1 = /x, and for all greater integers, l/mir is less than the radius of the circle. Thus 1/sin 1/z has infinitely many poles accumulated in the neighbourhood of z = 0, and the function cannot be expanded in the neighbourhood of z = 0. INTEGRATION. 185 The singularity of the many-valued function log z at « = is of a different nature. Within the circle | « | = p, the absolute value of log z > log p, and therefore log z cannot take all assigned values within the circle. The origin is here a branch-point at which infinitely many branches unite. § 148. A point c is a pole or ordinary singularity of the one- valued function, when we can transform the function by multiplying by a jjower of z — c, into one which presents no singularity at z = c\ otherwise z = c is an essential singularity. In the reciprocal of a one- valued function /(«), poles become zeros, but essential singularities remain essential singularities. When a function has infinitely many polar and essential singularities scattered over the whole plane, a complication may arise by the presence of infinite accumulations of these singularities at special points of the plane. This case has received treatment in some important memoirs (notably that of Mittag-Leffler, Acta Math., t. iv.), but the simpler case is that in which the places of discontinuity in fhe finite part of the plane are isolated. It is this latter case which we shall consider. Transcendental functions can be classified according to the num- ber of their essential singularities. Those with a finite number are more nearly allied to rational functions than those with an infinite number. After Weierstrass, we use the notation (?(z) to denote an integral transcendental function (see Ch. iii., § 99). We have used g{z) for functions holomorphic over the finite plane, or a region of it; while G{z) is reserved for functions holomorphic throughout the finite part of the plane. We shall understand by G( ) the v« — c a, , a. function defined by the series — ' — | 1 — to oo, where z — c {z — cy a^z + a^- -\ — defines an integral transcendental function. A theorem of Weierstrass^ s. A one-valued function f(z) which has only polar discontinuities in the finite part of the plane, and which has a single essential singularity at oc, can be represented as the quotient of two integral functions. The number of polar infinities may be infinitely great, but only a finite number of these are supposed to be scattered over the finite part of the plane. Let the poles be of orders wij, m^ ••• at points Cu Cj, •••, such that |ci|<|c2|<|c3| •••; i|c„| = oo. If a function Gi{z) could be constructed with zeros of orders m„ mj, ••• at Cu Cj, ■••, the functions Oi^{z) •/(«) would be finite for all 186 INTEGRATION. finite values of z, and would therefore be an integral function GJz), say. Hence /(z) = ^44- '-^^^.t the function G^{z) can always be constructed will be proved presently. In the special case when 2 — is a convergent series, Weierstrass has given the follow- er ing very simple proof : — Write, tentatively, G^(%) = (z - cO^K^ - c.o)'"^^ - Cs)"' - ; then ^L(iU5rl^''A»^ This product converges when n p~^^ j converges, where J»/is the greatest of the quantities m. This second product converges with jj/z-Ca\ ^^^^ . ^^,.^^ n^l+^^^) or with 2^^^^. But \z^ — cj V 2o — W Zo — Ca 2 ^ ~ ^" converges when 2— • — converges, and therefore 2o — Ca Ca 1 — 2o/Ca also when 2— converges; for, after a sufficient number of terms, 1 J 2o will always be less than some fixed finite number. § 149. Statement of the theorems of Wderstrass and Mittag-Leffler. I. Factors. This example suggests the problem : to represent as an infinite product any function which is holomorphic throughout the finite part of the plane, and has infinitely many zeros, of which only a finite number lie within any finite distance from the origin. This problem has been completely solved by Weierstrass, who has proved that such a holomorphic function can be expressed in the form where c„ Cj, Cg, ••• are the zeros of G{z) arranged in ascending order of absolute value, and G,(z) = z+|'+- + -. Z s INTEGRATION. 187 The simplest special case of Weierstrass's theorem is offered by the integral polynomial of degree n, which can be represented as the product of n simple factors. Another example, which will be proved later, is afforded by the formula sin7r2 = ttzU'Ii — -V" II. Partial fractions. We know that any algebraic fraction of the form ' ' ,"""*" — ± can be converted into partial frac- tions, and also that any meromorphic function h{z), with a finite number of poles at Cj, Cj, •••, c„ in a region r, = 2«iJ( |+a 1 \z — c,J function g{z) which is holomorphic throughout r, where R( \z — c represents an integral polynomial in l/(z — c). This theorem is capable of great extension. Mittag-Leffler has shown that a one- valued function can be constructed with infinitely many polar or essential singularities of assigned characters at places q, Cj, Cj, ■••, such that 1 Ci I < ' C2 < I C3 • ■ • , L'c„\ = x, and that the analytic expression for the function is In this theorem the number of discontinuities is infinite in the whole plane, but finite in the finite part of the plane. Weierstrass in a brilliant memoir (Zur Theorie der eindeutigen analytischen Functionen) proved the corresponding theorem for the special case of a finite number of singularities. It is to be noticed in AVeierstrass's and Mittag-Leffler's Theorems that, when c, is a pole, the series (?, ( ] ends after a finite num- ber of terms, whereas, when c, is an essential singularity, the series is infinite but convergent throughout every region of the plane which excludes the c,'s, and G^l ) expresses a transcendental function. \z — CkJ Further it must be understood that the points Ci, Cj, •••, in the finite part of the plane are at finite distances apart, so that 00 is the only limiting point of the infinite system Ci, c,, •••. • The conditions for convergence may or may not require that « sball be constant. Laguerre suggested a classification of integral functions baaed on this theorem. When s is constant he called G(z) a function of ' genre ' ». According to this definition sin s is of ' genre ' 1. Hermite, Coura d*AnaIysc. 188 INTEGRATION. Proof of Mittag-Leffler^s Theorem. § 150. We have to construct a one-valued function f{z) which has for its singular points Cj, c.j, c^, •••, and is discontinuous at these points like 6?/-^^ (k = 1, 2, 3, ••■)■ Expand gJ^^\ by \^ — c«. , , Maclauriu's theorem, in the form tto, , -I- ffli, «« + cu ^z' -\ 1- fly, ,2''-)- \ =/, (z) + \p, say, the circle of convergence extending as far as c,. For a given value of 2 we can, by taking p large enough, make the remainder X^, as small as we please. Let tj, £2, ••■ , t„,--- be positive quantities forming a convergent series, and let 2^ be taken so great that I gI^—\ -/, (2) j = j X, j < £„ where | 2 | < | c, |. \^ — c«. Consider now the series l\Gj-^\-f.{z)\. 1 \z — cj Whatever finite value | 2 I may have, there is a finite number of singular points c« which are less than z in absolute value, and an infinite number which are greater than 2 in absolute value. Suppose c„ to be the first singulkr point, such that ] 2 , < j c„ j. The first ?i — 1 terms of the series form a finite series with a finite sum. Removing these terms, we have a series whose terms are less than «« + fn+i H — . for all values of | 2 | < | c„ |, since, if | 2 | < [ c„ ', it must, a fortiori, < \ c„+i \, etc. The series <;„ + e„^i + £„+, H — being, by hypothesis, convergent, so is the series proving that 2 OJ )— /«(2:) is an unconditionally and uni- formly convergent series, when 2 is not situated at one of the points c. Now in a part of the plane which contains only the singular point c, we have assumed that/(2)= GJ )-f- a function which \2 — cJ IS holomorphic in the domain of c,. Hence, by subtracting the INTEGRATION. 189 function gJ ) from /(z), we render the point c» an ordinary \z — c2{z), •••■ Suppose that these branches can be represented by integral series in z, with radii of convergence pj, pj, ps, ■■■ such that then, by a suitable choice of the integral polynomials, /l(2),/2(»).-". the series 2(<^.(z)— /.(z)) can be made unconditionally and uniformly convergent throughout every part of the plane from which the singular points of 4>i{z), 4>2{z), ■■•, are excluded. Proof of Weierstrass' s Factor-Tlieorem. § 151. Let Ci, Cj, Cj, •••, differ from one another by finite amounts, and let |ci!^|Ca|<|c3|"-, i|c„| = cc; 190 INTEGRATION. also let G',(«) = lo^'( 1 V c« where that branch is selected which _ _ z ^' _ ^' _ ^~2^ SZ' ■■■' a series whose radius of convergence is | c, | . Then, by Mittag- Lef&er's Theorem, X(z)=i Jlog(^l-^j-/.(z)l can be made to converge unconditionally and uniformly throughout every part of the plane which contains none of the points c,, CjjCs, •••. This series will represent a branch of a many-valued function of z, for when z describes positively a closed contour round several of the points c„ each of the corresponding terms increases by 27rt. As the various branches of xi^) differ merely by multiples of 27ri, e<''> must be one-valued. Hence G(z) = e'<(') = n f 1 - - le-^.w is a one-valued integral transcendental function, which vanishes exclusively at the points c,. Cj, •••. In this way we have formed a product which is unconditionally and uniformly convergent, and whose factors (l )e"^<<'>, called by Weierstrass 'primary factors,' vanish at the assigned points Cj, c,, ••■ [Casorati, loc. cit.J. To complete the discussion let us consider a mode of construction of the functions /.(z), (k = 1, 2, •••). The problem is to determine integral polynomials /,(z) such that 2)log(?(z) = 2|--i--/,(z)} shall be unconditionally and uniformly convergent. We have r^ = -A.(^)+ ,/' . , 2— c« c^'iz-c^) where /.,.(z) = 1 + A + ^ + ... + ^. c* c« c^ c< Thus 2J-J-+/.,.(^)[=-S / . IXTEGRATION. 191 and the problem will be solved if this latter series be convergent. This will be the case for a, fixed value of s, if 2— ^^ be absolutely con- vergent. For many distributions of the points c, it is possible to find a fixed value for s which will make the series absolutely con- vergent. That there are distributions for which s must depend upon K, is shown by the series 1 + 1 .... (log 3)'+i (log 4)'-i ' which is divergent, however great s may be. Let us, then, suppose s a function of « ; as, for instance, s = k — 1. The series of abso- lute values is now « I yic-l I 2 , 1 |f/-'(z-c.)| and the ratio of the {k-\- l)th to the xth term is which is evidently zero when k = x . Thus the series is conver- gent. That the convergence of 2 — -; is uniform through- 1 c,«-'(2 — c) out every part of the plane, which excludes the points c, is evident from the consideration that, for each finite value of z, the series can be decomposed into two parts, 1 c««-'(3— c,)' n-lC," '(z — c,)' such that n is finite and z < ! c„+i |. For all values of z within a region of the z-plane, which contains none of the points c„ the lat- ter series is convergent and expressible as an integral series, and the former series consists merely of a finite number of terms. Kow that a satisfactory form has been found for /,,,(2), namely, 1 2 2- 2*— 2 - + — + — H +- — . C^ Cic Ck C^ we write Dlog(?(2)- 2 } -i- +A,,{z) \ = G,'{z), where Gf^{z) is holomorphic throughout the finite part of the plane. Multiply by dz and integrate ; the resulting equation is where Go(0) = 0. Cl 1 1 C2 1 1 Cs 1 • is divergent, l9 ' 1 1'' ' 1 |9 ' ci r 1 C2 |- 1 cj |- • convergent. 192 INTEGRATION. The reader should notice that the expression for a holomorphic function as an infinite product is fully determined, save as to an expo- nential factor which nowhere vanishes. Also that when r of the numbers c are equal, the corresponding primary factor must be raised to the exponent r, and that when r of the numbers c = 0, the preceding reasoning applies to the function — ^, so that G(z) now «'■ contains a factor a'. § 152. An example of the factor-theorem. Let G(;. ) = 5\T).-n-z/iTZ. The zeros of G{z), namely z=±l, ±2, • ■-, are distributed so that the series but Hence s = 1, and G{z) = n(l - z/c^)e"\ = ffxi - z/n)e'^", —x the accent signifying that n = is omitted. By combining each term (1 — z/)i)e''" with the corresponding term (1 + 2/?i)e~'^", we get G{z) =n(l-«VH2), or sin ttz = Trz{l - z'/l') (1 - 272^) (1 - 278=) ••• , or sin 2 = 2(1 - 2777=) (1 - 272V) (1 - 273V)- . The presence of the exponential in the primary factor ensures the unconditional convergence of the product. The value of n(l — z/k) depends upon the ratio n/m, when n, m tend to ao.* —771 An Example of 3Iittag-Leffler' s Tlieorem.f The series 2 {n = 0, ±1, ±2, ■••) is divergent, but if by 2' we denote that w = is not to be included in the summation, 2 {.Z— nir n-ir ) nTr{z — nir) * Cnylcy, Collected WorkB, t. i., p. 156. t Sec M. De Presle, Bulletin de la Sooi^t^ Math(5matlque, t. xvi.; also Hermite's Cours d'Analyae. INTEGRATION. 193 is convergent. Hence cot z — 1/z — 1.'z/mT{z — nn-)= a holomorphic function G(z). If it can be proved that G{z) does not become infinite as 1 ; tends to the limit co, G{z) must reduce to a constant (§ 140). (i.) Let z = x-\- iy, where 2/ ¥= 0. In the expression cot X cot iy — 1 1 .^ , 2 cot X + cot iy X + )";/ '^^^ (^ — '"^) the first two terms do not tend to oc with z, and the third term is finite. (ii.) Let ?/ = 0. The first term is infinite only when x = mir (m an integer), the second term is infinite only when a; = 0, while the third term tends to oc only for values of x which tend to m-ir. To examine the behaviour in the neighbourhood of m-n-, write x = x' + mir. The limit of the function cot(a;'4-mff) ■ — — is , when a;' =0. m-irX m-rr 1 X This shows that cot x 2,' is finite for every 1 „ X ntvix — n-n) value of a;. ^ ' (iii.) By combining (i.) and (ii.) we see that G{z) is finite throughout the plane and reduces to a constant. That this constant = is evident from the consideration that cote — 1/2—2' , ■ ■i.^ mr(z — mr) changes sign with z. ^ ' Hence cot z = 1/z + 1' - -xn7r(z — Jiff) Examples. Prove that CSC2 = -+ (-l)"! Z -» J(7r(2 — riir) tan2=2-— i -, r=± 1, ± 3, ± 5, • sec2=l + 2i'+' — — ! -, r=±l, ±3, ±5,. § 153. Tlieorem. In the neighbourhood of an essential singu- larity c, a one-valued function approaches, as nearly as we please, every arbitrarily given magnitude (Weierstrass). 194 rSTEGRATION. Let A be any arbitrarily assigned magnitude. We have to prove that within any circle, however small, roimd c as centre, there must be infinitely many points such that the value of the function at each of these differs from A by as small a quantity as we please. If infinitely many roots of the equation f{z) = A lie within the circle, the theorem is clearly true. If, on the contrary, the number of roots be finite, describe round c, as a centre, a circle C in which f{z) is nowhere = A. The function has an essential singularity ./■(2)--l at z = c, and no pole in the neighbourhood of c. Hence we may use Laurent's theorem : = P{z-c) + P\l/{z-c)l, /W--1 throughout the interior of C, the point c exclusive. The second series converges throughout the plane. Writing l/{z — c) = x, the second series becomes an integral series P{-i:) convergent for every point within a circle of radius R concentric with C. But, however great B may be, there are points outside C, at which the absolute value is greater than an arbitrarily assigned value. Let x be such a point ; to it corresponds a point z = c + 1/x, arbitrarily close to c, at which the series is greater, in absolute value, than any assigned quantity. Consequently f{z) has values which differ arbitrarily little from A. See I'rcard's Cours d' Analyse. i Picard has further shown that, as in the case of e' (§ 147), near an essential singularity of a one-valued function there are infinitely many points at which the function is exactly equal to A, except possibly for two values of A. Comptes Rendus, t. Ixxxix., p. 745; Bulletin des Sciences Math., 1880. The first theorem of § 145 can be extended to any one-valued functions. If /,(2), /^{z), two one-valued functions with arbitrarily many poles and essential singularities, coincide in value along a line L of length not infinitely small, they must be identically equal. For /i(z)— /j(2), =4>{z), is zero along L. Let C be a closed curve which does not include a discontinuity oi fi{z) , f.,{z) , but does include a portion Li of L. Within C, <^(«) is holomorphic ; but it = along Zi, therefore it must = throughout the interior of C. Xow C may be drawn as close as we please to a pole c^ of /i(z) or /^{z). The place Cj cannot be a pole of <^(2), for this would imply an abrupt change in value from to oc. Assume q to be an essen- tial singularity of each of the functions /i(«),/2(«). AH the points inside Cand close to Cj make <^(2) = 0, whereas, near Ci, <^(z) should INTEGEATION. 195 be completely indeterminate. It follows that c, is not an essential singularity of <^(2). Hence {z) < )i — 1, and that \p{z) has n simple zeros Zj, %, ■••,z„, none of which lie on the barrier. Let C consist of the two sides of the barrier and two small circles round a and B. Since the residue at z_ is ^' , we have r=2.i2^i^log^^'. Jc i//'(z,) Z, — a Again, the residues at a and 13 are zero, therefore j and | are zero, and I may be written jy{x^)dx^ +j°'f{xj)dx_, where /(.) = ^log^^, \li(x) X — a or j\Ax^)-f{xJ))dx, or, since the logarithm loses 2 Tri in passing from x+ to x_, o^iT^AAax. Jo. \\l{X) Accordingly C^^dx = 2^M log^-Il^, the formula for the integration of the rational fraction ^a^-. If by log ^, we mean that branch which is real when z is real Z — a and either >/3 or {z, w) be a function of the two variables z, iv, which is holomorphic in two circular regions r, Fi, of radii p, pi ; here z, tu are regarded as independent variables. The value of — ^\-^ — ^ at Zq, lOo (the centres of the two circles) dz'Siv" tf>(z, iv)dzdiv is given by — --=— = — I I hi (z-z„y+XM-«o)'"^' Write z — Zo = pe", tc — «;„ = pie»'i, then SZo' Z/5Wo iir-p'piJo Jo 200 INTEGRATION. and < r ! s ! -1^ , where ^ is the greatest absolute value of cfi{z, iv) for the regions T, Tj. "For purposes of comparison we shall need another function, 4>(!l', Z): •. z — Z,i Pi the partial differential quotient at Zg, tt'o is 82„'SmV p'pi* a quantity which is real and positive ; and therefore J^^IJ^^ (1). Szj8w„' I BzJ&ir„' This inequality plays an important part in what follows. Let us now consider the differential equation dw/dz= (f>{iv, z), where <^ is the function discussed above. Suppose that, when z — z^, a value of to is Wo; there will be no loss of generality if we write z„ = 0, iVg = 0, for this merely amounts to a change of both origins. We have to prove the existence of a solution iv — Pi{z). Write tentatively w = Pi(z), so that w = when 2 = 0. The coeffi- cients in Pi{z) must be the values of div/dz, —d^w/dz^, —d^w/dz^, ••• , when z = and lo = 0. We have for their determination a system of equations which are free from negative terms, w" = <^, -)- iv'tj>2, w'" = <^„ 4- 2 iu'<^,2 + w'-„, + w"4,2, J J (2), where <^i = 8<^/Sz, <\>, = B/hw, c^„ = S'+V/S^'Sw'. The series which is obtained in this way, namely, «;„'z + ic'o"zy2!+tf„"'^V3! + -, has, so far, merely formal significance as an integral. For it has not been proved that it converges in the neighbourhood of (0, 0), or that it actually satisfies the differential equation. It will be suffi- cient if we show that the series is convergent for a finite domain of (0, 0) ; for it results, from the method of formation of the successive INTEGEATION. 201 differential quotients of w, that (^ and tv', together with all their differential quotients with regard to z, are equal at (0, 0). The convergence for a finite domain is found by comparing the series to — Pi{z), derived from w' = and all its partial differential quotients are real and positive at (0, 0), all these coefficients are real and positive. Further the inequalities which are derived from (2) show, with the help of the inequalities (1), that, when z = 0,iv = 0, W= 0, 1 M)' I < *l. I W" I < *1 + I IV' I $2, and therefore | Wo'*' I < W"'', « = 1, 2, 3, •••. It follows that when the series Qi{z) is convergent, the series Pi{z) is also convergent. It will therefore suffice if we prove the convergence of Qi{z), or W,'z + W,"z'/2\ + T7„"'^V3! + .••, for a neighbourhood of « = which is not infinitely small. This is easily done, for the equation in 17 can be solved by the separation of the variables, one of the ordinary methods of differential equa- tions. Let W=0 when z~0; then the solution is ■^Pi V P. and the two values of TFare found by solving a quadratic. 202 INTEGRATION. That brancli of W which vanishes when z = is the radical being supposed to reduce to + 1 when 2 = 0. By expanding log (l—z/p), and afterwards the radical, as an integral series, we find that W is equal to an integral series Qi(2) whose circle of convergence extends to the nearest singular point. This singular point is the brancli-point given by l+^logfl-^Vo, Pi V PJ that is, z = p{l — e-f>i/-i^p) = p'i{ii: + w, z) — <^(m', 2). Since the right-hand side vanishes for (0 = independently of the value of 2, it must contain an integral power of iv. Thus diii/dz = (o''i//(2), where i//(z) is made a function of z only, by writing for w, w, their values in terms of 2. On separating the variables, this equation can be integrated along a curve (0 to ^) and gives K — 1 ,/0 )dz. Since the integral is finite and oo = 0, the equation is untrue. There remains the case « = 1 ; the value of u> is then INTEGRATION. 203 Since (uq = 0, w must vanish identicallj-. Thus the solution of the kind })roposed is not only existent, but also unique. The analytic function associated with the solution w=Pi(z) can be found by the method of continuation. To find a value at a place z', draw a path from to z' and interpolate points between z and z' in the manner explained in Chapter III., using each point as the centre for a circle of convergence. Progress towards z' may be barred by the presence of singular points in the region to be traversed. § 157. With the aid of the preceding theorem it is possible to prove the existence and one-valuedness of to as a function of z when cliv/dz = V(i(; — «i) (it! — a.j) {iv — a-i) {w — a^), Oi, flo, ttj, cii being four constants, supposed unequal. Suppose that ic = ifg when 2 = 0, «■„ being different from the constants «„ cu, «j, a^. Also let the sign of the radical be assigned when iv = u'^- The equation has an integral which is holomorphic in the neighbourhood of z = 0. By the "Weierstrassian method of continuation, this ele- ment defines an analytic function. When a place Zo is reached, for which w is equal to a^, cu, a.^, cij, or x , the expression on the right- hand side of the equation for dic/dz ceases to be holomorphic. That the integral of the equation is holomorphic in the neighbourhood of a point Zq, for which w = cti, can be shown by the transformation w = ttj -f W'-. The right-hand side of the resulting equation. 2dW/dz=y/{ai - a, + IP) {a^ - a.- + W-) {a, - a, + TF), is holomorphic in the neighbourhood of 1F=0, and therefore W and w are holomorphic in the neighbourhood of this point z„. Similarly the transformations xc = l/TV, z = z^ + z' enable us to determine what happens at a point Zf, for which lu^x. For the equation d W/dz' = - V(l - «i TF) (1 - a, W) (1 - a^ W) (1 - u,W) , when coupled with the initial conditions z' = 0, 1F= 0, has an integral TF" which is holomorphic in the neighbourhood of z' = 0, and z = 2o is a pole of iv. Thus, whatever be the values of lo at a point 2 = 2o, the integral of the differential equation is one-valued in the neighbourhood of 2 = Zg. Picard has pointed out (Bulletin des Sciences mathematiques, 1890; Traits d' Analyse, t. ii., fasc. 1) that the above considerations do not prove that the function w is a one-valued function of z 204 INTEGRATION. throughout the finite part of the z-plane. It has been assumed that the value of the integral is determinate at each point of the 2-plane. He makes the method of continuation apply rigorously to the whole plane by proving that the continuation can be effected by a circle of fixed radius p. Round the points a„ a.,, a-^, a^, co describe circles (a,), (fflj), (a^), (cit), (»). Within the region bounded by these five circles there is a finite radius of convergence at each point, and therefore a lower limit, which does not vanish, for the radii of con- vergence. When IV lies within (a), the transformation w = Oj + W^ transforms (cii) into a new circle, and so long as W lies within this new circle, there is a lower limit of the radii of convergence for W. Similarly for the other points rto, a^, a^, cc. The least of these six lower limits of the radii of convergence of the series which define iv is a number p, different from ; and when we assign to z, w any arbitrary values z,,, iv„ z^ being finite, w is one-valued and determi- nable within the circle whose centre is 2o a-i'id whose radius is the fixed number p. From this the desired conclusion follows. If we attempt to apjily the same argument to the equation dio/dz = y/ {w — a-i) {w — a«)---{w — a.2p+2)> we have, when iv = 1/W, dW/dz = - V(l - a, W) (1 - «2Tr)-..(l - a,^^,W)/W''-\ and the function on the right is not holomorphic in the neighbour- hood of W= 0, when p > 1. The inference is that w is not a one- valued function of z throughout the finite part of the «-plane ; in point of fact, w is an infinitely many-valued function of z, as will appear later. Beferences. On the general subject of this chapter, the reader should con- sult : Briot and Bouquet, Fonctions EUiptiques ; Cauchy's Works ; Hennite's Cours d' Analyse ; Jordan's Cours d'Analyse ; Laurent's Traite d' Analyse. Cauchy's fundamental theorem has been extended to double integrals of functions of two independent complex variables by Poincare (Acta Math., t. ix.). Algebraic functions of two variables are discussed in Picard's prize memoir (Liouville, ser. iv., t. v.). CHAPTER VI. EiEMAXN Surfaces. § 158. Hitherto we have considered the «-plane as a single plane sheet upon which the variable 2 is represented. In the Arganil diagram, to each point 2 are attached all the corresponding values of w. As long as tv is a one-valued function of 2, no difficulty arises ; for every path from z„ to 2 in the Argand diagram leads from the initial value w„ to the same final value w. But if iv be a many-valued function, a definitely selected initial value at 2, does not determine which of the values is to be chosen at 2, since different paths from 2o to 2 may lead to different values of w at 2. A familiar instance is afforded by the algebraic function given by the irreducible equa- tion F{iv", 2") = 0. Here each point 2 has attached to it n values w. When n > 1, two paths from z„ to 2 can always be found which will lead from the same initial value Wg at z^ to different values at z. In Eiemann's method of representation one value of w, and only one, corresponds to each point of a surface. Before considering the general problem we shall show how to form a Riemann surface in some simple special cases. Let tv = Vz. To one 2 in the Argand diagram correspond two values of w, whereas we wish to make two points correspond to two values of w. To these two points the same complex variable 2 should be attached. Instead of a single 2-plane we take two indefinitely thin sheets, one of which lies immediately below the other. For convenience of description we suppose them to be horizontal.* To the one value of 2 correspond two places in these sheets, one vertically below the other. Every place in the upper or lower sheet has one, and only one, of the two values s/z, — Vz, permanently attached to it. If, for a given 2, V2 be attached to the upper sheet, then — V2 must be attached to the lower sheet at the point 2 ; but we cannot infer, from * To avoid confUBion we flball speak of a point of a Riemann surface as a place. A pair of numbers {z, w) is attached to each place, and serves to name the place; but it is frequently con- venient to use a single letter for the pair («, w). When n sheets are spread over the Argand dia- gram for z, the vertical line through the point z meets the surface In n places {«, to). 205 206 RIEJIANX SURFACES. tliis alone, the value for the upper sheet at a second point 2'. "When 2 = or ex , the values of iv are equal, and only one place is needed to represent them ; hence we regard the sheets as hanging together at and x . But we require a further connexion between the two sheets. In the Argand diagram a closed path, wliich starts from z and passes once around the origin, leads from Vx to — V2. Tliere- fore, in the Riemann representation, if the path start from a place in the upper sheet, to Avhich a value -y/z is attached, it must lead to that place, in the lower sheet, which lies vertically below the initial l^lace. It follows that eitlier we must give ujj the idea of having only one jc-value attached to each place of the two sheets, or else we must make a connexion or bridge between the sheets, over wliich every path, which goes once round the origin, must necessarily jxass. If the bridge stretch from to cc , without intersecting itself, this condition will be satisfied for any finite path. In this example it should be noticed that and co are branch-points in the z-plane, and that the bridge extends from branch-point to branch-point. Secondly we know that the same path, described twice in the same direction, leads from Vz to Vz, i.e. from a place of the upper sheet to the same place of the upper sheet, assuming that the initial place lies in the upper sheet. The first circuit takes the place into the lower sheet ; in order that the second circuit may restore the place to the upper sheet, it is necessary that it should pass along a bridge from the lower sheet to the upper. This condition and the preceding one are satisfied by a double bridge, or branch-cut, which extends from to cc. Figure 48, a, is a section of the surface made by a plane perpendicular to the bridge. «i &i e, di >c as bs Oi di 0.1 h\ c[ di (a) v Z>CZZ~~(&) Oi 62 e'i di Fig. 48 Finally a closed path in the Argand diagram, which includes no branch-point, restores the initial value of iv. It remains then for us to show that such a path gives a closed path in the Eiemann surface. There is often an advantage in distinguishing by suffixes the places in which a vertical line cuts the sheets. In this notation z, and z, mean corresponding places z in the first and second' sheets. EIEMAXN SURFACES. 207 A closed path whicli starts at Sj must end at z^ and not at z,. With this notation, Fig. 49 shows that a closed path in the Argand diagram which does not enclose 0, is a closed path in the Eiemann surface. For instance, a point which starts at a place o, ant describes the path in Fig. 49, in the direction of the arrow, passes at h into the lower sheet, at c into the upper sheet, at d into the Fig. 49 lower sheet, and finally at e into the upper sheet, returning to the initial place aj. The figure also shows that the path gir^ga leads from qi to q.,, whereas qir.iq-2)\qi is closed. The Eiemann surface for ic = Vz is of the form given in Fig. 50. Looking from to a, the sheet 1 on the left continues along Oa into the sheet 2 on the right, and the sheet 1 on the right into the sheet 2 on the left. The branch-cut Oa extends from to cc, but is not necessarily straight ; the sheets 1 and 2 are nearly parallel, infinitely close to one another, and infinitely extended in two dimensions. The figure illustrates the way in which the surface winds round the point 0. The displacement of a branch-cut. The places Oj, h.2, are consecutive in Fig. 48, a ; but «„ by, or ao, b^, are not consecutive. Thus if the values of the function at a,, a,, be y/a, — Vn, continuity requires that the values at bi, b^, shall be — Vi, Vft; then the values at Ci, di, ••• are — Vc, — Vd, •••, and so on. The position of the branch-cut or bridge has been defined only to this extent, that it must pass from to oo, and never cut itself. Naturally any other branch-cut, subjecf'to these conditions, Fig. 50 208 RIEJIANN SURFACES. will equally serve our purpose. Let the cut be deformed from V to V, and consider the same vertical section of the surface as in Fig. 48. Fig. 48, b, represents the new state of things. A com- parison of this figure with Fig. 48, a, shows that the points which have changed sheets {e.g. by, b.^, which have become &o', &/) carry the values of iv with them. For the value at a^, in Figs. 48, a, and 48, b, being Va, the value at b.2 in Fig. 48, a, is V6, and the value at bi in Fig. 48, b, is also V&. Viewing the whole surface. Fig. 50, when the branch-cut is shifted to V, the part of the lower sheet between Fand F' becomes part of the upper sheet, and vice versd. These considerations show what is involved in the shifting of a branch-cut. Let us next examine the function w = V(z — a){z — b). This function is two-valued ; we wish to make one value of w correspond to one place of the Eiemann surface, and must therefore have a two-sheeted surface. The branch-points are a and b. In the Argand diagram, we know that a circuit round either branch-point, with an initial value «.', leads to a final value —iv ; while a circuit which includes both branch-points restores the initial value. This suggests that we draw a bridge from a to b. Any path, on meeting the bridge, has to change from one sheet to another, but the sheets are unconnected except along the bridge. § 159. The liiemann sphere. We have now had two simple illustrations of a Eiemann surface. Before giving other examples and the general theory, we point out that, as in the case of the Argand diagram, there is often a great advantage in substituting for a surface formed of infinite plane sheets a closed surface. Any surface may be selected which has a (1, 1) correspondence with the plane surface, the simplest closed surface being the sphere. Let us suppose, then, the Eiemann surface inverted with regard to an external point 0', which lies vertically below at unit distance. A two-sheeted Kiemann surface becomes two infinitely near spheres, joined together at the branch-points. These two spheres touch each other at 0', but by a slight displacement, this connexion can be destroyed, except when oo is a branch-point in the plane surface ; in fact, in the last example, the only connexion is along a line extending from the branch-point a to the branch-point b, whereas the Eiemann sphere for the function w = Vz has branch-points at and 0', which are connected by a bridge lying along any line, as for instance the half of a great circle, from to 0'. It is now clear that the function Vz is only a case of the more general function V(« — a)(« — 6). We may regard the sphere, in EIEMANN SURFACES. 209 the second case, as stretched, without tearing, until the branch- points are diametrically opposite, and then developed, by inversion, upon the tangent plane at either branch-point. It should be noticed that, in the case of ^ {z — a)(z — b), the method suggested by the first example, namely, the drawing of branch-cuts from a, 6, to oo , does not differ essentially from that adopted here ; for qo is an ordinary point on the sphere, and the two cuts, from a, &, to oo , amount merely to a single cut from a to h. § 160. The function w = z'''". To a given z correspond n values of w. "SVe wish to make one value of to correspond to one place of the Eiemann surface ; we must therefore have an 7i-sheeted surface. Let the n values of iv, for a given z, be «.'i = 2'^", iv^ = aivi, iv^ = o?iVi, '> o • •-, tc-„ = a"-"ii'i, where a = cos^^ — h » sin= — The way in which we ?i n number the sheets is immaterial. Let, then, iv^, u^, ■■•, «'„ belong, initially, to the first, second, •••, nth sheets respectively. In the Argand diagram, one positive circuit round the origin changes 2'^" into 02'^", i.e. ii\ into 1U2. A second circuit changes iV2 into w-^, •••, an nth changes ic„ into il\. Hence if, with the previous notation, z^, z^, ■•■, 2„ be n places in the same vertical line, to all of which 2 is attached, the first circuit must lead from z^ to z^, the second from 22 to 23, •■•, the nth from z„ 3 \^ ^ 3 4 ^^ 4 5- Fig. 51 to z,. A bridge must be drawn, as in Fig. 49. A closed path which passes once round the origin is an open curve in the Eiemann sur- face. Suppose that this path starts at the point z, and proceeds in the positive direction. The first description leads from Zj to z,, the second from 2, to z-^. ■■■, the nth from z„ to 2,. A negative descrip- tion of the pnth leads from Zj to z„, from 2„ to z„_i, from z„_i to 2„_2, and so on. By drawing the bridge, it is easy to see that any closed 210 EIEMANN SURFACES. path in the Argand diagram, which does not include the branch- point 0, gives a closed path on the Riemann surface, for sucli a path must cross the bridge an even number of times (see Fig. 49 for the special case n = 2) ; let a, b be two consecutive points where the path crosses the bridge. If, at a, the point pass from the xth to the A.th sheet, it remains in the Ath sheet till it reaches b, and then passes back to the Kth sheet. Thus, on the whole, it returns to the original sheet, and the path is closed. Fig. ol represents a section of the surface made by a plane perpendicular to the direc- tion of the bridge, and viewed from the origin. §161. The fnnction IV = Vz{l~z) {a — z)(b-z)(c~-z). The function is two-valued, with branch-points at 0, 1, a, b, c, x . The Riemann surface must be two-sheeted. It is required to find the connexions between the sheets. From the theory of loops we know that a closed curve which surrounds an even number of the six branch-points (infinity included), or which passes an even number of times round a single branch-point, will reproduce the initial value of w ; whereas, if the curve enclose branch-points an odd number of times, the initial value of iv is reproduced with a change of sign. The branch-cuts in the Riemann surface must be so constructed that a path, which is closed in the Argand diagram, shall lead from Zj to Zj or z.,, according as it includes an even or an odd number of branch-points, where, in estimating the number, we must count each branch-point as often as the path encircles it. A figure would show, immediately, that cuts along straight lines from 0, 1, a, b, c, to oo satisfy these requirements, but a simpler arrangement is to join to 1, a to 6, c to as , by branch-cuts. These branch-cuts must not cross themselves or one another. In Fig. 52 they are represented by straight lines. This figure shows that paths which enclose an even number of branch-points in the 2-plane are closed on the Riemann surface, whereas q./irq^, which encloses three branch-points, begins in the upper sheet and ends in the lower. In the same way the Riemann surface can be found for the square root of a rational integral function of degree 2 ?i — 1 or 2 Ji. It lUlOlANX SURFACES. 211 should be noticed that the former t'lmotiou is a special case of the latter, created hy one braiich-puiut inuviug off to iiitinity. § l(i2. Iticmunn surfaces fur irreibicible ahjehraic functions. We are now in a position to construct a Kieniann surface for the general algebraic function. Let the function be it-valued. Take n infinitely thin sheets, lying infinitely near one another. We first assign to the n sheets the branches which correspond to a given 2, say z = 2'"', which is not a critical point. When 2 describes a path from 2'"' to a branch-]ioint a, we determine bj' the principle of con- tinuity those values into wliich the several initial values of iv pass; by observing those which belong to the cj'clic systems near a we decide the connexion of the sheets at a. Through the sheets to which one and the same cj'clic S3'stem of values is attached, a branch- cut is drawn from a to cc , and this is done for each cyclic system at a ; the branch-cuts being distinct, though they may interlace (as in Fig. 53). a,h a,b' Starting afresh from 2'"', we repeat the process for every other branch-point. It will be convenient to suppose that neither the paths from z'"' to the branch-points, nor the branch-cuts from the branch-points to 00 , intersect one another ; also that none of the paths encounters a branch-cut. It is clear, as in § 1.59, that a new system of paths may lead to quite different connexions of the sheets. When the connexions of the sheets have been ascertained, the branch-cuts can- not be drawn at random, but must satisfy the requirements of the cyclic system or sj'stems at cc . That is, a very large circuit which begins at any place of the surface must produce the same effect as in the theory of loops ; this effect being nil when cc is not a branch- point. The s3-stem of branch-cuts adopted is merely one of infinitely many which are available when the connexion of the sheets has been established. For example, the cuts may be drawn to any finite point instead of to k , regard being paid to the requirements of the cyclic systems at that point ami to the branch-point at 00 . Sup- 212 EIEMAXX SUUFACES. pose tliat a non-critical point z„ is selected in tlie Argand diagram and that cuts are made in the z-plane from Zo to the branch-points of the function. Let these cuts, Vi, Fa, •■•, F„ be regarded as hav- ing positive and negative banks, which are met in the order Fi+FrF2+ ••• V*V~ on description of a small circle round Zy. By superposing it — 1 similarly cut sheets, called sheets 2, 3, •■-, n, on the z-sheet, and by joining the n positive banks of the cuts V, to the n negative banks of the same cuts, the n sheets become connected, and the sheets 1, 2, •••, n pass into the sheets «„ «.,,■••, «„, where («„ «2, •■•, a„) is a permutation of (1, 2, •••, n). Let the resulting substitutions along the ■>• cuts F, be S„ S.,,---, S,; then if the new surface is to be a Riemann surface in which it is possible to pass from any one place to an}' other by a continuous path, it is necessary that SiSi'-S^ be equal to 1. When an w-sheeted surface is con- structed by the process just described, the requisite data, in order that it may be a Riemann surface, are the positions of Zq, of the branch-points, and of^he lines F,, together with the substitutions 'S'l, jSj, •••, S„ where iS'i& ••• iS,. = 1. See Hurwitz, Riemann'sche Flachen mit gegebenen Verzweigungspunkten, Math. Ann., t. xxxix. A branch-point a in Cauchy's theory of loops is on the Riemann surface one or more branch-places (a, 6) with or without ordinary places (a, c) ; all these places lying in the same vertical. A loop to a from z'"*, in Cauchy's theory, is a vertical section of the surface, consisting of n loops on the surface which are closed or unclosed according as they pass near (a, c) or (a, b). When a branch-cut has been drawn from every branch-place to oo , any path C on the sur- face which begins and ends at places in the same vertical can be coh- tractefl into loops on the surface, precisely as in § 127. For there has arisen nothing to invalidate the theorem of that article. A closed loop on the surface is of no effect, and may be omitted. To each place of the surface there corresponds a pair of numbers (z, w) satisfying the equation F= 0'. How many places of the sur- face correspond to each pair of numbers ? Evidently only those places which lie in the same vertical, and which also bear the same value of w. First, when z is near a, let r values of w become nearly equal to b, and form a cyclic system. As in § 160 the )■ sheets in which these nearly equal values lie are regarded as hanging together at one place (a, 6), whether or not these sheets are consecutive. Kext, let there be a node at a ; then not only are two values of to equal to 6 when z is equal to a, but also two values of z are equal to a when w is equal to b. The series near a for these branches are integral (§ 113), and a circuit round a, which starts from a place EIEMAN'N SURFACES. 213 Fig. 54 near (a, b), must lie wholly in one sheet. There is therefore no connexion between the two sheets near (a, b). And, in general, when the pair (a, b) constitute a higher singularity, the sheets to which are attached those branches which are given, near (a, b), by integral series, have no connexion at (a, b) with other sheets.* "We have already had instances in which branch-cuts, instead of proceeding to the common point oc , are drawn directly from one branch-place (o, b) to another (a', b'). We have now to examine when this is possible. Since branch-cuts are not allowed to cross one another, the same sheets must hang together at the two branch-places, and a path C, which starts in one of these sheets and encloses these branch-places and no others, must run entirely in the one sheet. If, then, on crossing the original branch-cut a x , C pass from the Kth to the Ath sheet, it must on crossing the original branch-cut a'-ji , pass back from the Xth to the Kth sheet. This shows that if the positive loop from «"" to a permute the branches in the cyclic order (iCjif, ••• wv), the posi- tive loop from 2'"' to a' must permute the same branches in the cyclic order {iViIl\il\_i • ■ • n:,) ; that is, the substiUitions due to the branch-cuts at z'"* must be inverse. If not, aa' is not a permissible branch-cut. When the surface is two-sheeted, the condition is satis- fied by any pair of branch-points. § 163. Examples of Riemann surfaces for algebraic functions. The construction of a Riemann surface, in the study of the rela- tion of two complex variables, is an exercise similar to the tracing of a curve in the field of real variables. The present jiroblem is to map the jy-plane on the z-plane, when a relation F{w", z"") = is assigned. It should be noticed that there is not only an ?i-sheeted 2-surface, but also an m-sheeted zr-surface, the places (z, tc) and (lo, z) on the two surfaces being in (1, 1) correspondence, except possibly at special points. It is often advisable to construct both surfaces. The continuity of each branch, and the angu.lar relations near special points, as summed up in § 111, are the most important elements in the solution of the problem. In determining the path of iv in its ♦Instead of a penultimate form in the Cartesian sense, wec.in im.igine a liorizonlal displace- ment of some of the sheets, which necessitates possibly a stretching of tbi'se sheets, until there lie, in any one vertical, only two equal values of w. 214 EIEMANX SUllFACES. plane, for a given path of z in its plane, we must attend specially not only to the branch-points in the « plane, but also to those points in the «-plane which correspond to branch-points in the lo-plane. Cor- responding to a simple branch-point h in the w-plane, there is a point a in the z-plane near which m — 6 = P.(z — a) ; so that when z moves towards a in a given direction, makes a half-turn round a, and pro- ceeds in the same direction, a branch of w moves towards 6 in a cer- tain direction, makes a complete turn- round 6, and proceeds in the opposite direction. We shall speak of such a point as a turn-point. If (a, 6) be any pair of finite values, and if z describe a small circle round a, v: turns round k = b with the same direction of rotation. We shall suppose that the branches TOj, w,, ••-, w„, are initially assigned to the upper, second, •••, lowest sheets. Ex.(l). u.=f^ + lV^^ Initially let « = 0, Wi = exp'(7rt74), ii\ = iii\, ic^ = in:,, w^ = hcs- The series for v: near z = — l are iv = P^{z + l)''"*. Therefore when z, starting from 0, describes positively the circle whose centre is — 1, which by § 127 is equivalent to a loop from to — 1, the branches permute in the cyclic order (icurnic^iCi). Similarly, near z = 1, the series for tu are l/(f = Pi(z — 1)'''*, and when z describes positively a circle whose centre is 1, the branches permute in the order (ii',2i'4?''3'^"2)- -A- f(mi'-sheeted surface, with a branch-cut from — 1 to 1, extending through all the sheets, satisfies the requirements. Look- ing from — 1 to 1, the permutations, either at 1 or — 1, require that the right-hand portions of sheets 1, 2, 3, 4 shall pass into the left- hand portions of sheets 2, 3, 4, 1. This completes the construction of the surface. A contour on the surface, which encloses both 1 and — 1, restores the initial value (§ 127). Ex. (2). iv + l/w=2z. For a given z there are two values of w, say tv^ and Wj, connected by the relation Wizu, = 1. Hence the surface has two sheets. The branch-places are(l, 1) and (—1, —1). If ^('i describe an arc of a circle from — 1 to -f 1, the other root «'2 describes the remaining arc, and z, = (wi -|- Wj) /2, describes an arc from —1 to -1-1 in its own plane (Fig. 55). If Wj describe the circle C, whose centre is and radius 1, so does Wjj and z moves along the real axis from — 1 to -f 1. Let this line be chosen as the branch-cut. Since the two sheets are joined solely along the straight line from — 1 to -f 1, and since this line maps into the unit KIE5IANN SURFACES. 215 circle round the to-origin, it follows that to all points of one of the a-sheets correspond all points mside the circle C in the M-plane, and to all points of the other z-sheet, all points outside this circle. In the case of a circle in the ic-plane through — 1 and 1, to the arc inside G corresponds an arc in the one sheet ; to the remaining arc outside C corresponds the same arc in the other sheet. To the set of circles in the to-plane with limiting points ± 1, correspond circles with limiting points ± 1, de- scribed in both sheets of the 2-plane. Agaiu, if iv^ = pe'^, we have «;,=e-'Vp,2.r= (p + l/p) cos 6, 2y = {p-l/p) sin 6; therefore to circles round the origin in the w-plane, corre- spond ellipses with foci ± 1 ; also to lines through the origin in the zi'-plane, correspond the con- focal hyperbolas. The ?c'-plane is a conform representation of the two-sheeted Eiemann surface. The reader will find interesting discussions of this and allied examples in Holzmiiller's Theorie der Isogonalen Verwandtschaften. ■ Ex. (3). The relation considered here is a form of the relation aw- + 2biv + c + z (ttiic^ + 2 b^ic + c{) = 0, which defines the general (2, 1) correspondence. We shall show that this general relation can be brought, by (1, 1) transformations of both IV and z, to the form ?c'- = z'. The branch-points z^, z., are given by the equation (a -I- a,z) (c + c,z) = {b + b,zf. Let {z — Zi)/{z — z.,) =z' ; in the new equation between w and z', the equal values of w are given by z' = and 2' = oo ; and the equation is of the form {a'w + b'y = z'{ai'w + bi'y. 216 KIEJIANN SURFACES. If then we write (a'to + h')/{aiW + &/) =iu', the proposed reduction is effected. "Without going into details, we can infer at once that in the general (2, 1) relation, when we choose as branch-cut a circular arc in the z-plane, the two sheets of the z-surface map resjiectively into the interior and exterior of a circle i^i the w-plane. By the last example, any circle in the if-plane maps into the inverse of a conic in the a-plane (compare § 23) ; and by § 47, any circle in the z-plane maps into a special bicircular quartic in the w-plane, namely 'the inverse of a Cassinian. The z-circle is a double circle, lying in both sheets ; when it includes one branch-point it is a unipartite curve and maps into a unipartite quartic ; but when it includes neither or both of the branch-points, it is bipartite, and maps into two distinct ovals. Ex. (4). w'-3j« = 2«. Here &F/&W = gives 3w- — 3 = 0, i.e. w = ± 1. The equal values of iv arise from the z-values 1, — 1, co ; and the values of w which correspond to these special ^-points are — 1, — 1, 2 ; 1,1, — 2 ; oe , cc , X . We have to determine the manner in which the sheets of the Eiemann surface hang together. At the origin iu = 0, V3, — V3. Let these values be attached in this order to the three sheets. Calling the branches zi'i, tc.,, iv^, it is clear, from the Theory of Equations, that the following lemmas hold: — \ (1) Wi + iv., + u'3 = 0. (2) "When w is small, an approximate value is «• = — 2z/3. (3) All the roots are real, when z is real and j 2 | < 1. Using a w-plane, let z, starting from 0, move along the real axis to — 1 ; by lemma 3, Wi starting from moves along the real axis, and, by lemma 2, it moves to the right. In order to maintain the centroid at (lemma 1), iv^, which starts from — V3, must remain to the left of 0. Hence Wj and w, unite when z = — 1. In the same way it can be proved that u\ and lOj unite when 2 = 1. Accordingly, in the Riemann surface, sheets 1 and 2 join at z = — 1, sheets 1 and 3 at 2 = 1. At z = c(o all the sheets join. § 164. Ex. (5). The (3, 1) correspondence. Let zU+ V= 0, where [/"and Fare cubics in w. Any two cubics when rendered homogeneous by writing w/t for w, may be written in the forms 8Q/Sw, BQ/St, where Q is a homogeneous quartic in tv KIEMANN SrEFACES. 217 and t (Salmon, Higher Algebra, Fourth Edition, § 217), and the relation may be written z8Q/8w + 8Q/St = 0, the 7 effective constants of the original relation being reduced to 4 b}' means of the 3 disposable constants in the (1, 1) transformation effected on w. In this form z is the third polar of lo with regard to a fixed quartic, and accordingly the correspondence of a point and its third polar with regard to a quartic is the general (3, 1) corre- spondence. Let Q be brought to the canonical form lu* -f dciiy't- + t* (§ 44) ; then the correspondence is defined by z (w' + 3 «(.•) + 3 cic- + 1 = 0. The turn-points in the tt!-plane are given by the Hessian of Q (§ 3G), that is by JI = c{v:* + 1) + (1 - 3 c') w- = 0. The roots of H being a, — a, 1/a, — 1/a, we shall take the case when one root is real ; it follows that all the roots are real. Let a be that root which is positive and < 1. We have when z = 0, to = ± V — l/3c, oo, and when z is near 0, 18c^(ifT V— l/3f) = (1— 9c-)z, or tu = — 3 c/z. To real values of z correspond real values of iv, and also the points of the curve whose equation is tc^ + 3cio _ v:' + 3 ao 3cw- + 1 3cw- + 1 where w and to are conjugate, or 3c(iv-iv- + l) + w- + w- + ivw(l — 9 c=) = 0, or, in polar co-ordinates, 3c(p' + l) + 2p^cos2e + p\l-dc') = 0. Regarding a as given, c is determined by 3c=-l = c(«= + l/«=), and the two values of c have opposite signs. Selecting the negative value of c, let it'i, w^, ic^ be the 3 branches of w, and let tt'i be that branch which is positive when 2 = 0, tt'j that which is oo when z = 0, and zi'3 that which is negative. Also let these values be assigned to the first, second, and third sheets respectively at « = 0. 218 RIEMANN SURFACES. When z (Fig. 56), starting from the origin, moves to the right along the real axis, the approximate values show that Wj and ic, move to the right, inasmuch as 1 — 9 (r > ; while vi.^ moves to the left. Also the point at which xni starts lies between a and l/«. There- fore, at the first branch-point /?, sheets 1 and 2 join. When z describes ^ li l/,3 I . ! > I I ^i^^^- i- \37V. Fig. 66 a half-turn round /? and proceeds along the real axis, tw, and tv, make quarter-turns round the point !/«, and then describe the right-hand oval, meeting again at w = «. Therefore again, at the second branch- point 1//3, sheets 1 and 2 join. When z, after making a half-turn round 1/fi, proceeds to the right, Wj proceeds to the left to meet ic^, while u'l moves to the right. Therefore, at the branch-point — 1//3, sheets 2 and 3 join. When z, after a half-turn round — 1//3, moves to — (3, n\ and w^ describe the left-hand oval, and meet again at V] = — l/«. Therefore sheets 2 and 3 join at — p. The branch-cuts may be drawn from /3 to 1//3 in sheets 1 and 2, and from — y8 to — 1//3 in sheets 2 and 3. When they are drawn along the real axis, the whole of the first sheet corresponds to the interior of the right- hand oval, the whole of the third sheet to the interior of the left- hand oval, and the whole of the second sheet to the remainder of the ic-plane. • §165. Ex. (6). a-(iif-z-' + l) = {w + zY (1). The branch-points in the z-plane are given by 2= = (aV-l)(a=-22), and are a, — «, l/«, — 1/«, where a? + l/«2 = a\ We consider the case in which one, and therefore all branch- points, are on the real axis, and take « < 1. EIEMANN SURFACES. 219 We have, if il\ and tc^ be the branches, (aV -!)(«•, + «■,) = 2^,1 .... .ox (a-2- — l)u-iir, = a- — Z-. ) "When tui =h'2, there follows, on elimination of a^, either u- + z = 0, or icz' — 1 = 0. The case w + z = is extraneous, as it requires a = ; and there- fore, the symmetry of (1) being observed, the turn-points in either plane, corresponding to the branch-points a, — a, 1/u, — l/« in" the other plane, are 1/a^, — l/«^ «-^, — f-^- When z = 0, w = ±a, and near z = 0, v: = ±a —z. When 2 = X, ic = ± 1/a. Since a is real, to a real value of z correspond either two real or two conjugate values of w. Therefore to the real 2-axis corresponds part or all of the real w-axis, together with a curve whose equation is obtained by regarding zt'i and w, as conjugate, and eliminating z from (2). The equation is 4(a- + WiK',) {a-ii\W2 + 1) = {ii-\ + v.\^-{l — a*)-, and the curve is a central bicircular quartic. Starting from the point to the right, let z describe the whole of the real axis, making half-turns round all critical points (Fig. 57). -l/f» -n -l/r a? (t 1 ya- a V<^ .^-^-ir-i. 1 i 1 6 ^ Fig. 57 The figure shows the path described by the branch whose initial value is o.* Let the branch-cuts in the z-plane be drawn along the real axis from —a to «, and from —1/a via oo to 1 /«. A careful consideration of the figure shows that to these parts of the real 2-axis correspond * "When z is near - l/u^, the branch of lo in question is not near — a; hence to the balf-tura In the e-pliine round — l/a» corresponds a half-turn in the ui-plane. 220 EIEMANN SURFACES. the parts from — !/« via oo to l/«, and from — « to a, of the real tt)-axis. Let then the two-sheeted w-surface, which represents the dependence of z on w, have the same branch-cuts as those chosen for the 2-surface ; whenever (z, w) changes sheets, so does {ic, z). In this case, one surface serves to represent both the dependence of to on z. and that of z on w ; and this surface is mapped on itself by the rehxtion (1) without alteration of the branch-cuts. The most general quadri-quadric correspondence between iv and z involves 8 effective constants. The general sj-mmetric correspond- ence involves 5 effective constants, and can therefore be obtained from the unsymmetric relation by a (1, 1) transformation of z alone, inasmuch as the (1, 1) transformation places 3 effective constants at our disposal. The symmetric relation may be written aiv-z- + 2 bii:z{w+z) +c{i(r+z-+ 4 in) +2 d{K+z) + c+/i(w — zy= 0, where the first five terms are the second polar of ic with regard to the quartic az* + 4 hz^ + Q, cz- + -i dz -{- e ; and if this quartic be broiight to its canonical form Q = 2^ + Cc'2-+1, then also, by the same transformation applied to both w and z, the (2, 2) relation becomes iv-z- + c'(iC" + z- + -i KZ) + 1 + /x'(l(; — 2)2 = 0. The branch-points are given by (z= + c' + /.') [(c' + t^')z- 4- 1] = (2 c' - ;u')V ; that is, by H+^'Q = Q), where H is the Hessian of Q. [Halphen, Fonctions Elliptiques, t. ii., ch. ix. ; Cayley, Elliptic Functions, ch. xiv.J § 166. Ex. (7). vf-Zw = z. The branch-points, together with the values of w which become equal at them, are z = 4, w= -1,-1; 2=i4, w = -i,-i; z= -4, w=l, 1; ^ = — «-li ^<-' =h i; z = X, IV = X, ac, X, 00, 00. When 2 = 0, the values of w are and the four fourth roots of 5, and the approximate values are w=—z/o, and w — 5'-'^ = 2/20. EIEMANN SUEFACES. 221 2 -i 10 4; 3 1 5 sheeted z-plane Fig. 68 w-plane 222 KIEMANN SUKFACES. First, we trace the w-curves which correspond to the real 2-axis. These are the real axis and the curve IV* + ic* + u-ir(ic-+ Kic + ic'-) = 5, where iv, w are conjugate. In polar co-ordinates this curve assumes the form p^=5/(l + 4cos(9cos36'). The curve which corresponds to the imaginary axis is obtained by turning the curve already considered through a right angle. This may be seen directly, since, for each place {z, ic) there is another place {iz, iw). In Fig. 58 the second curve is dotted. Let us write a for the positive fourth root of 5. Assign the values vj=0, a, ia, —a, —ia to the iirst, second, . . ., fifth sheets. "When z moves from to the right, the approximate values show that the branch u\ moves to the left, while the other branches move to the right ; also, when z moves from into any quadrant, the branch u\ moves, in its own plane, into the opposite quadrant. As z continues to move from towards 4, the branch ti'i continues to move to the left along the real axis, and the branch w^. whose initial value is —a, to the right along the same axis ; and, when z is near 4, the two branches in question come near to each otlaer, so that a descrijition of a circle round z = 4 must lead from the first into the fourth sheet. "We therefore make a branch-cut from 4 to + cc along the real axis. If z move from to the left, the branches u\ and tt'j interchange round z = —-i, and a second branch-cut must be drawn to connect the first and second sheets. Let this extend to — cc. Similarly, branch-cuts from 4i to xi, and from — 4i to —xi, connect respec- tively the sheets 1, 5 and the sheets 1, 3. Since the complete «f-plaiie corresponds to all five sheets of the lliemaini surface in the 2-plane, there will correspond to the twenty separate (pxadrants twenty separate regions of the w-plane, and to the boundary be- tween two consecutive quadrants, the corresponding lines in the to-plane. "When two straight lines in the z-plane meet at a branch- point at which two branches permute, so as to enclose an angle of 180°, the paths in the lu-plane of the two branches in question intersect at right angles. Hence, when z describes that side of the positive part of the real axis which lies in the fourth quadrant and the first sheet, the corresponding branch u\ describes that part of the line (0 to —1) which lies in the second qiiadrant ; and, when z describes a small semicircle round 4 in the negative direction, after- wards continuing the description of the upper side of the positive real axis, il\ describes a small quadrant round —1 in the negative KIEMANN SURFACES. 223 direction, afterwards describing the lower half of the curve through — 1. Hence the straight line from to —1, and the lower half of the curve through —1, form part of the boundary of that portion of the w-plane which corresponds to the iirst quadrant of the first sheet of the five-sheeted Eiemann surface in the z-plane. Similarlj-, the remaining boundaries can be determined. Let a point starting in the quadrant marked 8 in the first sheet of the 2-plane describe a large positive circuit five times. Since, when z is large, the approximate value of ?« is z^'-', iv describes a large positive circuit once in the w-plane ; and since, Avhenever z moves from one quadrant to another, lu must cross one of the curves of the figure, the parts of the if-surface which correspond to the quadrants of the z-surface are readily determined. For instance, the region 18, in the lu-plane, lies in the second quadrant, and has for its boundary two portions of the axes and two portions of curves. It is necessary to test the system of branch-cuts by seeing that they produce the right results for circuits about the point cc. In the present instance, we have seen that when z is at a great distance along the positive direction of its real axis, and just above this axis, M'lhas assumed the position j Vz | exp (6 7r//o), and it can be simi- larly seen that, simultaneously, Mo is | -\/z | , w^ is it'j • exp (2 tri/o), iv^ is u:, • exp {-iTri/o), and iv^ is u:, ■ exp (S-n-i/')) ; that is, the branches are in the cyclic order {icm:^il\iviIc-). A large positive circuit in the z-plane should therefore permute the branches in this order, and the figure shows that this is what takes place. Ex. (8). u--'-oicz + z' = 0. Jjetw = tz; then z = 3i/(l + ^)- The surface for t is also the surface for ic, for to each pair (t, z) corresponds one w, and when two values of t become equal, so do two values of iv. "When z = a, (where « = V4), < = a/2, a/2, — o; when z = do, (where v = exp(2 7rt'/3)), t = va/2, va/2, — va ; when z = v-a, t = v-a/2, \ra/2, — v-a. When z = 0, ^ = 00, cc, ; and near z = 0, t = z/3 or ± ^'sjl- Finally at z = oo, i = — 1, —v, — v-. The real axis gives 3t 3t _, . ^ i.e. p' = ^ sec 6. Let c be a point on the positive side of the real axis near z = ; assign t^ = c/3 to sheet 1, U = VS/c to sheet 2, and <3 = — VS/c *° sheet 3. Xow let z move from c to a ; at o sheets 1 and 2 must hang 224 KIEMANN SUKFACES. / » \ /°\ / N F\g. 69 KIEMANN SURFACES. 225 together. Let z, starting from e, make a turn of 2 tt/S round ;. then t.,, t-i turn through — tt/S. Kext, let z move to va. We see that t^, t^ unite at va/2. Therefore at va sheets 1 and 3 hang together. Similarly at v"a, sheets 1 and 3 hang together. The branch-cuts may be drawn from the branch-points to cc, us in Fig. 59. It is necessary that the cut ••• co pass between va and v-'a, in order that a large cir- cuit may be closed, as it should be since oc is not a branch-point. The parts of the <-plane which map into the various parts of the sur- face are determined by observing that to a small circle round z = 0, starting from c, correspond a small circle round < = starting from t, and two large negative semicircles. § 1G7. Ex. (9). 32w!2 = (ic-f z)*. Let ?« + z = f, then F=?,2z{t-z)~t, and the Riemann surface, which repi'psents i as a function of z, will equally serve for to. The equations F= 0, hFjU = 0, give < = 0, KO ; r = 0, 4 ao jo ; where a = -^/'f' lo- = 1-^ •••> ^^''^^ " i^ 3."}' cube root of 1. The approximate values of t are (i.) When z is small, «= z or 2(2z)'/<; (ii.) When z— 4«a/5 is small, «— «a= j — 3«a(z— 4«a/5)/8i'^- The equations F=Q, hF/hz = 0, give 2 = 0, « ; t=0,2a; and the approximate values of z are (i. ) When t is small, z = < or fy32 ; (ii.) When i - 2 « is small, z - « = | - 3 «(f - 2 a)/2Y'\ When z = 00, f is also oo, and when t = x., z = oc ; the approximate relation is Z2z- = -t\ When 2 is a small positive quantity, let = (m — 1) (n — 1) — v — k. [This expression for p was given by Eiemann in his memoir Abel'sche Functiouen, Ges. Werke, p. 106.] § 171. The canonical dissection of a Rieinann surface. In § 130 we proved two theorems due to Liiroth and Clebsch. By means of these theorems we found that the n sheets of a Riemann surface can be coiniected in such a way that the branch- cuts between successive sheets are single links, except between the first and second sheets. In this exceptional case the number of branch-cuts is p + 1, when p is the deficiency of the algebraic curve F{ii:, z) = 0, obtained by treating iv and z as Cartesian coordinates ; it is assumed that the equation is prepared, and therefore con- tributes merely simple branch-points. There are thus jy superfluous connexions between the first two sheets. We have now to see that this ?i-fold sphere can be transformed, with- out tearing, into the surface of a body with p perforations. First take the case of a twofold sphere, whose two sheets are connected by a single bridge. In Fig. G4, Fig. 64 tne bridge is perpen- dicular to the plane of the paper, and a is its middle point. Let every point of the inner sheet be reflected in the diametral plane which contains the KIEMA^N SUKFACES. 235 bridge. Then the two halves of the inner sphere are interchanged, and the figure changes from A to B. This can be effected by a process of continuous deformation, the one hemispherical surface passing through the other, without any severing of the Eiemann sphere, and the order of connexion is unchanged. To an observer at e the appearance is that of a sphere with a hole in it. The inner sphere can now be pulled through the hole, and stretched, without tearing, into the form of a one-sheeted sphere, or of a doubly-sheeted flat plate without a hole. Xext take the case of a two-sheeted sphere with p + 1 bridges; the surface can be stretched, without tearing, until the bridges lie along one great circle. Apply to the inner sheet the same process of reflexion as before ; then the two spheres are connected along p + 1 holes. One of the holes may be stretched, as before, so as to form the outer rim of a two-sheeted flat plate ; and in the plate there remain p holes. The surface has become the surface of a solid box perforated by ]) holes. In the general case the inner sphere, and the one next to it, are connected by one bridge. The two can be replaced by a single spherical sheet. This sheet is connected with the third sheet by a single bridge. Eepeat the process uutil the first n — 1 sheets are replaced by a single sheet. This single sheet is connected with the outside sheet by p -f 1 bridges ; and the two can be replaced by a box with p holes through it. It is thus proved tliat the general Eiemann surface arising from an algebraic equation of deficiency p can be deformed, without tearing, into the surface of a box with j) holes. The principle of this method is due to Ltiroth ; the form of proof to Clifford (Collected I'apers, on the canonical form and dissection of ii Thiemann's sur- face, p. 241). See also Hofmann's tract, Methodik der stetigen Deformation \cn zweiblattrigen Riemann' schen Flachen (Halle, 1888). The flat two-sided sheet with p + 1 rims is used by Schottky in his important memoir on conform representation, Crelle, t. Ixxxiii. The following diagrams will illustrate these processes : Fig. 66 236 KIEMANN SURFACES. Figs, (a), (&), (c) show stages of the passage from the double Riemann sphere, with one branch-cut (Fig. a), into the double flat plate (Fig. c). In Fig. b, the reflexion has changed the bridge into a hole which connects surface II. with surface I. In Fig. c, the rim E of the hole has been stretched until it takes the position in the figure, the hole has disappeared, and the interior sphere II. has become the upper flat sheet, the exterior sphere I. the lower flat sheet. The process of conversion of a thin flat surface with two sheets, connected by p holes, into a double Kiemann sphere, can be illus- trated in a similar manner. We leave the drawing of the figures to the reader. Keeping the outer rim E fixed, we can make the two sheets pass from the flat form into one in which the two sheets form two infinitely close hemispheres connected along E and along the pi holes. These holes may be regarded as places at which the outer hemisphere passes into the inner, and £ is a place of like nature. By a contraction of E into a small circle, these two hemi- sjjheres can be made to pass into two infinitely close spheres con- nected by j5 -f 1 holes, namely, the original holes and the new one supplied by E. Arrange these p + 1 holes along a great circle, and employ the reflexion process as described above. Each hole changes into a bridge, and the final form is a two-sheeted Eiemann sphere with p -\-l bridges. § 172. Klein's normal surface. The box with p holes can be deformed, without tearing, into a sphere with p handles,* a convenient normal form used by Klein. ■ b' When p = 0, we have the ordinary sphere ; when p = l, the anchor- ring can be selected as the normal surface, for it can be deformed into a sphere with one handle. Fig. 67a is the normal surface for the case p = 2 ; the curves A, B are latitude and meridian curves respectively. The rim of KIEMANN SURFACES. 237 Clifford's box has become the curve E in the plane of the paper, which can be called the equator. Dissection of the normal surface. — It is to be understood that throughout what follows the surfaces employed are punctured at a point ; otherwise there is no boundary, and the definition of multi- ple connexion has no meaning. A surface is said to be dissected when it has been made simply connected by a system of cross-cuts. Let us consider the mode of (C) ■<^ K> & Fig. 67 dissection of the anchor-ring (Fig. 66). Let P be the puncture. Make the cross-cut Pa' along a meridian curve. The cut surface can now be deformed into a cylinder (as in § 168). As a second cross-cut, take bb'b ; the cut cylinder can now be deformed into a rectangular sheet. Thus two cross-cuts, along a meridian curve and a latitude curve, have made the surface simple. 238 KIEMAXN SXJKFACES. In exactly the same way, the surface of a sphere with p handles can be rendered simple by drawing a meridian cut through each handle, and a latitude cut round each handle. "We have ^j pairs of cuts A^, B,,{k = 1, 2, ■ ■ •, p)- 'J-'o dissect the surface, take a point s, on either ^1, or B^, say on B^ ; make cuts C^ to the p points s, from the puncture. The cuts have now become cross-cuts ; p of these are sigma-shaped, being formed by the combination of C\ and B^ ; and there are also jj cross-cuts A^. In all there are 2p cross- cuts. Fig. G7 shows the process in the case j) = 2. After two meridian cuts B„ B.,, the section along the equator of the sphere with two handles takes the form represented in Fig. 67 (b). The latitude-cuts, and the joins to P, piass into the heavily marked lines of the figure. The surface is now dissected ; for Fig. 67 (r) can be derived from Fig. 07 (&) by pulling out the tubes in the directions of the arrows, and the cylinder, if cut along the heavily marked line, is evidently dissected. In the dissection of the box with p holes, cuts were made along curves A, B, C, the curves A, B being respectively latitude and meridian curves. In dealing with ?i-valued algebraic functions, it is often necessary to retain the ?i-sheeted sphere, and to dissect it by 2p cross-cuts.* The easiest way of finding out how this can be done is to retrace our steps and change the multiply connected surface with ^) holes into the Eiemann sphere with n sheets, con- nected by branch-cuts as in § 171. Xow the j) cuts A must change into curves round branch-cuts. Also the branch-cuts round which these curves pass can only be those connecting the first and second sheets ; for a cut round the single branch-cut which connects the first and third sheets would sever the third, since the third is linked to the remaining sheets merely by this bridge. Thus the cuts A must pass into curves such as A (Fig. GS). The curves B passed originally through the holes for which the ^'s were latitude curves. Hence the B's, after the change, must traverse the p bridges round which the ^"s have been drawn. This shows Iiow it comes to pass that the (n — 2) bridges between the first and third, fourth, •••, ?ith sheets, are unrepresented in the diagrams of the cross-cuts. The equator E becomes a branch- cut over which all the B'r pass. Clifford's box is deformed by the same process ; the p holes pass into p bridges, the rim into a (;?-f l)th bridge. The meridian curves can be drawn in many ways from hole to hole ; the most • As before, the equatiuu between w nnd z ie in ibe pieimied form. KIEMANN SURFACES. 239 C,//, // V natural construction is to draw each round one hole and the rim. Returning to the Eiemann sphere, this gives cross-cuts iJ„ B.,, •■•, B^,, which all intersect one and the same bridge, namely, the (j) + l)th; thus the cut B^ / passes over the rth and / {p + l)th bridges (r=l,2,...,p). In a 5-ply connected Eie- man surface, four cross-cuts reduce the surface to simple connexion. In Fig. G8 the cross-cuts are C\ -f B^ A^, C-, + B,, A^ ; sigma-shaped. By following the arrows in Fig. 68, it is easy to see that the complete boundary of the dissected Riemann surface can be described by the continuous motion of a point, the two banks of a cross-cut being described in opposite directions. Fig. 68 Ci + Bi and C, + B, being § 173. Systems of curves irhich delimit regions. The closed curve j4, of Fig. 68 does not by itself delimit a region of the surface ; in other words, it is possible to pass along a continuous curve from a point on on'j bank of A^ to the point immediately opposite, without cutting the rim or passing out of the sur- face. £| is such a curve. Again, J,, B, in Fig. 08 do not conjointly delimit a region of the surface. The dissection of the Kiemann surface thus raises the question as to the maximum number of closed cur\-es, which neither singly nor in sets delimit a portion of surface. Also it is evident that the cross-cuts are capable of deformation. To what extent can such deformation be effected ? In Riemann's memoir on the Abelian functions these questions are discussed and various theorems are arrived at, which are of importance in the subject known as Analysis Situs. These theorems were used by Riemann as a basis upon which to build up the theory of the dissection of a multi- ply connected surface. In this chapter we have accom- plished the dissection by the use of theorems due to ^'9- ^^ Liiroth and Clebsch ; but as Riemann's theorems are interesting in themselves and throw valuable light upon the effects of cross-cuts, we shall give a brief account of his method. 240 RIEMANN SURFACES. I. Instead of a multiplj' connected Riemann si'rface with only one rim, we can employ the multiply connected single sheet of § 108 with several rims. The curve 1 of Fig. 60 does not by itself delimit a region ; when taken in conjunc- tion with 4 it delimits a region. It is true that it is impossible to get from one side of 1 to the other, without crossing the boiindary ; but nevertheless 1 does not delimit a region. To meet the case of a surface with more than one rim, we say that a curve does not delimit a region, when it is possible to reach the boundary from each of two opposite points of the curve. II. Let w = \/{z — '01-- — i')', t'le corresponding Riemann surface is two- sheeted, with a bridge from a to b. Let A be an oval curve round the bridge ab, and let c, d be opposite points on the outer and inner banks. If the punc- ture be made at a place p, outside A, it is impossible to connect e with d, except by a line which crosses A. The curve A delimits a region. III. If the puncture extend through both sheets, A does not delimit a region ; if, however, a second curve A' be drawn vertically below A, it will no longer be possible to draw a curve from d to the puncture without crossing A or A' ; therefore A, A' conjointly delimit a region. IV. The systems (1, 4) ; (1,2,3) ; (2, 3, 4) delimit regions (Fig. 09). V. Let K, K (Fig. 70) form the boundary of a surface. The curves 1, 2 delimit a region, and so do the curves 3, 4. The four curves divide the surface into two sets of points : — (1) points in the shaded portions, which have the property that lines which join them to the boundary must meet the four curves in an odd num- ber of points ; (2) points in the unshaded portions which have the property that lines which join them to the boundary must meet the curves an even number of times, or not at all. Points of the former kind are known as interior points ; points of the latter kind as exterior points. The curves 1, 2, 3, 4 taken together separate the interior points from the exterior. The collective space filled by interior points is said to be ' completely bounded ' or ' delimited ' by the four curves, and the curves are said to be delimiting curves. With this convention as to interior and exterior points, the theorems, which we are about to prove, hold good even when the curves intersect. However, the reader will find it easier to follow the argu- ment by drawing a figure in which the curves do not intersect. Theorem. In any surface T, let A, B, C be systems of circuits. If A and B be delimiting curves and if A and C be delimiting curves, then B and C are delimiting curves. Formal proof is superfluous; the following considerations will suffice. Since A can be deformed into B without papsin'i over the boundary, it must, in the process of deformation, trace out a continuous surface, which lies wholly within T. This region of T is delimited by A and B. In the same way A and C delimit a region of T. But the two regions, so obtained, adjoin each other Fig. 70 EIEMANN SURFACES. 241 along the lines A. Therefore, if these lines be suppressed, there is a region delimited by B and C Theorem. Let J.,, A.,, •••, An be a system of n circuits on a surface, of ■which neither the whole nor a part delimits a region, but of which some or all, taken in conjunction with any other circuit, do delimit a region. Let B^, B.., •••, B„ be another system of n circuits, possessing the first property of the A'f ; namely, that neither the whole nor a part delimits a region. Then this system, taken in conjunction with any other circuit, delimits a region. That is, the B's possess the second property of the ^'s. Let us here understand that ' all of n things ' is included under ' some of n things,' but that 'none of n things' is excluded. By the hypothesis, B„ with some of the ^'s, delimits a region. Let A^ be one of these A^s. Take any cir- cuit C which is not a boundary. C also, with some of the ^'s, delimits a region. There are two cases : — Case 1. .4i is included amongst these ^'s. Case 2. A^ is not included. In Case 1, A^, with some of C, A.^, A^, ■■•, A„, delimits a region ; ^„ with some of U;, A,, A^, •••, An, delimits a region ; hence a system taken from C, i?,, A,, X, •••, ^1„ delimits a region. We say that Cmust belong to this system ; for if not, some of B^, A.^, ■••, An delimit a region. But Bj, Ai by themselves, or with other ^'s, delimit a region. Hence, by the previous theorem, some of A^, A^, ••-, An delimit a region, which contradicts the hypothesis. In Case 2, C, with some of A.^, A^, ■■■,An, delimits a region. Hence, a for- tiori, C, with some of Bi, A.,, A^, ••-, An, delimits a region. Thus, in either case, the system B^, A^, ••-, An, which has the first property of the ^'s (that is, that neither the whole nor a part of it delimits a region), has also the second property of the .^'s. Continuing the reasoning, all the A''s can be replaced by iJ's ; which proves the theorem. Theorem. Let ^,, A,, •■•, An be as in the last theorem ; and let C„ Cj, • ••, C„ be another system of circuits, such that Ar and Cr (r = 1, 2, ••■, n) delimit a region ; then the A's may be replaced by the C"s. To prove this, we must show that no system of C's can delimit a region. If possible, let some of the C's, say C,, C.^, ■■■, C, delimit a region, and let this system be called a delimiting system. Since A^, C, form a delimiting system, it follows that (A^C.Cs ■•• C,) is a delimiting system. And since A^, Q form a delimiting system, it follows that (^,^,C, ■■■ C) is a delimiting system. Eeplacing in this way each C by A, we see that the original supposition demands that A^, A,, ■•■, Ak shall delimit a region ; but they do not. Hence no system of the C's delimits a region, and therefore, by the last theorem, the C's may replace the A^s.* *The8P tlienrems on dclimilinz curves were stated by Riemann; the proofs in the text were given by Kiiiiigsberger, Elliptiscbe Fnnclionen, pp. 47 et seq. We may also refer the reader to a memoir by Simart, Commcntaire sur deux m^moirea de Riemann; and to Lamb's Hydrody- Damics, Note B. 242 lUEMAXN SURFACES. Riemaiin's theorems on delimiting curves are immediately applicable to a Riemann surface. If, after a certain number of cross-cuts have been made, ■without severing the surface, we can draw a circuit on the surface such that it ■ is possible to get from the interior to the exterior region without cutting the boundary, the surface is not yet simple, and more cross-cuts will be needed ; but if no such circuit can be made, the surface has been dissected by the cross- cuts. Riemann defined the order of connexion of a surface by this maximum number of cross-cuts, increased by 1. We have seen that on a Riemann surface of deficiency j), 2p cross-cuts can be drawn, that these 2p curves neither singly nor collectively delimit a region, and that some of them delimit a region when taken with any other closed curve. Hence, by Riemann's definition, the order of connexion of a Riemann surface of deficiency j) is L'j.) -f 1; a result which agrees with that of § 170. For the 2p curves which serve the part of cross-cuts can be substituted any other 2p curves, of which neither the whole nor a part delimits a region. Thus the meridian and latitude curves of the normal surface can be deformed into new meridian and latitude curves, when the new meridian and latitude curves B^', Ak are such that Ak, AJ and £«, Bk', (k = 1, 2, ■■■,p) form -Ip delimiting systems. When a cross-cut has been made in an {n + l)-ply connected surface T, the new surface T' is, as we know, only I^-ply connected. It is interesting to show, graphically, that only )i - 1 curves of the type A can be drawn on T'. A ^^ ■ <^ A — ■ -r ^~- "*■«., K^ ■-^. __^ Fig. 71 The figure is that of a triply connected and two-sheeted surface, but the argument applies generally. A and B are circuits on the triply connected surface. Neither separately nor together do they delimit a region ; for the line K connects the points k, k', on opposite sides of B, with the puncture P. Treat K as a cross-cut, and, when it is made, let the new surface be T'. ^ is a closed circuit on T', which does not delimit a region. AVe wish to prove that every other circuit A', which does not of itself delimit a region, does so when taken in conjunction with A. Now A' is a circuit on the uncut surface T. Therefore A', A, B delimit a region Tj on T. We prove, in the first place, that B must be omitted from the boundary of Tj. For if B were part of that boundary, it would be impossible to get from one side of B to the other ; but the figure shows that K does pass on T from Ic to k'. Hence A, A' must delimit a region on T ; that is, there is a region T, on T, from no point of which can a curve be drawn to P, without meeting A' or A. But further, A' and A also delimit a region Tj on the surface T' ; that is, it is impossible to pass from a point of the region T, to either P or the cross-cut A'. For K does not meet either A' oi A; and RIEMANX SURFACES. 243 therefore if a curve, starting from a point of T,, could once get to any point of K, it could be continued along if to P; but we have already seen that P cannot be reached. § 174. Algebraic functions on a Riemann surface. It was proved in Chapter V., § 154, that when w is, throughout the 2-plane, an ?i-valued function of z with a finite number of poles and algebraic critical points, and free from essential singtilarities, this function is connected with z by an equation zC + ri(z)if"-' + ro(z)w'"--2 -\ h r„{z) = 0, where j'i(z), rjz), •••, »■„(«) are rational fractions. Let the denomi- nators be removed and let the resulting equation be F{vf-, z'") — 0. The corresponding Eiemann surface T is n-sheeted. If the resulting surface be connected, it will be possible to pass, by properly chosen paths, from any value of Wi to each of the ?i — 1 associated values «'2> ""31 •••) w-„. That the equation F=0 is, under these circum- stances, irreducible can be shown by the following considerations. Assume that F( if, 2) = Fi{ic, z) ■ F.^{n; 2), where Fi(!i% z) is irreducible. At an ordinary point 2 = a in the Argand diagram there must be some branch, say ic^, which makes Fi vanish at a and throughout a region which contains a. In the neighbourhood of 2 = « the function Fi{u\. 2) can be represented as an integral series P(z—a), and this integral series vanishes identically throughout its circle of conver- gence in virtue of the theorem of § 87. The method of continua- tion shows that the analytic function derived from this element is everywhere zero ; but by suitable closed paths of 2 in the 2-plane, u\ can be changed into jcj, iv^, •••, w„. Hence Fi(iVi, 2) can be made to pass into Fi{u\,, 2), Fi{u:j, 2), •••, -Fi(w„, 2). This proves that when Fi(tc, 2) vanishes at 2 = a for w = iv^, it also vanishes for jcoi i^s) •••)«'„; in other words, an equation Fi{iv, a) = 0, of lower degree than n in w, is satisfied by n values. This being impossible, the original equa- tion F=() must be irreducible. It is evident that any rational function of lo and 2, say R{io, 2), is one-valued upon the Riemann surface T for F^w", 2"") = ; we shall prove the converse theorem that every function one-valued upon T and continuous except at certain poles and branch-places, at which the order of infinity is a finite integer, is of the form R{iv, 2). Let the n values of w which correspond to a given 2 be u\, tOj, •■•, w„; and let the corresponding values of R be R^, R^, ••-, R„. The symmetric combinations s = .BiWi'-i + R.jivr."-' +■■•+ RnW^"-^ {k = 1, 2, ..., n), 244 KIEMANN SURFACES. are one-valued functions of z, for they take the same values at the n places attached to 2. Now a one-valued function of 2 which has only polar discontinuities is a rational function of z. Choosing Xj, ^2, ■••, K-\ so that iv^"-^ + XiiV + -Votiv""' + - +-^"-1 = (/^ = 2, 3, -, n), we get -Ki(!i'i"~' + XiWi"-= -f- A,Wi"-2 -^ h X„-i) = s„ + XiS„_, H h X„_iSi- The form of the equations in tw^ shows that these quantities are the roots of an equation w"-> -f Xi^u"-^ + - +X„^i = 0; but they are known to be the roots of Fiv:", 2'")/(?(; — ?('i) = 0; that is, of (/,!o" +/ii(;"-i + •■• -f /„)/(w - ?i'i) = 0, or of /o")— + (/oi«i + />)»""' + • • • + (/o't-i""' + /i"'i""' + ■ • ■ + /,.-i) = ; hence, x, = (/oH'i + /i)//o; •••; K^i = {Uor' +fiior' + - +L-i)/f{z) be one-valued and continuous in T, we have when each rim is described positively. j:< (z)dz = 0. Let {z) = u + iv. Then j {z)dz= j (udx — vdy) + i C{vdx + vdy) . Suppose that the surface r is cut by two sets of vertical planes parallel to the axes of x and y. A strip made by two consecutive parallel planes may run entirely in one sheet, or it may meet a KIEMANN SUEFACES. 247 brancli-cut and pass for a part of its course into another sheet ; there may be several strips between the same vertical planes (in Avhich case the strips may be said to be in the same vertical) ; and further, the boundary may consist of various rims. But however Fig. 72 many times strips in the same vertical may meet rims, and however many times they may change sheets, it will always be true that (^^dy =U,-Uy+U,-U,+-+ U.2, - U;,_i, J hy where J7 is a real function of x, y, which, with its first differential quotients, is continuous throughout V, and 2t — the number of points in which the strip meets the boundary. Hence, dxC^dy = {U2- Ui)dx + {L\- U,)dx + - + {Uu-U„_,)dx; J by that is, the parts of the integrals | I —dxdy and — | Udx, which the strip contributes, are equal. Hence for the whole region these integrals are equal. Similarly, Hence, by writing U= u, v successively, we get /(z)dz= ( {ttdx — vdy) + i ( {vdx + udy) =-//(M)"'+'//(M)-»- 2-1:8 KIEMANN SURFACES. But |^'-f^ = 0, and|-' + |^ = 0. Sx By &y 8x Hence, |<^ (2)0/2 = 0. In this proof the restriction that ~, -^ are to be continuous Sx by throughout T, may be to some extent removed. For example, if ^, ^ be discontinuous at a finite number of places of r, while V Sx 8y rm . is one-valued and continuous at these points, the integral J g^ ^^ -—da; become infinite for one value a; = c between x = a and x = 6. Then P^^dx = limpfdx-.flimf l^dx Ja hx e=U Jo. OX 1=0 .^t+i) OX = lim lu{c-e,y)-U{a,y) + U{b,y)-Uic + r,,y]. e=0, 11=0 (. ) But U'is one-valued and continuous, therefore lim U{c — t,y)= lim U{c + -q,y), and the integral —U{h, y) — U{a, y), a quantity independent of c. [See Kouigsberger's Elliptische Functionen, p. 69.] § 177. A function {z), which is one-valued and continuous at all places of a Eiemann surface, is a constant. For let the function take the values ^1, <^2> •••) n at the n places which lie in a vertical line ; then each of the symmetric combinations <^l'^ + <^2' + - + <^/, where k= 1, 2, •••, n, is a one-valu.ed and continuous function of z, for all values of z, and is therefore a constant (§ 140). Hence the equation whose roots are c^„ <^2) •••> <^» lia-s constant coefficients. Hence the roots are themselves constant for all values of 2, and since they are continuous, they are equal to the same constant. It follows that an algebraic function of the surface is defined, except as to a constant factor, when its zeros and poles are assigned. For if two such functions have the same zeros and poles, with the same orders of multiplicity, their ratio is everywhere one-valued and continuous. Compare § 145. EIEMANN SXTEFACES. 249 § 178. The theorems which relate to the value of an integral Jf{z)dz, where f{z) is a many-valued function, are greatly simpli- tied by the use of the Eiemann surface for /(«), but the methods employed by Eiemann are, in essence, identical with those of Cauchy. The fundamental theorem in integration is that the value of the integral j f{z)dz, taken positively over the rims C of a delimited region F, is zero when/(2) is continuous in that region. To fix ideas let us consider the integral of a function E{iv, z) which is rational in iv and z, where tv is an algebraic function defined by F{tv", 2"')=0. This integral is called an Abeltan integral. The advantage which results from this limitation of the field of discussion mainly arises from the fact that we know the nature of the branch- ings and discontinuities in the case of the algebraic function and the order of connexion of the corresponding Eiemann surface. When R{ic, 2) is continuous throughout a region T of finite extent upon the surface T, delimited by a system of rims C, the value of I Edz, taken positively (with regard to T) over all the rims C, is zero. Suppose that the region T is delimited by an exterior rim C and by interior rims Ci, C^, •••, (7^, we have the extension of the theorem of § 135, the difference being that a rim may pass several times round an interior point before returning to its initial point. Suppose that the region in question contains an infinity c ; this infin- ity must be cut out by a small closed curve described r times round c, say an 7--fold circle, where r is the number of turns that must be made before the curve can be closed. This curve (c) must be added to the other rims, and the theorem runs CBdz = 2 f Rdz + C Rdz, where the curves C, C„ (c) are described positively with regard to the region delimited by them. When r is a simply connected region on T, such that R has no infinities within T, the value of j Rdz taken over any closed line in r must necessarily vanish. As an immediate consequence, it follows that when I, II are two paths which lie in F and go from a place s„ to a place s, the value of the integral is the same which- ever path be chosen. This theorem may be stated somewhat differ- ently. When, on the Eiemann surface T, a path /which connects a place «„ "^ith another place s is deformed continuously into a second path II between the same two places, the value of the 250 EIEMANN SURFACES. integral is unaffected so long as the path does not cross a place at which R is infinite. To complete the theory, we require some way of dealing with the discontinuities of R. Let the infinities witliin a simply connected region of finite extent delimited by the paths I, II he Ci, Cj, ••■,c^; these places must be cut out from the surface T. The simply connected region has now be- come a multiply connected one within which R is everywhere continuous. Accordingly, CrcIz = CrcIz + 2 f Rdz, where the integration is effected in the sense indicated in Fig. 73. On the simply connected surface T', which results from T by the drawing of 2p cross- cuts, the integral | Rdz may be finite. This will be the case when the infinities of R are not infinities of the integral. In the most general case, however, the infinities of R affect the value of the integral, as has been already explained in speaking of Cauchy's theory of Residues. The surface T' must be changed again into a multiply connected surface by cutting out those infinities which affect the value of the integral, and this new multiply connected surface must be reduced to simple connexion by fresh cross-cuts. On the surface T" the value of the integral J Rdz is completely independent of the connecting path. § 179. "When iv is replaced by its values in terms of z, let the expression R{w, z) become the ?i-valued function {z). The theorems which follow apply also to functions of a more extended character than R{ic, z); but we shall not have occasion to use these in this treatise, and consequently omit them from consideration. When r is a region of T which lies entirely in one sheet, and encloses no branch-place, the same deductions can be made from Cauchy's theorem as in Chapter V. Since that branch of the many- valued function <^(2) which is represented on T is, tliroiiglioxit V, a holomorphic function of z, Cauchy's methods are immediately appli- cable. For example, if the infinities be excluded by small closed curves C", C", ■••, C''', then J_ r^(2)cfe_J_| r 4>{z)dz 2niJc z-t ~2ni<^Jai^>~z^IT^*'''' EIEMANN StTEFACES. 251 where ^{t) is the value of <^ at a place (t, tv) of T. When (f>{z) has no infinities in r, this equation becomes IttiJc z — \ and the same consequences follow from this formula as in Chapter V. The most important of these is that at any place («i, Wi) of the region r, ^(z) can be represented by an integral series P(z — z-^, with a circle of convergence not infinitely small. Hence it follows that <^'(z), <^"(«), etc., are holomorphic throughout the same domain. Hitherto the supposition has been that r contains no branch- point, and one of the results arrived at has been that in the neigh- bourhood of each place (Zj, jy,), ^(z) can be represented by an inte- graZ series vaz — z^. We now make the supposition that (z„ w-^ is a place at which r sheets hang together, and seek to find the expan- sions in the neighbourhood of this branch-place. Let a circle C pass ?• times round (z,, it'i), so as to limit a region V within which there is no branch-place of <^(z) other than (Zj, tt'i), and no infinity. The transformation z-z,= l^ transforms T into a circular region r, in the ^-plane, and the ?--fold circle C into an ordinary circle Cj. Since the function ^(z) recov- ers its initial value after a description of C, the function <^(z, -f f'), or ^{jOi must recover its value after a description of Cj. That is, •//(O is holomorphic in Tj. Applying Cauchy's theorems to the function i/'(^), we have for the region Fj, X' ^^r^(Od^=0 (/t = l,2,3, .-), and where ^' is a point inside Fj. Hence r<^(2) (z - z,)-i+<»+"'''-c?z = (7i = 1, 2, 3, •••), ana ^^ >- 2r7riJc{z-z,y-''^ ' {z-z,y^ - {z' -z,)"^' where (z', tv') is any place in F. By the use of the binomial theo- rem, the quantity under the integral sign becomes an integral series in (x' — Zi)''", and therefore {z') =P{z'- z,y'^ = a, + P,(_z' - z,y\ 252 EIEJIA^'N SUKFACES. Here the r values of <^ attached to the r places of r, which lie on a vertical line, are associated with a cyclic system of r expan- sions. The differential quotients of ^{2') can be found from the series. For example, omitting accents, This result shows that c^'(*i) is infinite when A < ?•, although the corresponding value of (^(x,) is finite. The only places at which i^\z) can become infinite are either branch-places or infinities of <^{z). Next let (zi, tt'i) be an infinity as well as a branch-place, but otherwise let the suppositions with regard to T and G be the same as before. By an application of Laurent's theorem to the function '/'(O) '^^ have where q is finite, since it is assumed that ^ = is not an essential singularity. Hence The expansions at z = 00 are found by putting z=^/z', i.e. by replacing 2 — » by I/2'. If r sheets hang together at gc, r sheets of the Eiemann surface for <^i(2'), = is an infinity, but not an essential singularity, of ^(2), the expansion runs <^(2) = 2"'-P'o(2"'''')- The reader will observe that the results obtained in Chapter IV. can be derived from Cauchy's theorems. They state that if w be an algebraic function of z, the expansions for R(w, 2) at various points are of the forms P(2-a); (2-a)-»P„(2-a); P{l/z); z-P,{l/z); P{z-ay"; {z-a)->"P^{z-aY"; •P(2"'"') i 2"'Po(2"'^0; where g, r are positive integers, and ?• > 1. § 180. Inter/rals of algebraic functions. So long as the path of integration is not permitted to pass over a cross-cut of the surface EIEMANK SURFACES. 253 T" (§ 178) the integral is one-valued. The cross-cuts serve as barriers ■which separate the branches of the many-valued integral. But now it is possible to remove this restriction that the barriers are not to be passed over. The cross-cuts required to make T simply con- nected were 2p in number, and further cross-cuts connected the curves round the infinities of j Rdz with the boundary of T'. No curve is to be drawn round any infinity of R which yields no loga- rithmic term to | Rdz. For instance, if (2;,, lUy) be a place at which R={z-z{)-^''P,{z-z,y'% (q>r), the integral I Rdz is logarithmically infinite when, and only when, Pq(z — z,)'/' contains a term (2 — ^l)~'+«^^ When this term occurs, the value of j Rdz, taken r times round z^, is called the residue 2 ttU at the point (§ 146). Thus the residue is r x the coefficient of the term. Let T" be the simply connected surface derived from the canoni- cally dissected surface T', by the drawing of cross-cuts from the curves C"', round logarithmic infinities, to the boundary of T'. Let I{z) = C' Rdz, where the path between («o, Wo), {z, w) can be drawn freely on T; J{z) = the same integral when the path is restricted to T". Then J{z) is a one-valued and continuous function on the surface T". Any difference which may exist between I(z) and J(z) must arise from passages over cross-cuts. In order to dis- tinguish the two directions across a cross-cut, an arrow-head is attached to the cut. The cut may be regarded as a stream with right (or nega- tive) and left (or positive) banks (Neumann, Abel'sche lutegrale, ch. vii.). Two places a, infinitely near but on opposite banks, can be distinguished as a+ and a_. Now in passing from a to a' (Fig. 74) the values of dz and of R, at opposite " f/o. 74 places of the cut, are equal. ~Rdz, along the right bank, = I Rdz, along the left bank ; or ~ J{a ') - JiaJ) = J{oJ) - J(((+). 254 RIEMANN SURFACES. Hence J{aJ) - J{a_') = J{a^)- J{a_) ; that is, the difference between the values of J at opposite banks of the cross-cut is constant. This constant is called a modulus of periodicity, or iwriod, of the integral. To each cross-cut there belongs a period. The full importance of this idea will be seen later (ch. x.). To connect I(z) with J{z), suppose that the path is s„a6s (Fig. 74), ■which crosses the cut A^ from right to left, and the cut jB, from left to right. Then f Rdz= r Rdz+ f Rdz+ f Rdz, the infinitely small paths a_a^ and hj}_ being neglected ; they con- tribute nothing to the integral, as R is supposed finite in their neighbourhood. Each of the three jiieces s^a^, a+&+, 6_s lies on T"- Therefore J Jtrfl Rdz = J{a_) ; f Rdz= r Rdz- C Rdz, u/2 be assigned to the upper sheet. The places oo being cut out of both sheets, the cross-cut may be drawn along the imaginary axis, coinciding with the positive part of this axis in the upper sheet and with the negative part in the lower sheet (Fig. 76). We must now assign a value to / at a given place . Let 7 = at the place s_, which is infinitely near (0, 1) on the right of the cut. The value of / is now everywhere determinate ; at the place s^. on the left of the cross-cut, the value is 2w, as appears or by integrating round the branch- + s+ Fig. 76 either from the residue at co points as in § 138. If 7 be the value at any place {z, w), the value I' at {z, — w) is TT — I. For dz/Wj Xz, ~w dz/w, =j dx/+Vl-ar' + J^°' ''dx/- Vl^^' + J"' '"dz/w, = 77-7. In such cases as we have been considering, the z-paths which correspond to lines parallel to the axes in the i-plane may be deter- mined directly from the differential equation. For instance, in the case dl= dz/VT^^, KIEMANN SURFACES. 259 let / move parallel to the real axis. Then, equating amplitudes, we have where <^i, c^,) ^ ^^re the amplitudes of the strokes z — l,z + l, and of the tangent of z's path. Hence the tangent bisects the angle — 1, z, 1 ; and the path is a hyperbola. For developments of this idea, see Franklin, American Journal, t. xi. § 182. (4) Let w' = (z- a,) (z-a,)--{z- a,^^,) . The 2-plane is in this case covered by two sheets ; and the points a< are branch-points. The bridges may be drawn from Ui to a^, from Fig. 77 ttj to (ii, etc. The cuts A^ may pass round a2,_, and a™,, where ic=l, 2, •■■,p, and the cuts B^ may all pass over the remaining bridge from 02^+1 to aj^+j. See Fig. 77, which is drawn for the case p = 3. When the cuts C< are drawn, the surface becomes simple.* * Id Fig. 77 the cute C, are drawn to the cuts ^^. This U convenient in a complicated 260 RIEMANN SURFACES. Consider the integral 1= | zHz/w. It is finite when 2 = oo if A 0. There are therefore p such integrals which are finite at all places of the surface, for we may take X = 0, 1, •■■, p — 1. By Riemann's extension of Cauchy's theorem a cut A^ or B^ may be contracted till all its points lie arbitrarily near the lines which join the included branch-points. The places of the cut pair off (see Fig. 78) into places a, u' at which the values of w are arbitrarily nearly / " ^ ? ' ^ opposite ; therefore, in the descrip- ^—^ a: tion of the cut, z'-dzjiti runs twice p_ -jg through the series of values which it takes in passing from, one branch-point to the other. It follows that the integral taken from one branch-point to the other is half the period due to the description of the cut which encloses those branch-points. Let the periods across A^ be o), (k = 1, 2, •••, p), and those across B^ be tiij. Then Fig. 77 shows that the periods round A' are w,', while those round B,, are — to,. It follows at once from what has been said that J'd7 = ^^. § 183. Birational transformations. The subject of birational transformations of curves dates back to Eiemann's memoir on the Abelian functions (Werke, 1st ed., p. 111). KIEMANN SUKFACES. 261 It was first presented in its geometric form by Clebsch in a memoir on the applications of Abelian functions to Geometry (Crelle, t. Ixiii.). Let the curve Fi{ W, Z) = arise from the curve F{ic, z) = 0, by the elimination of iv and z, -where W and Z are rational algebraic functions of tc, z. "When, conversely, F{iu, z) arises from Fi( W, Z) by the elimination of w, z, where lo and z are rational algebraic functions of W, Z, the curves correspond one-to-one, and the trans- formation is said to be hirational. By means of the birational transformations W=R{ic,z), io = r{W, Z), Z=S{k,z), z = s{W, Z), the points of the surface associated with F=0 correspond one-to- one with the points of the surface associated with F^ = 0. Therefore the maximum number of retrosections must be the same for the first surface as for the second ; this shows that j:i is the same for both surfaces. That is, p is an invariant with regard to birational transformations. It is not true, conversely, that two Riemann sur- faces, with the same order of connexion, can always be connected by a birational transformation, except in the special case j) = 0- The number j) is an invariant of the surface T not merely with regard to birational transformations, but also with regard to con- tinuous deformations. For a description of the possibilities as regards continuous deformation we refer the reader to two memoirs by Klein, Math Ann., tt. vii. and ix. ; and to Klein's Schrift, p. 2o. That two surfaces must have the same j), if they are to correspond point to point, can be proved as above. Jordan has proved the less evident theorem that the sufficient condition in order that two surfaces may be deformable continuously, the one into the other, is the equality of their p's (C. Jordan, Sur la deformation des surfaces, Liouville, ser. 2, t. xi., 1866 ; Dyck, Beitriige zur Analysis Situs, Math. Ann., tt. xxxii., xxxiii.). Thus the number p is the only invariant of a surface in the Geometry of Situation, a theorem of importance in the ulterior Eiemann theory. Suppose that W and Z are /n-placed and v-placed algebraic functions of T. By reason of Z=S{iv, z) the surface T can be represented conformly on an nj-sheeted surface Tj spread over the .Z-plane, so that the points of T and Tj correspond one-to-one. Since the function T^is one- valued on T and cc^ at fi places, the function must be one-valued also on Tj, and continuous at all places other than the ju. places wliich correspond to the ;u. infinities on T. At 262 BIEMANK SUEFACES. these IX places it is oo\ Hence to a given T7 correspond ^ values of Z. Similar reasoning shows that to a given Z correspond v values of W. Accordingly the new equation which connects W, Z must heF,{W\ Z'') = 0* "When this equation is irreducible, all functions which are one- valued on Ti and continuous, except at isolated places at which the singularities are non-essential, can be represented as rational integral algebraic functions of z^ and w,, and the transformation is birational. § 184 Any rational fraction Gi(iv, z)/G.,{iv, z) can be expressed in the form '•0(2) + r,{z)io + n{z)iv- -\ h r„_i(z)ir-\ where the expressions r{z) are rational fractions in z. For gl(HV. z) Gr2(w«, Z) _ g,(TOl, Z) ••• g;(w.^i, z)g,(w,^i, z) ••• G,(«'„, z) _ g,^^^^,^^ ^^_ IlG2{V-\, Z) The denominator contains symmetric combinations of the w's, and therefore reduces to a fmiction of z only. The numerator con- tains symmetric combinations of Wi, •••, u\_i, w^+i, •••, w„. and can be expressed in terms of tv^ only. Hence, using F{iu^, z) = 0, Gi(!(^ = u{z) + »-,(z)w« + - + K_,{z)ic,--\ Go{w^, z) The n equations formed by giving k the values 1, 2, •••, n can be replaced by the single equation ^'^^ = r,{z) + r,{z)io + ... + r„_,(2)i«-i. G2{u; z) The following pages will contain a brief account of Kronecker's methods, as developed in his memoir Ueber die Discriminante alge- braischer Functionen einer Variabeln, Crelle, t. xci., p. 301. (See also the memoir by Dedekind and Weber, Theorie d. algeb. Funct. einer Veranderlichen, Crelle, t. xcii., p. 181 ; and Klein-Fricke, Modulfunctionen, t. ii., p. 4SG.) •Owing to the exceptional case mentioned in §167, it may happen that F^iW, Z) is reducible. Riomann has proved that in this case F^ ia of the form [*( /r''i, ^^i)]'*, where * ia irreducible (Werke, p. 112J. The surface T, for i^^ = is then spread k times over a surface of n^K sheets. EIEMANN SURFACES. 263 "When the irreducible equation i^= is in its most general form it may be written go{z)vy + c?i(z)i(;"-' + ... + £,^(2) = 0, where the quantities g are integral polynomials in z. In what follows g and ?• are used for integral polynomials and rational fractions in z, in much the same way as F{z) for an integral series in Chapter III. The functions of the system R{w, z) are called algebraic functions of the surface T. That the system is a closed one is shown by the fact that the functions which result from the operations of addition, subtraction, multiplication, and division, when applied to members of the system, are themselves members of the system. Call this closed system ( R) . We have the theorem that every function of the system (R) is expressible uniquely in the form niz) + ri(z)w + r,{z)iv' -\ h ?-„_,(z)i«"~', and that conversely every function, of the kind just written down, belongs to (R). Here the members of (R) are expressed linearly in terms of a system 1, ?t', w^, •••, m"~\ and the coefiBcients in the linear expression are rational fractions in z. The n functions, 1, w, la', •■•, w;""' are said to form a basis of (R). Kronecker has pointed out that the limitation that all functions are to be expressed in terms of this basis solely is partly needless, partly injurious. We can choose any other set t/i, tjj, •••, •>;„, which satisfies the equations ,. = ,•„<'' + n<"i« + - + r„_i<"M;»-' {k = 1, 2, ..., «), where the r's are rational fractions in z, with a determinant which does not vanish identically ; and every function R {u; z) is then expressible in the form R{W, Z) = piT/i + p2'72 + ••• + Pn^n, where the p's are rational fractions in z. The necessary and suffi- cient condition in order that jyi, jjj; •••> ''in '^'^1 form a basis, is that no combination P\^\ + P2I72 + ••• + PnVn is to vanish identically. [See Dedekind and Weber, loc. cit., p. 186.] § 185. Integral algebraic functions of the surface T. When (7o(z) = 1 in the irreducible equation F=0, the function w is said to be an integral algebraic function of the surface. The function w cannot be infinite for any finite value of z; and con- 264 KIEMANN SURFACES. versely every algebraic function of the surface, which is finite for all finite values of z, is an integral algebraic function of T. It can be proved that the sum, difference, and product of two integral algebraic functions of T are themselves integral algebraic functions of T. Calling the system of the integral algebraic functions of T, for shortness, the system (G), we have the theorem that an integral algebraic function of any member of ( G) is itself a member of ( (?) . Thus the system {G) is closed. Suppose that ^i, 4) •••) L form a basis, the members of this basis being selected from the system ( (?) . All functions of the system 9ii^)L + 92(^)^2 -{ \-9„{z)L, where the g's are any integral polynomials in z, belong to ( (?) . But it is not true conversely that every member of (G) can be written in this form. Let us assume that there is a member of ((?) which can be written 9iii+ 9-2^2 -i \-9,.L (z-c) where the g's are not all divisible algebraically by « — c. Suppose that the remainders of g^, ..., g„, after the division, are Oj, a.^, ..., a„; then a,^i + a24H |-«„L is an integral algebraic function of the surface. We know that at least one of the a's is different from zero. Assume ai=?i=0. The n functions ^, 4, ..., ^„ of the system (?, form a basis of (B). Denot- ing the values of ^, ^„ (k = 1, 2, • • • )i,) at n associated places of T by ^<'>, V^\ ..., V"\ and ^,1, Z;,2, ..., f,„, we have, by the theory of determinants, •) till (z-cy tll> t2I) ■■•) ^nl 4l2> 4221 • • •) 4n2 4lnlfe2/i) •••)4ni say A (4 ^2, ^3, -, L) = -~^M^^< 4, ..., L). Here both determinants are integral rational functions of 2; hence the determinant on the left-hand side is of lower order than the determinant on the right. By proceeding in this way, we can remove the factors (z-cy from A (^1, 4 ..., f„) up to a certain point. EIEMANN SURFACES. 265 Then no further factors can be removed. Suppose that ^i, ^2') •••) U are the ^'s when this stage has been reached. Omitting accents, these new quantities form, in Klein's nomenclature, a minimal basis. [See Dedekind and Weber, loc. cit., p. 194 ; Klein, loc. cit., p. 493.] Since there are no longer any available denominators 2 — c, every algebraic function of the surface T can now be expressed in the form giL + 9A2-\ 't-g^L, ■where the g's are integral polynomials in 2 ; and, conversely, every function of this form is a member of (G). The analogy to the theorem for functions of the system (R) is evident. Let us now construct a minimal basis ab initio. We know (§ 184) that every function R{io, 2), which is at the same time an integral algebraic function of the surface, is expressible in the form n{^) + )-i(2)iy H 1- r„_i(2)2o"-\ in which all the denominators are factors of D{z), where Diz) = F'{w,).F'{ic,)...F'{io^), F'{w^) standing for 8F/Sw,. In other Avords, the denominators are factors of the discriminant of F. Select from the integral algebraic functions of the set ro + nwH h?-«w', that which has the highest denominator in the coeflB.cient of w'. Let So + SiW + S.2W- + •■• +s«w* be -a function which satisfies this requirement. Suppose this denominator to be d,+i, with a corresponding numerator n,+i. As the expression s, is in its lowest terms, it must be possible to find integral polynomials p,+i, a-^+i, such that p,+in,+i + o-,+id,+i = 1. The integral algebraic function of T, Pk+1 lSo + SiW-\ h S,W" \ + (T,+,W', begins with a term w". Call this function ^^+1. 266 EIEMANN StIEFACES. Let R{iu, z) be an integral algebraic function of the surface. When it is expressed in the above form, let its order in w be k. The denominator of the coefficient of w" is a factor of d.+j. For, otherwise, R{w, z) + d^+i is an integral algebraic function of the surface, such that the denominator of its highest term w' is of higher order than that of d„^i. This is contrary to supposition. We shall prove that the 6's form a minimal basis. § 186. The expression for R{w, z) in terms of the 6's. The func- tion R{w, z) is expressible in the form fi'«+l(2). -10" -f- terms in iv 1. Hence R(w, z) —g^+iiz) ■ O^-n = an integral algebraic function of the surface, of order not higher than k — 1 in w. This expression in turn can be changed, by subtraction of g^{z)d^, into an integral algebraic function of the surface, of order k — 2 in w, and so on. Thus, R{iv, «) = g'«+i(z)e,+i-)- (7,(2)61. H h 9-1(2)611. Let the values of 6^ at n associated places (2, iv^), •••, (2, w^ be ^«i> 6k2, ••■, 0^„, and form the determinant of the basis, viz., 9ii ^21 ■•■ ^„i |. ^12 ^1'2 '*■ ^n2 I The square of this determinant is a symmetric function of-w„ mjj, • ••, w„, since the determinant itself is an alternating function. Call the square of this determinant A (2) ; we suppose that the ^'"s have been multiplied previously by such constants as will make the coefiicient of the higjiest power of 2 in A (2) equal to unity. The discriminant D{z) of the equatioii ;J'(ii'\2"')^0, in which tRe coefB'ci'ent of w" is 1 and the coefficients of the remain- ing powers of id .are integral polynomials, is the square of the deterpiinani is the determinant of the system 1, W, TP, •••, W~^- Hence A, as well as D, is divisible by A(z). The expression A (2) is a divisor of the discriminants of all the irreducible equations of order n which define functions W of the system (G). The expression A(z) is called, by Kronecker, the essential divisor of the discriminant. The remarkable property of the essential divisor is that it is unaffected by a birational transformation which transforms F{w", z) = into Fi{ W", z) = 0, where the coefficients of iv", W" are 1. The remaining part of the discriminant is a complete square ; it is called the unessential divisor. A factor of A(z) is an essential factor ; a factor of the square root of the unessential divisor is an 268 KIEMANN SURFACES. unessential factor. The essential divisor is equal, save as to a con- stant, to the square of the determinant of any minimal basis. Corollary. The square of the determinant of the coefRcients a in 1 = onfiii + a.rfi is equal to the unessential divisor. Conversely, the square of the determinant of the coefBcients when $1, 6-2, ■••, 0„ are expressed in terms of 1, ic, w', ■••, to""', is the recip- rocal of the unessential divisor. But we know that every integral algebraic function of the surface is equal to 1 hence, there is no integral algebraic function of the surface r^(^z) + ri(z)iv + •■• + 5"„-i(z)zo"~^, which contains in its denominator factors distinct from the unessential factors of the discriminant. § 187. Kronecker has proved that the essential divisor of the discriminant of F=0 is the highest common divisor of the dis- criminants of the integral algebraic functions of T. Let where the X's are arbitrary coefficients. The function v satisfies an irreducible equation of order ?i ; and in this equation the coefficients are integral polynomials in z, except the coefficient of v", which is unity. The discriminant Z>„ must be divisible by A(0), and the remaining part must be a perfect square. Suppose that where O is an integral polynomial in z and the X's, which contains no factor independent of the X's. "We have to prove that 8(«) reduces to a constant. Assume, if possible, that 8(z) contains a factor z — c, where c is a constant independent of z and the X's. In the system (G) there must be a rational function R{v, z) v,-hich is of the form go(^) +9A^)v + g2(^)^'- + ••• + g„-i(^>''-\ z — c EIEMANN SURFACES. 269 Suppose that the lowest order in v for such functions is s, and that one of these functions of order s is ^^ ' . Then g,(z) is z — c not divisible by « — c. Let t be an undetermined quantity. Since tv — ^ ' is an integral algebraic function of T, it may be written z — c A^A + lJ-2^2 + ••■ + ^„^»7 where the /j.'s are integral rational functions of z, Xi, X2, •••, X„, and symmetry shows that its discriminant must be A(z)j8(z)psG(«,Mi,/^2, -, /^„)r But the discriminant of tv — - — - is equal to the product where «, ^ take all pairs of values, from 1, 2, •••, n, such that a>)3. This product is equal to Equating the two values for the discriminant, we have ± G{z, /*], IX 2, •", ^„) = 6(z, Ai, Xo, •", X„) • n I i — _° _ j- . The arbitrary quantities X can be chosen so that the two func- tions G are finite and do not vanish when z== c. Hence the above equation cannot be true if the second factor be infinite for 2 = c. The initial supposition requires, then, that ^{Va) — i/'(«s) ^ ^ t {Z—C){Va~Vp) ) be finite for z = c, and therefore also for all values of z. Hence $(«) =0 is an equation in t with coefficients which are integral polynomials in « ; it defines integral algebraic functions of the sur- face. It follows that 1 » ^(^),) - ^(v,) Z — Cr=2 Vi — V, must be an integral algebraic function of the surface. The follow- ing considerations prove that this is impossible. The function just written down is equal to z — c ' « — Cr=l V' ~V, 270 KIEMANN SURFACES. and the term which involves the highest power of Vj in this expres- sion is ('I - s)5r.Vi-'. This is contrary to the supposition that s is the lowest order in v for the system of expressions z — c hence Z{z) must reduce to a constant. To return to the discriminant Z>„. Choose the arbitrarj' quantities X so that G(z, Ai, X2, •••, X„) shall not contain any essential factor. As this is always possible, it follows that the essential divisor A(«) is the highest common divisor of the discriminants of all the integral algebraic functions of the surface, which satisfy irreducible equa- tions of the Hth order. The square root of the unessential divisor 1 6(2, Xi, X2, •••, X„)5- may have, for special values of the X's, repeated factors ; or again the unessential factors may be partly the same as essential factors. Kronecker has proved the highly impor- tant theorem that it is possible to choose the X's so that G has no factor in common with A(2) or with hG/^z. "When the quantities X are left arbitrarj-, G has no divisor in common with SG/Sz. Assume, if possible, that this is not the case. Let II(z, X], X2, •■■, X„) be one of the common irreducible divisors; and let G = HK. Then the equations G = HK, G'=HK'+KH', show that K must be divisible by H. When G is divided by H' the quotient is an integral rational function of 2 and the X's. Let the quotient be Q. Since G = H-Q, it follows that any divisor of G, G' is also a divisor of SC/SX, (k= 1, 2, •••, n), But the product-expres- sion of the discriminant shows that GVW) = ±nix,(^,. - e,^) + \,{K - 6,^) + ... -f x„(^„. - Ml, where «, /? are taken from 1, 2, ..., n, and a> P; hence one of these linear factors must be equal to another of these factors, multiplied by some constant. Suppose that the factor which contains suffixes a, y3 is equal to the factor with suffixes y, 8, multiplied by a constant ^•1 ; and use the equations Ci = l, e, = a + a^w, e3 = b + bj,w + b^w', ^4=0 + 01^ + Cjw' + c^to^, •■•. EIEMANN SURFACES. 271 Then Aai(iVa—iv^)=a^(iVy — iVs), Ab.2{it\- — Wf) + Ahi(iv^ — wg)=b..{Wy- — lo/) + bi{Wy — Wj), ^CsCr'/ — lu/) + Ac.{ii\- — iL-f) + ACi{w^ — wp) Since Oj, ftj, Cj, •••, are different from zero, it follows that 10 J- + w<,i(j^ + iv^- = iVy' + iVyivs + Wi'. Eliminating ivs, Ave have the relation (iUa — iCy) (iv^ — lU^) = 0. This equation is not satisfied in general. Hence the initial assumption that G and SG/&Z have a common divisor is disproved. Here it is assumed that the A.'s are arbitrary. Suppose that the X's are integral polynomials in z with arbitrary orders and coefficients. The expression which results from the elimination of z from G, G' does not vanish identically ; therefore special values can be found for the orders and coefficients, such that G and G' have still no common factor. Also values of the X's can be found which ensure that Gr shall not contain any of the essential factors. Suppose that these values are X/, Xj', ••■, X„'. Hence we have found a function, which belongs to the system (G), and which has a discriminant whose unessential factors are unrepeated, and distinct from the essential factors. § 188. Kronecker has shown that this separation of the dis- criminant into two divisors can be applied when the algebraic function to is not integral. Let F(ic% z-") =/oW" +/i?(;-» +f.,'V-'' + - +/„ = 0, where /„, /„ /o, •■•,/„ are at most of degree m in z. When two finite values of w become equal, F'{iu) = 0. Hence if w^, w^, ■■-, iv„ be the n branches, all the branch-points must be included in the equation say /o"~-£(2) = 0. The function E{z) is one-valued because the roots Zi, «2, •••, z„ enter symmetrically. When 2 is finite, ^^(2) can only be made infinite by choosing w = 00 . When tc = x, /o = ; 272 KIEMANN STJKPACES. but /oW is finite, for the coefacient of vf~'^, namely, fyo +/i, is zero. Since f^w is finite, the factor /(,"-" makes E{z) finite, for when one of the w's is infinite, that one of the factors i?"(t«) which becomes infinite oc /oiw"-'. Tu^t fa''~^E{z) = Q{z). The function Q(z) is one- valued and finite for all finite values of z. Its degree is at most m(n — 2) + mn or 2in{n — 1). Without loss of generality it may be assumed that /^, /o have no factor in common ; for the discriminant is unaffected when w + 6 is written for w. Write to' =fyo. Then F, (w', z) = w"' + g.io'"-^ + . . . + 5r„ = defines an integral algebraic function of the surface T associated with Fi = 0. Since SF^/Sw =f„''-^F/Sw, the discriminant D{z) of i^i = satisfies the equation ^(^)=/o'"-"'"-='-Q(^)- Kronecker has proved that /^^("-iX"-^) enters into the unessential divisor of D{z), so that Z»(2) =/„("-»(»-« . {8(«)P. A(2); hence Q(z) = {8(z)P • A(2). The expression A(z) is called the essential divisor of Q, and {8(z) p the unessential divisor. § 189. 7%e geometric aspect of the discriminant. Let (Id 4, 4) » ('?!) V2i Vi)' ^® ^^^ centres of two pencils of rays where {i-q) = 0, (iC) = Oj (vO = 0, are the equations of the lines connecting |, r; with each other and with a third point ^ whose co-ordinates are (^i, 4, ^3). If w and z be connected by an equation F(w, z) = 0, the intersection of corresponding rays of the two pen- cils traces out a plane curve. Starting with any plane curve C of order a and class ft, it is possible to associate with it an equation F{iv, z) = 0. The form of F will depend upon the choice of the points I, 7j. Assume that |, 7; have no specialty of position with regard to C; this means that |, 17 are not to lie on C, or on any of the singular lines associated with C, understanding by singular lines (1) the join of two singular points, (2) singular tangents, (3) tan- gents at singular points, (4) tangents passing through a singular point. Further, the line (I?;) is not to touch C, and is not to pass through a singular point (see Smith, loc. cit., § 170). Projecting KIEMANN SUBFACES. 2/3 the line (^rj) to infinity, ^ve can now treat iv, z as Cartesian co-ordi- nates. The order a oi F =0 is also the degree in tf, z separately, the line at infinity meets the curve in a distinct points, and therefore the multiple points of the curve lie in the finite part of the plane. It should be especially noticed that when a line z = a touches the curve, its contact is simjile, and that it passes through no multiple point ; also that no two multiple points lie on a line z = a. "When the equation F{ic, 2) = has been put into this special form, it is easy to assign a meaning to the factors of the discriminant; and when the geometric significance of these factors has been found, the extensions to more complicated cases (which arise when the restric- tions are removed as to tangents z = a having simple contact, tangents at the multiple points being oblique to the axes, etc.,) are readily effected. The discriminant Q{z) of F(il\ z) with regard to to is found by the elimination of iv from F=0 and BF/&iv = ; that is, the roots of Q(z)=0 correspond to the points of intersection of F=0 with the first polar of the point at infinity on the la-axis. Eemembering that Q{z) contains ll{ii\ — ic,)-, it is easy to see what factors are contributed by those branch-points which are distinct from the multiple points (i.e. the points of contact of the tangents from $ to C), and by the multipde points of the curve. (i.) If z = a be an ordinary tangent at (z = a, w = b), there are two series w. - 6 = A(z - a)V', w^ - b = Q,{z - ay', and (w, — it\)-xz — a. (ii.) If (a, b) be a node, u\-b = Pi{z-a), io^—b=Qi(z-a), and (to, — iv^)-x (z — a)-, (iii.) If (a, b) be a cusp, w^-b = Ps{z- ay', w,-b= Q,(z - ays and (u\ — it\)- x{z — ay. (iv.) If (a, b) be a multiple point which can be resolved, by the methods of Chapter IV., into " nodes and •>. cusps, then the corre- sponding factor of the product n{tr, — iL;)-x{z — a)-''+^''* (see Chapter IV., § 121). * The equation F= has been bo arranged that no line z = a cuts the curve at more tban two consecutive points; in a more general case, it might happen tliat a tangent z = a cuts in r consecutive points, the remaining a—r points of inttreection being in the finite part of the plane and distinct. In this case, those terms of the product U{Wj. — U'g)^ which correspond to thepoint of contact contribute to ^ a factor {z — ay~^. 274 EIEMANN SURFACES. Thus Q{z) can be resolved into two parts. The first part con- sists of linear factors (z — Zj) (z — Zj) ••■ (2 — 2^) which arise from the ordinary vertical tangents ; the second part consists of multiple factors (2 — 2,')=''.+"i(2 — 2,,')2"2+'''..-.(2 — z/)->-c+2' as the period par excellence. As an example of a singly periodic function, consider exp 2 Triu/ot. Mark off in the 2-plane the series of points «0! «o ± *>; Mo±2 1. (i.) Let X, = 0)2/0)1, be real and incommensurable. Convert it into a continued fraction, and let ni/»2. ni/n^ be consecutive convergents. Then \ lies between Hi/»2 and n^/n^, and therefore difiers from either by less than l/iun^. Hence (^_ni , 0)1 n^ n^n^ where 6 is a proper fraction, and ! ?iiO)i — n20)2 1 < -^ : thus ) jiio)! — n^o)2 \ can be made as small as we please. ELLIPTIC FUNCTIONS. 279 (ii.) Let 0)2/(01 be commensurable and =p/q, where p, q are integers. Because wo/wi = p/q, the expression ?»ii But p, q are relatively prime, hence it is possible to choose mu wij so as to make niiq + mo}) = 1. Lemma II. If ^/{mi + nij + m3) ; and since | m' — m f < £2 and I M I < Xti, therefore | m' | < Xti + c^. Hence | »ni(Oi + m.,u>.2 + "13(03 1 < \'m,i + m.i + m^\ (Xei + e,), which proves the lemma, since mi, m,, "13, \ are finite. It is assumed that the zero-point is not on the join of (Oj and (03 ; if it were, the preceding lemma would apply. The first lemma shows that if /(m) be a one-valued doubly periodic function, with the periods (Oi, (02, the ratio (02/(0, must be complex. A real commensurable ratio would involve the existence of a period (oj/g, contrary to supposition. And a real incommensurable ratio would involve the existence of arbitrarily small numbers amongst the numbers mi(Oi + m2ii>i, and therefore also the existence of a one- valued function /(m), which can be made to assume the same value, as often as we please, along a line of finite length. Such a function must reduce to a constant. Hence the ratio (02/(01 is complex. 280 ELLIPTIC FDNCTIONS. Theorem. There cannot be a triply periodic one-valued function of a single variable. For, if wi, u)2, W3 be the three periods, which are unconnected by a homogeneous linear relation, the corresponding points form a tri- angle. By the second lemma it is possible to choose mj, m,, m^, so as to make wiju), -|- him., + ms^s arbitrarily small. Thus within any region of the plane, however small, the function assumes the value /(!() an infinite number of times. Such a function reduces to a con- stant. The same conclusion holds for a function with a finite num- ber of values for each value of the variable. For the behaviour of a function with an infinite number of values, see a paper by Casorati (Acta Math., t. viii.). Throughout this chapter we shall consider solely one-valued doubly periodic functions which have no essential singularities in the finite part of the plane. It will (lemma I.) always be supposed that tn^/mi is complex. It is an evident consequence of the theorem, that there can be no one-valued function with more than two periods. The inversion problem in Abelian integrals leads to Abelian fnnctions of p variables, which are one-valued in those variables and 2jt)-tuply periodic. By a 27)-tuply periodic function /(!(,, Ji.;, •■•, Up) is to be understood one which is unaltered by the addition to w,, u,, -.•, xtp of the simultaneous periods w,,, u,,, ■ ■■, iiiKpiK = 1, 2, •••, 2p). It can be proved that such a function, if one-valued, cannot have more than 2p sets of simultaneous periods (Riemann, Werke, p. 270). To return to functions of a single variable, some misconceptions on the subject of multiple periodicity are widely spread. Giulio Vivanti has pointed out that Prym, Clebsch and Gordan (Theorie der Abelschen Functionen) and Tychomandritsky (Treatise on the Inversion of the Hyperelliptic Integrals) have assumed that the points in a region at which a K-tuply periodic function /(?;) takes a given value fill that region. Let k be one of these points ; the other points are given by tt + 7K|W, -I- •■• -f m,u<. Now it can be proved that these values of u can be arranged in a determinate order «,, u.^, t;,, ..., so that each value of u occurs once, and only once, with a determinate suffix. Hence although the num- ber of points «,, «2, ?(3, ■•• within the region is infinitely great, the known theorem that the points of the region caimot be numbered in the way just described, shows that tlie region contains points which do not belong to the system u„ ti.^, u.^, ■■■. See Liiroth and Schepp's translation of Dini's work : Grundlagen fiir eine Theorie der Functionen einer veranderlichen reellen Grosse, p. 99. Gopel (Crelle, t. XXXV.) appears to have been the first to point out that Jacobi's theorem on the non-existence of triply periodic one-valued functions of a single variable, does not cover the case of infinitely many-valued functions. To judge by his reply to Gopel, Jacobi would seem to have denied the possibility of the occurrence of such functions. Vivanti's paper is in the Eendiconti del Circolo Matematico di Palermo, Sulle funzioni ad inflniti valori ; there is an accompanying note by PoincarS. ELLIPTIC FU^TCTIONS. 281 § 194. Primitive 2'>airs of periods. If (2 is a period, provided wii, m, be positive or negative inte- gers. Also all the periods of the function are included in the system iiiimi + moojj ; in other words, a primitive pair of periods is one from ■which all other periods can be built up by addition or subtraction. Taking any initial point Mo, mark off in the «-plane the points «o + miuii + moO)2, where mi, 7)ij are integers. We shall speak of the system of points as a network. All the points for which iiii is constant lie on a line ; and all these lines are parallel. Similarly, in., = a con- stant, gives another sys- tem of parallel lines. The two systems of par- allel lines divide the plane into an infinite number of equal parallelograms. Any such parallelogram is called a parallelogram of periods. A point u' = Wo -I- ?jiio)] -f vi„t02 is said to be congruent to Ua, and the equation is written u' = Uo (mod. (1)1, wj), or shortly u' = u^. It is evident that the number of primitive pairs of periods is infinite. For instance, w^ and wj being one primitive pair, uj and oji 4- Wo form another primitive pair. But it must not be inferred th.Tt every pair of primitive periods is a primitive pair. For instance, from oii + o}., and wj— 0)2 we cannot, by addition and sub- traction, recover the original pair (o)], o),). Taking, generally, Fig. 79 (i-) (1)1 = TOi2 IO2 ^ TliWi -\- 71120)2 ) where the coefficients are integers, the pair (a>i', ouj') will be primi- tive, if (ii.) ti)i = /aicji + /U2CD2 ") <02 = VlUli + Vnttl. ' where the coefficients are integers. 282 ELLIPTIC FUXCTIOXS. Now, solving the first system of equations for i/iDi is {u — ifi)/{, OT,)i2 — m.^n, = ± 1, )i.,u,, ) let TOi/jii be converted into a continued fraction. First let the convergent immediately preceding m^/n^ be m^^jn.^. In this case (iii.) is satisfied. Let the convergents, in reverse order, be »»l/»li '"Hhh^ TO3/"3. •••• mr+l/lr+l, and the corresponding quotients g,, q.^, g^, ••., gr+i, so that "h = 9l™2 + "'3' '"2 = 22™3 +TO4'"-") , , \ ■ • ■ ■ (v.). «i = 9i«2 + "ai «2 = 92n3 + "i. "• ■• Then a-^ — m,w, + m.^i.^ — m.^{q^-^ui-^ + Wj) + TOjW,, uj' = ni«i + jijU; = n2(3i«i + W2)+ n^ui^. For the first transformation let us take "1 = "1 •) ELLIPTIC FUNCTIONS. 283 so that Wj' = nijoij + nijUj ■> "We have now smaller numbers to deal with. Proceeding in the same way, we get, if w^ = q.^u^ + Uj, and so on. Now, in the last convergent, either the numerator or the denom- inator is 1. If )ir+i = 1, nir+i = Q'r+i ', the final equations in (v.) are mr = qrrrir+i +1,' Ur = 2r, and the final change of periods is Similarly, if mr+i = 1 ; but if m,> ?i,, this case cannot occur. A simple example will make the process clearer. We wish to change the periods from w,, u^, to 07 a^ + 30 Wj, 29 Uj + 13 w,, by changing only one period at a time. Now 67/29 = 2 +^^^^ ]^- Here s,=2, q,=A, q,=S, q,=2, w,=2^,+u,„ u, = 4 Uj + w„ u^ = 3 Mj -r w., Wg = 2 Uj + Uj, and the pair (0)5, Wj) is primitive. Secondly, if m^/n.^ he not the convergent preceding m^/n-^, let /i/v be that convergent. Then the equations m-^n.^ — m.^)i, = ± require that m^v — Jii/n = ± : either m^ = M + m/, «2 or m, = m,t — 11, n. : >• + n-f, ■ "1' — "1 t being an integer (see Todhunter's Algebra, p. 389). We have, then, Wj' = nijUj + TOjWj = ni[(uj + toj) i/toij -v Wj' = »,u, + njMj = nj (uj + Jwj) ± j/Uj / Thus the single change from «„ Uj, to w, + toj, ± Wj, reduces this case to the preceding one. It may be remarked that in a change of the kind exemplified by (iv.), the new parallelogram and the old have a side in common, and lie between the same parallels, so that they are equal in area. Hence we see again that all fundamental parallelograms are equal in area. Corresponding to the parallelogram of periods, there is for the 2p-tuply periodic functions of p variables a parallelepiped of periods in 2j3-dimensional space. § 195. A one-valued doubly periodic function, whose only essen- tial singular point is at eo , is called an elliptic function. In the present paragraph we shall build up a fundamental elliptic function. A primitive pair of periods will be denoted by 2o)i, 2a)2, although in 284 ELLIPTIC FUNCTIONS. the general treatment of periodicity we shall continue to denote a period by w. Consider the series 2'|--i--A| (A), i (u — ivy iv > where w = 2 miwi + 2 m2<02, u ^ iv, and mj, vi.i = 0, ±1, ± 2, • • •, the accent signifying, as in Chapter V., that the combination nil = m, = is excluded. Here the remark may be made that an accent attached to 2 or 11 always signifies the exclusion of an infinite term or factor. We shall denote the series obtained from (A) by omitting all terms in which ] m; | < ] m | by <^(m), and the series of excluded terms by ip{u). It is first to be proved that the series <^(m) is absolutely convergent. The general term is " '' '■ ~ "/ ' "/ • Comparing this M^(l — ulwy with 2 ufxi?, we see that <^ (m) is absolutely convergent if 2i 2 ujv? be absolutely convergent, where Jj denotes summation for all values of w such that | w | > 1 1( ]. It was proved in § 77 that the series 2'2m/w', of which 1-fiulw? forms a part, is absolutely convergent. Hence <^(m) is absolutely convergent. The terms in i/'(m) are finite in number, and yield an ordinary rational function of u. Hence (A) is absolutely convergent. Expanding each term of <^(m), we see that 1 1 ^2u 2,u^ (u — tuy w^ vf w* and that the series for <^(w), is still absolutely convergent. For if accents denote absolute values, _^ 2u'(l-u'/2iv') ' iv"(l -u'/w'f = a convergent series. Rearranging the series for <^(m) according to ascending powers of M, it is evident that ELLIPTIC FUNCTIONS. 285 71-1-1 ■where c„ = 1^ ~ ■ This equation holds for all values of I m I less than the minimum \w\ in 2,, say {u)+ tp{ji)= an integral series -|- a rational fraction, when | ?t | < p. That is, the series (A) represents an analytic function oiu, and the convergence is uniform, since <^(u)-|-i/^(m) does not depend on p. Hence the differential quotient of the series (A) = ^'(w) + 2ic„ • nw"-' = i/.'(u) -t- 2, ■ Because il/{u) contains only a finite number of terms, 4'\u) = 2 {u — ivy where the summation extends over all the values of lo, which enter into i/'(m). Hence f(M)+S. ~"^ =2' (u — iuy {u — ivy where tu takes all possible values, except w = 0. We have proved — 2 that the series (A) gives on differentiation 2' ;; {tc — wy therefore the series gives j,.(„) = _22^^3 (2). From the composition of J)(m) it is evident that the function has infinities at all points of the network 2mia>i + 2'm.^u)2 (»?ii, mj, =0, ±1, ±2, •••), and that each infinity is of order two. There are no other infinities ; further the function is even, because the change of u into — u does not affect the value of the right-hand side. Similarly )fi'{u) is an odd function of u. We have in the function J3(m), obtained as above, an example of Mittag-Lefler's theorem (§ 150). The series 2' rx is condi- (u — iv)^ tionally convergent, inasmuch as 2'— „ is conditionally convergent. w But the series 2' -I = ; [ is unconditionally convergent. ( {u — wy vr ) 286 ELLIPTIC FUNCTIONS. The integral series for 2' j- -, -„ [ , namely contains only even powers of u, because 2'-j^ = 0, the terms can- celling each other in pairs. To prove the double periodicity of •^{u), we observe that the right-hand side of (2) is manifestly unaltered by the substitution of m -t- 2 i)-hC, and C = 0. Hence J>(w + 2'{u), i3'(M -I- 2a.2) = X,'{u), ^3'(m -I- 20,, -f 2.U2) = J)'(«), give i>'(«>,) = 0,l3'(u,2) = 0, 13' («,,+ «;,) = • • • • (4). The doubly periodic function '^(iC) is the simplest doubly periodic function. Its introduction was due to Weierstrass. While Jacobi's functions sn(?t), cn{v,'), dn{u) are recognized as valuable for nu- merical work, it is now generally conceded that Weierstrass's func- tions ^)(m) and (7-(m) — the latter function will be defined later — form the proper basis for the theory of elliptic functions. rt will be noticed that in one sense four infinities 0, 2a)i, 2o)„ 2(1)1 -|- 2(U2, occur in the parallelogram {2ia^, 2u),), one at each corner, but three of these must be regarded as belonging to adjacent parallelograms. ELLIPTIC FUKCTIONS. 287 § 196. The best insight into the nature of the properties of doubly periodic functions, with periods cu„ m, is probably obtained b}- the use of Liouville's methods. Liouville's theorems are referred to by their author in the Comptes liendus, t. xix., and were com- municated to Borchardt and Joachimsthal in 1847 (see Crelle, t. Ixxxviii.). Cauchy anticipated Liouville in part. Our account of this theory is based on Briot and Bouquet. We denote by A the region bounded by a parallelogram of periods, whose angular points are «„, «,, + ui, Vo + wi -f ^.nd conversely. Therefore the order of f(u) is the number of zeros of /(?t) — k. The theorem states that a doubly 290 ELLIPTIC FUNCTIONS. ' ' ' ' periodic function /(?(), of order m, takes every value m times within A. In particular, the number of zeros of /(«) is the same as the number of infinities. Theorem V. If c,' be a zero and c^ an infinity of f{u) , both situated within A, 2c,-2c/ = 0. More generally, if d, be one of the points at which the function takes a given value k, 2c,-2d, = 0. To prove this, we make use of a theorem concerning residues. If g{u) be one-valued throughout the finite part of the plane, du k~ r 9(tt)f'iu)du ^^ r g{u)f'(u)du ^ r g{u)f{u) Ja f{u)-k J(v f{u)-k Jic,) f{u)-} r)f{u)-k J{d,)f{u) — k *^('r)/(w) — i = 2naig{d^)-g(c^)l. Let g{u) = u, and let Uq be a corner of the parallelogram. Then r nf'(u)du ^ r».+"i f iif'(ti)dn (u + ,^,)f'{u + w^^du ] ^. r"o+"^ ( (u + uy,)f'(u + a>,)du uf'{u)du > A 1 f{u + w,)-k /(M)_fc|' or, by virtue of the periodicity of /(m), = - ">2 r°°"'"'d log [/(m) - fc] + 0)1 r"°'*""''d log [/(m) -A;] = -a.2 log /('to + '-'O-fe , lo„ /K + '"2)-fc /(«„)- fc ^ /(«„)- fc = 25ri(>niO)i + mjMj), where Wi, m^ are integers. Hence 2rf, — 2c, = mi; ~ 2 ' 2 ' 2 ■ If Ci, Cj coincide, so that /(«) has now a pole of the second order in A, /'(m) becomes a function of the third order, with zeros con- gruent to Ci + ^, Ci + ^, Ci + "*' ^ "' . If the origin be taken at a pole, the zeros are congruent to ^', ^^, "^ ^ "" ■ The function ii ^ z f\u) now bears to '^\u) a constant ratio, if the periods be the same. The theorem shows that when the centroid of two non-congruent poles of an elliptic function of the second order is taken as origin, f(u) =/(— m), and the function is even. § 198. Theorem VII. Two doubly periodic functions /u f^, whose networks of periods have a common network, are connected by an algebraic equation. Let wii, mj be the orders of the functions in the smallest paral- lelogram formed by a pair of common periods o)i, ojj. Let ^(m)= 2 2a„,„,/iY2"' (i-) where fi^, fi^ are positive integers at our disposal, and the a's are constants. The function <^(m) is doubly periodic, with periods 0)1, 0)2, and with the same poles as those of /i, f^. If it be possible to choose fij, fi2 so as to make all the infinities disappear, without reducing all the coefficients a, other than a^, to zero, then <^(m) must be a constant, and an algebraic relation connects /i, f^. 292 ELLIPTIC FUNCTIONS. Let c be a pole of <^(it) ; then, in the neighbourhood of c, ^(«)= ^' + ^'-\ +...+J^ + IXu-c). {u — c)" (m — c)«-' u — c The coefficients X,, X,_i. ••■, Aj are linear and homogeneous expres- sions in the coefficients a, as can be seen by substituting in (i.) the expansions of /„ /j, in powers of u — c. Equating to zero all the coefficients \, we get as many equations as there are infinities of 4>{u) in the parallelogram o),, id,; i.e. it.imi + fi^m^. Now the number of a's is (|ixi + 1)(/a, + 1), and ju,i, /aj can always be chosen so as to make (/xi + 1) (^i,+ 1) > fiim, + /ijTOj ; therefore /«,„ ^j can always be chosen so as to make the number of the equations in the a's less than the number of the a's. This proves that there exist functions <^(?() which are nowhere infinite, and consequently reduce to constants. The number of these alge- braic equations in /„ f., is infinite, but the corresponding expressions must be composed of powers of a single irreducible expression and of irrelevant factors. In this unique irreducible expression ;u,, ^ m^, ju,2 must be a constant ; that is, /' is an integral algebraic function of /in the sense of § 185. Corresponding to that infinite value of / which arises from u = c, there are k^ expansions for/' in descending powers oi p'", the initial term being /'«'''''^'. To the kj + kj +•••, = m, infinities ot f{u) correspond Ki + ^2 +•••!=■'"'! branches of /'. Thus/' occurs to the order m in the equation (^(/, /') = 0. To a given value of f{u) correspond m values of w within A, say M„ w,, •", ?f„, and all the remaining values are congruent to Ml, u.,, ■■-, «„. Hence, when /is given, the doubly periodic function /' takes m values f'itii),f'{ih), ••■;/'("».)• Any symmetric combi- nation of the m branches /'(wi), ••■,/'("„.) of <\>if,f') = is an integral function of/. Therefore where S;^f'{Ur) means the sum of the products of the m branches, X at a time, and gi, g^, -•■, (/„ denote polynomials. We have still to determine the orders of gi{f), g-zif), ■■■,9,t{.f) i" /■ Since, when/= oD,the order of each branch /' of /is of the form (k + 1)/k, ^ 2, Sif'{u^) is at most oo- wheu /= x, Sif^u,) is at most oc* when/= cc, S^{u^) is at most cc-" when/= oo. Hence the equation <^(/, /') = may be written /'"- ^iC/)/'-' +•••+(- i)'"-Wi(/)/' + (- 1)""9-»(/) = ... (ii.), where gXf) contains/ at most, to the power /^'. The coefficient 5r„_i(/) vanishes identically ; for Ml + Ma + ■ ■ ■ + ''a = ^ constant, therefore du^/df-ir du^/df + • • ■ + dujdf= 0, or 2 !//'(«,) = 0, so that J7»^i(/) /?-(/) = 0- § 199. A special case of great importance is that in which /(m) is of the second order. Then the equation (ii.) becomes /'- = a quartic in /. 294 ELLIPTIC FUNCTIONS. The importance of this formula renders a more direct proof desir- able. Let the poles Cj, c^ of /(«) be distinct; and let (Cj -f C2)/2 = c. Consider the expression xp, where [/(W) -/(C + o.,/2)] [/(«) -/(C + a,, + cu,/2)]. Evidently tp{u) is doubly periodic, with periods K«)= [/(«) -/(c+a>,/2)][/(i.) -/(c-h<.2/2)] [/(«) -fic+l^:+^,/2)-], and the quartic reduces to a cubic. § 200. Let/(M) be an elliptic function of the second order, and let F(u) be any elliptic function of order m, with the same periods. Then J'(«) = (<^„ + iA».-2/)/x»> where <^„, i/'„_2, x™ a-re polynomials in / of degrees m,m — 2, m. Let us take as origin the centroid of the two poles of /(m), which lie in A. Let the poles, in A, of F{u) be q, c^, ■■■, c„. Let F{u) be separated into an even and an odd part by means of the equations F{u) + F{-u) = 2F,{u), F(u)-F{-u) = 2F^{u). The poles of Fy{u) are both those of F{u) and those of F{— u); hence Fi{u) is an elliptic function of order 2 m, with poles congruent to ± c, where k = 1, 2, ■••, m. From § 197, F^{u) has also 2m zeros, say ± Ci', ±02', •••, ± c„'. The function n!/(«)-/(c;)|/S/(w)-/(c.)| has the same periods, the same poles, and the same zeros as F^{ii,). Hence the ratio of these functions is constant, and we can write ELLIPTIC FUNCTIOXS. 295 The function J'sC'*) has the same poles as Fi{u), and has there- fore 2 m zeros. Four of these are 0, 1. Hence, if s^ = 2'^, we have from equations (1) and (2) of § 195, ^M = 1/m^ -I- SSiW^ -|- 5s^u* + ••• jj'm = - 2/m= -h 2 • 3S4M -I- 4 -Bs^u^ -\- Now if we assume, from § 199, the relation ■f'-u — (4 bfu+6cp^u+ 4 dpw-f- e) = 0, and determine the coefficients from equations (5), so that the relation holds when m = 0, then the equation with these coefficients *It will becoDvenient to write pu, p%, etc.,in9tead of p(?(), [p(u)]', etc., where no Ambiguity arises. This is the custom in the case of such familiar functions as sinx. Further, where only one argument is in question, we shall on occasion write p for pu. The formulae in Weierstrass's theory are from Schwarz. Formelu und Lehrsatze, and for the proofs we are largely indebted to Halphen, FoDctioDS Elliptiques. } (5). 296 ELLIPTIC FUNCTIONS. will hold for all values of u. For since the expression on the left- hand side of the equation is a doubly periodic function, which has no infinities within the parallelogram of periods, it must ^•educe to a constant. This constant must be 0, otherwise the expression could not vanish for u = 0. Now 'p'^u begins with i/u", therefore 6 = 1. Equating to zero the coefficients of l/u\ 1/u-, and the absolute term, we have 6c = 0, 4-3% +4c? + 2'-3-S4=0, 4-3-5s6 + e + 2'' • 5 • Se = 0, so that c = 0, id = — 2^-3-5Si, e = — 2^-5-7ss. Instead of 4d and e, we shall write — g-,, — g^, so that g, = 2^.3.52'^J ?3 = 2^5.72'-„ (6). These are called the invariants of ^m. The relation between pu and its derivative is now p'-u = ip^u - g^fiu - gs (7). We know that the zeros of p'u are cui, cdj, {p) be another polynomial of degree m, Multiplying by p and then making p infinite, we have f'iPr) f{P) so that %±iPA = o, f'iPr) when m + 1< n. In the case under consideration n = 3, and 2-£^ = 0, S^— =0: f'iPr) fipr) therefore 2 -^^; = 0, apr + b or SdWr = 0, or Swr = a constant. Suppose that initially a = & = 0, and therefore that pi = *l! ^^2 = ^2) p3 = ^3- We can take for the corresponding values of u Ml = (Oi, Mj = •=!, 2, 3) we get = Again, we get 1 Vi ^i 1 V2 P2 1 h P'^' ih' - h')/(V2 - ^53) = a ; and since ^51 + p2 + P= = a'/i, Vi + h + h= ip2 - h'y-/i(p2 - hY (12). or, writing u^ = u, Us=v, Ui= — {u + v), and noticing tliat \>u is an even function, , 1 /b'm — t)'v\2 (13). The relation (13) is known as the addition theorem of )pu. Changing the sign of v, we have 1 /b'u 4- B'y\2 i\)()U—pV The addition of this to the previous equation gives (14). ;(pM-pi')' 2i)pu + pvy, jj(M + D) + KM-^) = r,j but ^j'-M + )?"v= iCp'u + fo) - g.,{);su + pv)-2 g^ hence ^3 (m + v) + (3(m — 1)) = 2{pu + »)d) (pHpv - ^§2) - 93 Again, the subtraction of (14) from (13) gives •p'up'v p{u + v) — p{u — v) = - {pu — pvy The equations (15) and (16) give {\)u + pv) {2 pvlfiv — lg.2) -9-3 - p'up'v p{u + v): 2(pM-pi;)2 (15). (16). (17). Notation of periods. To give symmetry to our formulae we shall usually denote by 2 o),, 2 w^, 2 w, three periods whose sum is zero, and which are such that one of the three pairs is a primitive pair. ELLIPTIC FUNCTIONS. 301 It follows at once that all three pairs are primitive pairs. We shall use X, /J., 1/ to denote the numbers 1, 2, 3 in any order. We have always <«A + (O^ + ^. Then pv = e^ p'v = 0, and ^(''+<-^)-iG-^3 '"^ '--' or, since p'h>. = 4:Cpu-e,)(pu-e,)(pu- -63), K" + <«a)- „,-, „ ^«- -Sa e^)pu + e^e^ + e^^ „ , Se^^^^e^e, pu -e^ pu-e^ Therefore \p(u+w^)-e^\lpu-e^l = {e^-e^){e,-e^) .... (18). Again, in (13), let v = u. Then b 2 m + 2 tow = lim - /^ ^'''~^'^ Y, v=u iypu — pv J 4 \ v=™ pM — J)V / ~l[p'u)' therefore p2«= iM^^);:^Mii!ziM^3) _ >3^+L92t)^ + 2g3^> + A.92^ ^p'^-92p-93 Now if we regard Ap^ — gsp —g^ as a quartic in p whose leading coefficient is zero, so that one zero is at infinity and the others at fii, gj, 63, the invariants of this quartic are g^, g^ {% 42). Also the Hessian is (§ 41) H=-{p' + ^g.p' + 2g,p+ i^g,') ; hence H=z-p2u-p"u (19). * When it l8 deelrable to specify a primltiTe pair, we ohall select 4ii>i, 2uj, In conformity with our previous notation. As Welerstrnss selects 2u)„ 2u„ the formulsB will occasionally diverge from those in the Formeln und Lehrsatze. 302 ELLIPTIC FUNCTIONS. It is now proper to see how the Jacobian J is expressed in terms of \i. Writing, for an instant, the quartic as a homogeneous func- tion of. two variables p, q, we have or, since pU^-\-qU, = 4:U and pH^ + qH^ = AH, J=\ U, U In our case U = ^'hi, U, = ^{ip'- g,p -g,) = 2p", hence J= 8p p"u )p'ht — )j)'2u-)p'u—p2u- p"u,—p2u-p'-u = p'^u-p'2u (20). The well-known relation (Salmon, Higher Algebra, 4th ed., § 204) J' = -4m + g,HU'-gsU% which connects the invariants and covariants of the quartic, follows at onoe from the identity p"2u = ip''2u-g,p2u-g,. The zeros of the Jacobian. Formula (20) shows that the Jacobian vanishes when 2u is half a primitive period ; i.e. when u = .2/2 + u}i,3m.2/2 + u>i, (a)i-|-<02)/2, 3(^) = pu, we deduce |-s(;(M + 2<.,)-^t| = o, du and, therefore, ^{u + 2u>)) — t,u = ^ constant. Write M = — (0^; therefore the constant =2^ -, (- l)"«-'(m. - 1) ! /(M - c,)"". The function /(«) - [«!<"'{(« - cj - a2''>r (u - c,) + ... + (_ l)"«-'a„;''(;'"'«-i'(«-cj/(m. - 1) !] has therefore no pole at c,, but it is not an elliptic function, since U" — c.) is not elliptic. But if we remove in this way all the ELLIPTIC FUNCTIONS. 305 irreducible poles of /(m) (i.e. all the poles within the parallelogram of periods), we arrive at the expression K K which is an elliptic function; for ^'(m — c,), V{'^ ~- '^k)) '"' ^^^ elliptic functions, and 2ai''''^(M — c,) is also an elliptic function, because where A = 1, 2. Further, it has no infinities within the parallelogram of periods ; hence it is a constant. Therefore /(w) = a„ + 2a/«'^(M-c,)-2a2"'r(M-c.) + ... + 2(-l)'»«-'a„J''^"»«-"(M-cJ/(??i,-l)! • • ■ (24). Instead of ^', ^", •••, we can of course write —p, — ^5', •■•• This theorem is assigned by Briot and Bouquet to Hermite. [Fonctions Elliptiques, p. 257; Halphen, t. i., p. 461. J § 206. The function (tu. The general term in (21) gives, on integration, log (1 — u/iv) + u/w + ?(Y2'.(;-, = - u'/Stv' - My4M)* . The series just written down is convergent if | m | < | w |. Now however large be the finite quantity u, it only needs the exclusion of a, finite number of values of w, to secure that, for every remaining value of w, the condition of convergence is satisfied. Hence the complete series 'X'\\og (1 — u/w)-\-u/w-\-ii,^/2vf\ \s convergent. By the use of exponentials, and by the introduction of u, we get the function o-(m): — (rw = ?crM (27). Take a triad of periods 2u>^ 2a)^, 2(j^, such that <"a + % + <«^ = 0- By a repeated use of (27) we get o-(m + 2 0);^ + 2 0)^ + 2 (0 J = — e''''A"'+«'x+^^+2«v'+%(»+v+*°F'+*'/»+"^' . o-it, ELLIPTIC FUNCTIONS. 307 Hence Vx + Vf. + V,> = ^ (a relation which can also be deduced from § 204) ; \ and = {2r-l)^i/2 . . (28), when }• is an integer as yet undetermined (see p. 320). Again, from (27), (t{u + 2mi(0i) = — e'''V"+='»i'"i-'"iV(w + 2»n,,(oi — 2o)i) = ( — l)"'ie-"ili'""'""i'"i'crw. Therefore, if ai= inii + n^oti, and ^ = mir/i + ^1.27)2, o-(m + 2 a) = ( — l)",e-'"i''."'+-"2"2+'">"i'cr(jt + 2 7712(02) = (_l)'»,»!+"x+»!e2^("+"'o-u .... (29). From (23), by integration with regard to v, log o-(m + v)+ log o-(m — v) —2 log (TV = log {fu — tjiv) + const, so that o-M, ■write u for u + m^; the resulting formula is i) + m.i{t^.M)^ — 77^(i)o), the factor e- is, from (28), (— 1)"' when \ = 1, (— l)""! when X = 2, and (— l)"*!"^"*! when A. = 3. In (30), write v=b>)^. Then ^u^e, = - "^" + "P"/" " "^^ = f^f ■ - . (35). Writing — ^ = i^, we have from (3o) the differential equations (d|,/d«)^ = (^,= + e,-e^)(^/ + e,-0 . • - (36).* Also from (35), o-/-o-/=(e,-e^)o^, | ^ and (e^-O<^/+(e.-e^)v+(e^-e^)(7/=0J Multiplying together the three equations of the set (35), we have ■^'-U = 4 (o-iMo-jMcrg w/(t^m)^ In taking the square root, the sign is determined by supposing u small. The principal values of \i'u, au, v^u, being — 2/m', u, 1, the value of ^j'm must be '^'u = — 2 iTiU;„, e^ be real and e^ lie between C;^ and «^, «2<1. The reader la referred to Weierstrass-Schwarz, p. 30. 310 where f{u) = kU ELLIPTIC FUNCTIONS. (r(M-c/) exp 2{ixir]i + lJ^ri2)u 2 {cj — cj = 2(/i]«)i + ix^u).,), and /aj, ;U2 are integers, (39), is the most general form for an elliptic function of the mth order, with zeros c,' and poles c,. For from formula (27), when ?i becomes rt +2 a);^,cr(?i — c,')/o-(it—c,) acquires the factor exp 2 7;;^(c, — c/) ; therefore /(m) acquires the factor exp [2 >;^2(c, — cj) + i{iJ.i-r)i + p-2'/2)<"aJ or exp [4 fi.i{rii'u,-f,^"-'^-u 1, fu.2, p'u2-"p <»-2)„ ("-2)„ 1, pu^, ij'M„---p where Mj, u^, •■•, u„ lie within the parallelogram of periods A. This expression, regarded as a function of u^, is doubly periodic, and infinite at m = in the same way as :p'"' ^'m^; that is, in the same way as ( — )"("• — 1)1/11". There are no other poles within A, and there- fore A„ is an elliptic function of ?'*o-(?t — Wi^)/''e~''x"o-(D^o-;^M/o-(o^, = Ve„ — e^ei^" • trcu^o-,?*. AVhen m is replaced by — m in these formulae, a- changes into — o- unaffected. Hence cr{u± (D^) = ± e-iAVu^o-^K, and o-^ is unaffected. Hence «^a(m ± ""a) = T V(e^- Ca) («.- Ca) • etixVa,;,cnt, (42). 312 ELLIPTIC FUNCTIONS. From these, by division, these formulae are equivalent to special cases of (34), namely (43). o-^ (m + 2 o) J = e='!-<"+'".''o-^u ) We see from these equations and (27) that o-^w/cru and a^u/a^u are doubly periodic functions, and that 2(o^, 4(u^, 4to^ are primitive periods. These facts can be also deduced from (35). Example. Prove that o- (m + 0)^ — 01 J = (e^ — e^)e< V~i^*"o^co^o^,(a' + b'-c'-d'), b= ),{a" + b"-c"-d"} c' = iia-b + c-d), c"= i(a'-bi + c'-d'), c= i(a"-b" + c"-d") \ d' =},(-a + b + c-d), d"-l(-a' + b' + c'-d'), d =J(-a" + 6" + c"-d") J and (46), (47), a" = ]{a+b + c-d), a = i{a' + b' + c'-d'), a' = l(ai' + b" + c" -d") b" = \{a + b-c + d), b = i{a' + b'-c' + d'), b' = iia" + b"-c" + d") c" = iia-b + c + d), c = Ka'-i' + c' + (i'), c' = i{a" -b" + c" + d") d" = i(a-b-c-d), d=l{a'-b'-c'-d'), d' = i(a"-b"-c"-d") and also a^ + 6^ ^ c^ + (£2 = a'-' + 6'2 + c'' + d'- = a"' + V" + c"^ + d"' = 2{u- + u,' + u/' + u,'). The equation (45) becomes o-ao-&(rco-d + ;^ d—w^ for a, b, c, d. After removal of a factor, there remains (e^ - e,)a^c<7f,d + (e, - e^)a-^a'a^b'<7/^, b + u>^, c + o->=0, o-;,(«+?;)o-^(M-i;) -o-/M(r/'y+ (e;^-e^) (e;^-eJ(T=MT2(;=0. Again, if we put a" = 0, b" = 0, c" = u + v, d" = u-v, so that a = u, b = — u, c =v, d = v, a' = V, b' = —V, c' = u, d' = — w, we obtain only one new formula, from (51), a^ucT^v — (xJ'va^hL + {e^ — ej)ar{u + v)(t{\i — v) = 0. If we put a = 0, b = u + v, c =u — v, d =0, so that a' = u, b' =v, c' = — v, d' = u, a" = u, b" = V, c" = -V, d" = - u, we obtain only one new formula, again from (51), • • • (58), a-^{u + v)cT^{u—v) = a^-uor^-v — {e;, — e^)aJ'ua--v- ■ • (59), a^{u + v)cr(ju ~ v)= cr^UcTUcT^Vo-^V — a-^Ua-Jlcr/^VaV • • (60), cr{u + v)crf^{u — v) = cr^ucrU/ + ,T,V)^j ; therefore (t3u= o-(o-2a-3 + CT-jO-i + CTiO-a) (— cr^Tj + 0-3 Tj + o-iTj) (o-jO-j — o-jCTi + ctiO-j) (0-20-3 + 0-3O] — 0-10-2) . The formula for o-4?« in terms of o-, o-j, 0-3, 0-3 can be obtained from o-4« = 2o-2uoi2Mcr2L'Mo-3 2M. Example. Prove that, if T;^ = o-;^ 2 u, or 3 W = 0-(r2T3 + T3T1 + T1T2), To express pnu in terms of pu, we have pnu — pu = — If we write ij/, = a-ru/u' u, o{n + l)u ■ o-(n — l)u ahiu • a^u then j„,w _ (3,( = _ li^l^i (66). Again, the equation (63) becomes the factor (r2('»'-^»'^-') dividing out. Noting that \pi = 1, we infer, as on the preceding page, that We have now to express iZ-j, 1//3, 1/^4 in terms of Jj. We have already il/2 = — p', from (31). Again, p2u-'pu = - ,f,s/p'\ and from (19) p2u = - H/p'^ ; therefore i/tj = Zf+ jjp'^. } • • • (67). ELLIPTIC FUNCTIONS. 317 If in formula (16) we replace u and u by 2 m and u, we obtain The left-hand expression is \pi\i' /'fit, and the right-hand one is - Jp'"-/tpi% by formula (20). Hence ,p, = -Jp'. The equations determine i//,, when n > 4, in terms of \j/i, xj/^, 1/^3, 1//4 (Halpheu, t. i., p. 100). § 213. Expression of (ju as a singly infinite product The doubly infinite product in (25) is absolutely convergent, and the factors may be taken in any order. The factors for which wij = give Mn'(l — M/2m,o)i) exp (M/2mi m-^wi), mi = ±1, ±2, •••. 00 Now 11' (1 — x/fi.) exp (a;/ju,) = (sin ■!rx)/Trx ; therefore Mn'(l — it/2«iii also n'e"'''*'"i'"'i' = e"'/''"i'- = e"'"'^''^.'. Hence the selected factors give 2a)i/7r'Sin7rw/2(Ui-exp(7r^My24i • csc^ 7rm2.^ \ ^ jl + (2m;a,;-M) /2i> i,a), j exp j - (2m,o,,-«)/2mi«>i j ^:[^;;:^;^^^;:^^ - ^^_ \1 + ?)l2(U2/?)li(UiJ • exp \— 7)l2,) = Cm + 2,i. In the formula cot J^^ + -^Vcot -f^^ -- derived from (68) by logarithmic differentiation, write u = (dj. Hence 71,0)0 , ff ^ TTCOo TT " (2m, + l)u)2 (2 m2 — 1)0)2' cot n- r. i— — cot TT- K — 4 (Ul J ,. IT ^ (2 7?l., + 1)0), TT . and o)]i;2 — woiji = lim -cot tt-^^ ^^ ■" = ± „ ' ; mo=+oo ' ijiojo — 7/20)1 = - i, when the imaginary part of — is + U)l = — -i, when the imaginary part of — is — 1 0)1 (70). The former assumption will be made (see § 194). This formula is the Weierstrassian analogue to Legendre's rela- tion Eir + E'K- KK' = % (Cayley's Elliptic Functions, p. 49). The formula (68) can be written in a form which is better adapted for applications. Let 0)2/0)1 = lu, m2 = m. Then / . j-TTM \ 2o), . ^M „ Jl «L ''"20,1 o-M = — ism- — eiizo,, n 1 — — 1 I TT ^0)1 m=i\ sin-^Trmo)/ „ sin Ttib) — -^ — \ir J^ sin ^ e"',- n -X ~-^. e--/2.-. TT 2 0)1 '„=i sinTrmo) f 0)j sin( wio) + „^i sin irmo) Writing * e"'"^^"^ = z, e"'" = q, we get . / u sm ir nio) — sin jrmo) g-" — 1 ♦ WeicrBtraaa usca h for Jacobi'a q, and Kk-ln usea r for q-. 320 ELLIPTIC FUNCTIOKS. Hence cm = — ' sm — e'"2w, n -j -^ n -^ 1 — 2o''"cos — + o«" = sm^ e''i2u,, n -5—2 • (71). Logarithmic differentiation gives o^-sin — Cm = ;^— cot ^— + 1?, - H 2 . Za)i /0)i <0i . • /. -= 22r"sinne. 1 — 2 r cos 6 + r- i This is an identity which holds for complex values of r as well as for real values. Let r = q^", where | g | < 1 ; then 1 1 — 2 5"'" cos 6 + 5*"' „=i „=i ^ The condition | g | < 1 implies that the imaginary part of 0)2/0)] is positive. Now let the summation be performed for m, and we have the result r,2n 'l-q" 2i -ssinw^- Thus Cm = ;5— cot;^ h-^H 2 ., „2„ sm — . (72). 2 0)1 2 o)i 0)1 o)i 1 1 — q' Differentiating with regard to u, we have . TUTU 2 o ? ng cos ,„„, ^.=^2csc-^-^-^4i , .r • ^^^)- 4 0)1'' 2 0)1 0)1 a)f 1 1 — g^" Expand in powers of m : we have ELLIPTIC FUNCTIONS. 321 The development of esc" 7rw/2 mi to the lirst few powers can be found by the following considerations : — Let r ( nV 3 15 4 a,/ 27 • 7 16 0.,^ I Equating the coefficients of the powers of u, we have IJ (Dj O)] 1 1 5 ©-4.--^i^. <->■ Many other formulae of this kind will be found in Halphen, t. i., ch. xiii. § 214. Expression ofa^u as a singly infinite product. From formula (71), TT 2 i 1 1 — g^" 1 1 — g^". From this can be readily deduced analogous expressions for the allied functions o-,m, o-jM, o-jM. When m changes into u + w,, 2 be- comes iz, and the exponent iy,?tY2(i)i becomes 77i?i + (rj2 + 7r(/2o)i) (m + 0)2/2) = i/im'-/2 0)1 + t/jM + -^ i/joji + ff m/2 0)1+1 niui. 322 ELLIPTIC FUXCTIONS. Wken u changes into u + wj, z becomes —iq~^^% and riin'/2ui.^ becomes i7,My2 coi + ?7iW3/<"i ■ (w + W3/2) = ■>7iM-72 . 2- ' . VtV TT _ 9-''"g + 9'''-z-' \ fi (1 + ^-"+'2-') (1 + g-^-'z-) _ _ V?(Oi g 2^^ +i=«+ii3"3 » (1 + g'"-'2-') (1 + q^'-'z') ~ Trq^^* 1 (l-g-"')2 ' so that, putting OrJ Qi* Let the discriminant of the cubic 4 ^j' — g^ys — ^3 be A ; that is, let A =gi - 21 gi = 16(e, - 63)^(63 - e,y{e, - e,y. From the preceding equations A = ^ q Qo'' But Q0Q1Q2Q3 = n(i - g^'»)n (1 - q*-"-') = Qo. 324 ELLIPTIC FUNCTIONS. Hence QiQiQs='^, and A^fLY'qma-q'-y' (82). In this formula take the logarithms of both sides and differ- entiate with regard to q. This gives rilog(Acoi'0 ^ ? _ 48 2 ^ dq q m=i 1 - g"" dlog(A..")^,_^g| mg^. dlogg m=il — r" Comparing with (74), we have* _7r' d log (Aa,i") '""^-24 dlog9 ' or, since log q = ttim, TT dlog(Aa)i'^) '''"'=24i — d;;^ — ■ ■ If i 0)2 r»72. i-i. • SloffA , SlosfA ^n we obtain <"!— r^ \- 1^2 — =-= = — 12, 8(1)1 Scu, which can be written down at once from the fact that A is a form of degree — 12 in 2i Vi> V2f 92' 9i, log A, we refer the reader to Halphen, t. i., ch. 9, and Klein-Fricke, Modulfunctionen, t. i., pp. 116-123. * For other proofs of this formula, eee Halphen, t. i., pp. 259, 321. ELLIPTIC FUNCTIONS. 325 § 215. The q-series for uu and a-^u. Let <^„(s) = n (1 + g2»-is) (1 + g2„-ig-i) fn=l = do + ai(s + s-') + ■.. 4- a„(s" + s-"), where o„ = gScsm-D _ ^n!_ For s write qh. Then (1 + q'-^h) (1 + g-'s-') '^»(9's)/<^,(s) = (1 + qs) (1 + q"'-'s-') 1 + 9^"+is gs + q'" Multiplying up, and equating in the two series the coefficients of s"+^, we have or a„ =«»+i- ^.„^i_^w whence a.-i = a„ . ^,„_r ^^,„^i = g'"-"' • yz:^> 1-9^-^ .„-„^ (1-9^) (1-9^-^) ^m2 (1 _ q*") (1 _ g^"--') ... (1 _ (ftm+n+l)~^ When n is made infinitely great, and m ^ an assigned integer k, lim a„ = g'-yQo; and when m> k, the absolute value of the denominator lies between fixed finite limits, namely, the least and greatest terms of the sequence \i-q'\, \l-T\\l-q'\,-. Therefore <^.(«) - SI + q{s + s-') + ... + q'-'is' + S-«) j/Qo 7=1 where | 6,+, | also lies between fixed finite limits. The series ^q"' (s" + s"") is absolutely convergent, for it sepa^ rates into two absolutely convergent series ; hence the right-hand 326 ELLIPTIC FUNCTIONS. side of the last equation can be made as small as we please by taking k large enough. Therefore Q„n(l + g2'»-'s)(l + g="'-V) = l + ig""(s"'+s~") • ■ (86). This important algebraic identity appears in Gauss's Works, t. iii., p. 434 ; and in Jacobi's Fundamenta Nova, §§ 63, 64. Cay ley's statement of Jacobi's proof will be found in his Elliptic Functions, § 385. See also Briot and Bouquet, p. 315 ; Clifford, Works, p. 459. Tlie above method is due to Biehler, Sur les Fonctions doublement periodiques considerees comme des limites de fonctions entieres, Crelle, t. Ixxxviii. See Enneper-Miiller, pp. 97, 113; Jordan's Cours d' Analyse, t. i., p. 148. Let s = z^ = e"'"''"! ; the right-hand side of the above identity becomes + 25003 |-29*cos h •••, or, in Jacobi's notation, 63^^. 2 0)1 Therefore, from (79), 2(01 When « = 0, this equation becomes and therefore ^^u=evy^^eJ^ye,{0) (87). If in the preceding -work we write iz instead of z, and therefore M 4- wi instead of u, we obtain in the same way „,v = e^^"'^'''.ef~\/e{0) (88), where 6i-= 1 — 2^ cos2 y + 25*0034^ — •••. Again, in (86) let s = qz-. Then Qo(l + 2-') n (1 + q'-^z-) (1 + q'-'z-') = 1+2 (g'-'+^gS™ + q^'-^z^-^), or Qo{z + 2-»)n(l +q'-"'z') (l+q^-"z-') = z+z~'+ 2 g'»c»+»(22'»+i+2-2"'-i) m=I = q-w 2 qy^ cos ^ + 2 g'/^ cos ^ + 2 g^/- cos ^ + 2a,i 2a)i ' ^ "" 2 2, the hyjierelliptic integrals form only a portion of the complete system of Abelian integrals. The values (x, y) which belong to F{x, 7/) = 0, and all rational functions R(^x, y), form what is termed by Weierstrass the algebraic formation or configuration ('Gebilde'). The properties of the Abelian integral and of every function of the integral must be ultimately dependent upon the properties of the algebraic configuration. The method which we have adopted in this chapter starts with the elliptic functions and leaves out of account the connexion with the configuration. In the next few paragraphs we shall follow another order, and endeavour to indicate the connexion of the elliptic function with the elliptic integral. § 217. If F{x, y) = be of deficiency 1, the corresponding Rie- mann surface is triply connected and is reducible to simple connexion by two cross-cuts. Is the equation reducible in all cases, by bira- tional transformations, to one of the form y- = Co(a; - a^) (x - a,,) (,r -a^) (x - a^) ? In other words, can there be found transformations Xi = Ri(x, y), >/i = E.2{x,y), which, give x= i?,'(a-„ y,), y = RJ{Xi,y^), and trans- form F{x, y) = into y,^ = a quartic in x^ ? Before showing that this can be done we must explain briefly how the theory of the algebraic Eiemann surface can be connected with the theory of the algebraic plane curve. Among the rational functions on the surface, we are entitled to select two, x and y, such that the equation between x and y, when regarded as a plane curve, presents no other multiple points than simple nodes (§ 189). We can further assume that the nodes and ELLIPTIC FUNCTIONS. 329 branch-points lie at a finite distance. This curve will be called the basis-curve. , , Let any rational function of x and v be , , where <6, and d, are rational polynomials. Instead of the jjlaces on the surface at which this function takes an assigned value iv, we can speak of the points on the curve F, which are cut out by the curve <^i — ivcj).; = 0. "When 10 varies, we obtain different groups of points, in which the curve F is cut by the curves of the pencil <^i — iv4>-, — 0. Any two of these groups of points are said to be equivalent or coresidual. Through any point of the curve F there passes generally but one curve of the pencil, and the rational function has a unique value. But when a basis-point of the pencil is on the curve F, <^i/<^.; takes at this point the form 0/0 ; and its true value must be determined as the limit of the values of 4>i/4>i at adjacent points of F. That is, at a basis-point on F, the value of i/i is obtained by taking that curve of the pencil which touches F at the point. When a basis- point coincides with a node of F, there will be two curves of the pencil which touch F at the node, and two distinct values of <^i/4>2- Particular imirortance attaches to those pencils of curves which pass through all the nodes of F. These are called adjoint curves. It will be noticed that we are not concerned with the pencil i — tc2 = as a whole, but only with its intersections with F; these alone correspond to places on the Eiemann surface. And also that imaginary intersections are to be included as much as real ones. It must not be supposed by readers acquainted with the theory of curves that the connexion with the theory of Riemann surfaces is solely in the interest of the latter ; the curve is by far the greater gainer. § 218. Let the curve F{x, j/) = be of order k and deficiency 1. The number of nodes is ^ k (k — 3) ; and the number of points through which a curve of order k — 2 can be drawn is ^(k — 2) (k -f 1). Hence, since -J-(k — 2) (k -f 1) — ^ k{k — 3) = k — 1, a pencil of adjoint curves i — wt/), = can be drawn through the nodes and through K — 2 other assigned points of F. Each curve of the pencil has, with F, k(k — 2) — k(k — 3) — (>-• — 2) remaining intersections. Therefore the function lo or <^,/<^2 takes an assigned value twice ; that is, we have a function which is one-valued and two-placed on the Eiemann surface which represents ?/ as a function of x. Let this surface have m sheets ; as j/ is to be rational in x and lu, it is necessary that to the m places in a vertical line be attached m dis- tinct values of w. Thus the equation between w aad x must be of 330 ELLIPTIC FUNCTIONS. order m in w; and it is of order 2 in x. Let it be Fi(3?, zi-°') = 0. The following considerations show that y is now a rational function of X, v). Given a jiair of values x, iv, there is only one correspond- ing y. Hence the two equations w = —, F{x, y) = 0, when treated as equations in y only, have a single common root. They therefore give on elimination ?/ as a rational function of the coefficients, i.e. y as a rational function of x and ic. Thus the necessary condition for passing birationally from F{x,y) = to Fi{x,w) = is also the sufficient condition. The new equation Fi(x-, «,•"*) = can be transformed, in a similar manner, into F,{z-, w-) = 0, by the use of a function z which is two- placed on the surface which represents a; as a one-valued function of w. The new equation is f,ir + 2f,w +/, = 0, where /o, /i, /a are polynomials in z of orders ^ 2, and leads to /o where Q is a quartic in z. Let y^ = VQ, Xi = z; then the equation between w and z passes into 2/r=<2(a--i). Here the last transformation is birational, for t/, satisfies the necessary and sufficient condition ; namely, it takes two values when z is given. This equation, which is the standard form for the case j9 = 1* can be written 2/i" = Co(a;i — aj) (x^ — a,,) (Xj — a^) (Xi — a^); and this can be brought into the form used by Weierstrass by the birational transformations Xi - a^ = - 1/Zi, the new branch-points in the z,-plane are e^, e„, e^, oo, where ^A - «4 = - l/e^ X = 1, 2, 3 ; and the new equation is - cht\' = Co(z, - e,) (z, - e,) (z^ - e^/e^e^^, or, if _ 4 c' = Co/eifoes, w/ = 4 («, - ei) («i - e.,) (2i - e,). ELLIPTIC FUNCTIONS. 331 "We can suppose the origin in the Zj-plane transferred to the cen- troid of the points e^, so that €i-\- e.2 + e« = 0. The transformations of x^ and Zj have been bilinear. Hence if the quartic Q(j'i) be e„V + 4 Cjcr/ + 6 c^^ + 4 c^x^ + c^, whose invariants are fj., = c,/4 — 4 C1C3 + 3 ci, the invariants of the transformed expression will be of the forms 0-2 = firg.,, (ji = itPcj,^. But in the transformed expression Cq' = 0, c/ = 1, c>' = 0, and therefore cjj = — i.cJ, rjJ = — c^. Hence the new equation is iv' = -iz'-rj.:z-rj.^, where fj.! = ,x"rj.,. gJ = ^"r/j, or, if we replace 2 by ix.z, and w by V/a^it', • il-^ - £73- If- = 4 ar' ■ The conclusion of this i^aragraph is that the values of x and y which satisfy an equation of deficiency 1 can be expressed rationally in terms of z and y/'iz^ — g.z — g^, and therefore also they can be ex- pressed rationally in terms of pit and p'ii. [Klein, Einleitung in die Geometrische Funktionentheorie, pp. 208, 209 ; see also Thomae, Abriss einer Theorie der Funktionen, p. 122 ; Riemann, Werke, p. Ill ; Clebsch and Gordan, § 20 ; Salmon, Higher Plane Curves, § 366 and § 195.] § 219. EUiptic Integrals. The consideration of integrals of the form | R{x, y)dx, where x and y belong to the elliptic configuration, is now reduced to the con- sideration of I R{z, w)dz, where v:'- = 4 r — g.z —g.. The simplest integral, setting aside those which are expressible /dz — If we suppose that M = when 2 = cc , we have, as implied in the last paragraph, 2 =PM, 10 = p'm. 332 ELLIPTIC FUNCTIONS. Hence i R{z, iv)dz— if{u)du, where /(h)= ^(pM, ^3'«)p'«• Since this last expression is an elliptic function of m, the elliptic integral is also the integral of an elliptic function with regard to the argument. In § 205, any elliptic function was expressed by means of t,{u — c^) and its derivatives, where c, was a pole of the function. Therefore in the integral of such a function there will occur : — (1) a term agii ; (2) the terms 2 a/"' log log ~~- - 2a,<«)^< +/i(«), where /i(«) is an elliptic function with the same periods as f(u). The integral m or I dz/iv is an integral of the first kind. Its distinguishing characteristic is that on the surface T, which represents TO as a one-valued function of z, it is everywhere finite. The func- tion ^u or — I pndu takes on the surface T the form -f j ' zdz/iv, and is an elementary integral of the second kind. It has the single infinity z = co, and if we write z = l/l^, it behaves near t = like -l/t. By § 204, we have l{u - c) - ^u + ^c= {^'u + ))'c)/2(^3M - pc). ELLIPTIC FUNCTIONS. 333 On integration, .2. To map the z-sitrface on the u-plane. From § 157, z is a one-valued function of w, and it has just been seen that at two places of T' which have the same z, the values of u are in general different. Therefore to a given u corresponds only one place of T', and therefore only one sheet is required in the u- plane. In general du/dz, = l/ic, is neither nor 00, and there is isogonality. But near e^ we have du/dz =■ — pj-sJ^. — eA, Vz - e^ X = 1, 2, 3, and therefore k — w^^ = P](vz — e^), where 0)3 = 0)1 + 0)2; whence it follows that while z turns roand e^, u turns half as fast, in the same sense, round W)^. Kear z = x, m = jPi(1/2'^-), and again the isogonality breaks down. The region of the t<-plane into which the surface T' maps is bounded by that curve into which the boundary of T' maps. Let (z, It') start from the junction at C3 and describe the positive bank of A upstream ; and let L^ be the corresponding p)ath of 11. When (z, ic) continues from Cj along the negative bank of B downstream, let M_ be the path of u. While (2, w) describes the negative bank of A downstream, m describes a path L_, which is obtained by dis- placing i+ through — 2o), ; and while (z, w) describes the positive bank of B upstream, u describes a path M+, which is obtained by displacing M_ through 2a)2. Since (z, iv) has now reached its starting- place, the whole w-path is closed. Thus the boundary of T' maps into a curvilinear poraKe?ogfra9?i. Since u is finite for all values of 2, the whole of the surface T' is mapped on the interior of the parallelogram. By giving a suitable form to the cross-cuts, the parallelogram can be made rectilinear. It is convenient to suppose that the bridge from e^ to e^ is shifted with the cross-cut A, so as to remain within it. ELLIPTIC FUXCTIOXS. 335 Now let us remove the restriction that z is not to cross A or B. Suppose that z crosses ^1 positively ; then the integral has a new range of values, which are obtained by increasing the old range of values by 2 wy To represent these we require a new parallelogram in the !(-plane, obtained by shifting the old parallelogram through o ^0),. In this way we require, to represent all possible values of the integral on T, a network of parallelograms, which cover the «-plane completely and without overlapping. A path on T which starts from a place (Zo, ?('(,) with a value u^, and returns to this place after crossing A «i+ times positively and ?n_ times negatively, and B m^' times positively and m_' times negatively, gives as the final value of the integral Wo +2 mo)i + 29ii'cD2, where m+ — m_ = m, m^' — in J = m' ; and therefore the pair (Zf,, ir„) is represented by infinitely many points of the w-plane, all congruent to Uq. Tlius z and to are one- valued, doubly-periodic functions of u. When (z, ?(,■) is restricted to T', the correspondence of tt and (2, ic) is (1, 1), and the correspon- dence of z and to is (3, 2); therefore the correspondence of u and z is (2, 1), and that of u and w is (3, 1). In other words, the orders of the elliptic functions 2 and iv are respectively 2 and 3. In passing, it should be pointed out that a new idea makes its appearance here. Since u is an infinitely many-valued function of 2, the ordinary Riemann surface which represents u as a function of 2 would cover infinitely many sheets of the 2-plane ; but the infinitely many parallelograms serve equally well to show the connexion be- tween z and u. The importance of this idea, which consists in the replacing of sheets lying one below the other, by regions of a plane which lie side by side and cover the plane without gaps, has been shown by Klein in his memoirs on Modular Functions in the ]\Iathe- matische Annalen. See Klein-Fricke, Modulfunctionen, 2yaf!sivi, and Poincare's papers on Fuchsian and Kleinian Functions in the Acta Mathematica. In this paragraph it has been assumed that the ratio (oo/wi is complex. This fact follows from a general theorem on the periods of Abelian Integrals, which will be proved in Chapter X. It will appear, in the same chapter, that on a surface of deficiency p there are p linearly independent integrals of the first kind ; whence it will follow that in the present case every integral of the first kind is of the form au + b. 336 ELLIPTIC FUNCTIONS. § 221. It follows from the equation du = f?2/2V(2 — e,) {z — €2) (z — €3), as in § 181, that to straight lines in the w-plane there correspond in the z-plane curves which obey the relation T — ^- 2<^^ = constant, where t is the inclination of the tangent at any point 2, and <^^ is the amplitude oi z — e^. These are the curves along which the cross-cuts must be drawn, when the parallelograms in the it-plane are to be rectilinear. In general, these curves are transcendental, but they contain a system of algebraic curves when 02 and g. are real. In this case, either the branch-points Cj, e,, 63 are all real, or one is real and two are complex and conjugate. When the discriminant > 0, the branch-points are real. Let e.22a\, or, from (30), = cr{3 a + v)(r{3 a — v) /a-{a + v)iT{a — v)cr* 2 a. In § 211, let a, b, c, d =2a, 0,3a + v, 3a —v; then a', b', c\ d' = 4 «, —2a,n + v,v — a, and a", b", c", d" = « + v, a - v, 4 a, -2 a. Then (49) gives a^ 2 a V/((/) which is oc- at {y, V/(^)) only. Let '^'^^' y' be the function. Since this function is to be cc^ for Mx, y) (a;, y/f(x)) = {y, -\/f{y)), assume -'p'{u ; J/s, f/s)' Thus J), ^' are also covariants with regard to the two sets of variables Xi, x^ and t/i, 2/2- They are of orders 0, 0, and of weights 340 ELLIPTIC FUKCTIONS. 2, 3. It will be remarked that u itself has equally a covariantive character, and is of weight — 1. Since the expressions V/(a;) V/(.V) + ">/ , ^u, are covariants of weight 2, c and c' must be invariants of weights 0, 2. When / is written in Weierstrass's normal form we have V7(2/) = 0, «/«/ = 2 x^x,. Hence ay Xo Therefore c = 2, c' = 0, and these are the values in the general case, since c and c' are invariant. Therefore vjM = v:rv)y/x y)+«.v For other proofs of this formula see Haljihen, t. ii., p. 358 ; Enneper-Miiller, p. 27 ; Klein, jNIath. Ann., t. xxvii., p. 455. Eeferences. In this chapter we have made constant use of the work of Sohwarz, Formeln \\aA Lehrsatze nach Vorlesungen und Aufzeichnungen des Herrn K. Weierstrass ; and of Ilalphen's Fonctions EUiptiques. We are also indebted to Miiller's edition of J]nneper's Elliptische Functionen, Theorie und Geschichte (where also an abundant bibliography will be found) , and to Briot and Bouquet's Fonctions EUiptiques. On the important subject of modular functions the reader will not fail to consult Klein-Fricke, Theorie der Elliptischen Modul- functionen. Weierstrass's theory was first made accessible to the English-speaking public by Daniels (Araer. Jour., tt. vi. and vii.) and Forsyth (Quar. Jour., t. xxii). In Greenhill's recent work on the Applications of Elliptic Functions this theory is developed side by side with the older theory. Among other books whose main object is the theory of elliptic functions we may mention those of Bobek, Cayley, Durege, Konigsberger, Thomae (Theorie der Functionen einer complexen vcran- derlichen Grosse), and Weber. Cayley 's work has been translated into Italian by Brioschi. Keference must also be made to the great originators of the subject, — Euler (Institutiones Calculi Integralis), Legendre (Fonctions EUiptiques), Abel (Works), Jacobi (Works), and Gauss (Works); to whom we may now add Hermite and Weierstra.ss. For information on the early history of the sub- ject the reader may consult Konigsberger's valuable work, Zur Geschichte der Theorie der Elliptischen Transcendenten in den Jahren 1820-9, and Enneper- Jliiller's book alreadv mentioned. CHAPTER VIII. Double Theta-Fuxctions. § 223. Consider the double series ^ i_^ ^ 2^ eKipni[Tn{n,+g,/2y+2r^{n, + rj,/2) {n,+rj,/2) +T,,{n2+9-2/2r+2(n,+g,/2){v, + ]i,/2) +2{n,+g,/2)iv,+h,/2)^ (1), where g^, g,, hi, h., are integers, and n^, 71^ are supposed to take all integral values from — co to oc independently of each other. This series converges absolutely when the quantities t„, = a„ + J/3„, are chosen so that /?u>0, fe>0, /3,,^1, by the extension of Cauchy's integral test which was employed in § 77. The terms of the series are the ordinates of the surface 4 = (A.^f + 2 MA + M-r +2 2/:li + 2 2/,4)-' at the points of a network formed by unit squares in the plane i^^.,- The ordinate ^3 is infinite only at points in tlie plane $^$-2 which lie on a conic ; and the condition /3i/ < finP-n ensures that this conic is an ellipse. "When this ellipse is referred to its axes, the surface takes the form is = iyuVi" + y22V-2' + yo)"') where yn and 722 have the same sign. Let a contour be drawn in the plane |,fo, which encloses this ellipse, and conceive a cylinder drawn through the contour, parallel to the axis of ^3. The volume contained between the surface and the plane, exterior to the cylinder, is V=ffuir„dr„; or if Vyui^i = p cos 6, Vy2>i?2 = p sin 6, pdpcie -ff: Vy,iy2o(p- + yo)'' V is finite if the integral remain finite when p = 00 ; that is, if k > 1. Now the sum of the ordinates is equally the sum of parallele- pipeds whose bases are the unit squares of the network ; and if we assign to each square that ordinate which is furthest from the origin, these parallelopipeds lie within the volume V; and their sum is convergent when V is finite. Within the cylinder there is only a finite number of squares, and therefore when those ordinates are excluded which stand on the ellipse, the sum of all the ordinates is convergent. In the case when the ellipse is imaginary, the cylinder may be dispensed with. The excluded ordinates annul the real part of the exponent in (1), but do not affect its convergence. Accordingly, the series (1) is absolutely convergent. [Picard, Traite d'Analyse, t. i., p. 265 ; Eiemann, Werke, p. 452; Jordan, Cours d'Analyse, first edition, t. i., p. 1C9.] DOUBLE THETA-FtJNCTIONS. 343 § 224. Subject to the inequalities already stated, the series (1) defines a function of v^, v.,, which is one-valued and continuous for all finite values of the variables. This function is called a Double Theta-F unction, and is written in one or other of the forms e 9i (v^ Vo); e 9i 92 h,. (Vl, V2, Til, Tia T22); the latter form indicates that the moduli, or parameters, are th, tj,, T22. The symbol " is called the mark, or characteristic, of \_Jh IhJ the function ; and the quantities g^, g.,, hy, lu are called the constit- uents, or elements, of the mark. We shall often write the theta- function more shortly (v.)- § 225. Differential equations. When the general term in (1) is differentiated twice with regard to v„ it is multiplied by a factor l2Tri{n^ + g^/2) s^ ; and when it is differentiated once with regard to T„, it is multiplied by ■7ri{n, + g^/2y. The ratio of these factors is 47rJ, and is therefore independent of n^, n^. Hence a-(9/Siv- = 4 7riW/hT„ (?• = 1, 2) ; and similarly, h-e/hv^hv,=z2wih6/hT^. § 226. The addition of even integers to the constituents of a mark. In the series (1) replace (/, by g^ ± 2. The effect is the same as that produced by writing ?!, ± 1 tor ?ii. But ni±l, like ?ii, takes all integral values from — a: to +00. Hence the series is not altered, and we have the theorem that e 9-2 h. (i;,v,) = e 9i 9-2 ho. (I'l, fo). Similarly, g., may be increased or diminished by 2, without affect- ing the value of the function. Next rei^lace h^ by li^ ± 2. The general term of (1) now acquires a factor exp ±2T:i{ny-\-qi/2), whose value is (— !)'■. Similarly, when h., is replaced by /t^ ± 2, the general term acquires the factor (— I)*-'. By a combination of the preceding results, it results immediately that 6\9, r + ^K K + ^l^r {V,) = (-l)^r>'re (Vr) (2). 344 DOUBLE THETA-FUNCTIONS. With the help of this formula it is always possible to replace the integers g^, \ by or 3 ; this leaves the original theta-function unchanged, except perhaps as to sign. The general formula for such a reduction shows, for example, that e 3 8' 4 15. {v„v.^=6 1 .0 1. (■Ul, Vs)- § 227. Even and odd functions. Changing the signs of Vi, v.,, 9r the general term in d {-Vr) is exp ^i[(rn, r^, r„) {n,+g,/2, n,+g,/2Y+2^^{:n,+gJ2){-v,+K/2)'\. This may be written exp iri[(Ti„ T^, T2.,)(— «i — gi/2, — 712 — 9'2/2)- + 2 2^( - n^ - g,/2) (tv + ^2) + 2 I Ov + 9r/^)K]- Now exp iri 2 (2?i,7t, + g^li^) = (— l)-^!-, and the remaining factor differs from the general term of (1) only as regards the signs of ''h + 9'i/2, M2 + g>/2. But — («, + gr/2) runs through the same set of values as n^ + gr/2, of course in the reverse order. Hence we have \k (^-Vr) = {-iygJ'rd (r.). This theorem shows that a theta-function is even or odd, accord- 2 ing as "^g^ is even or odd ; it leads to the division of marks into even or odd, according as the corresponding functions are even or odd. § 228. Tlie theory of marks* The symbol '91 9-2-Up posed of the 2p real numbers g„ Ji„ is the mark associated witli a ^-tuple theta-function. We shall be concerned solely with marks whose constituents are integers. When the constituents of one mark differ from those of another mark by multiples of 2, the marks are said to be congruent to each other. Thus gi' 9-2 ■ •T/; = 9i 92- -9, h' h,'. ••vJ bh h,. ..h * Clifford, Matbcmutical Papers, p. 366. DOUBLE THETA-FUNCTIONS. 345 when g^ = gr/ (mod 2), and /i, = hj (mod 2), r= 1 2 ••• » Every mark is congruent to a reduced mark, whose constituents p are 0, 1. The mark is said to be even or odd, accordiug as 2gr/i, is 1 oven or odd. The ninnber of eveii and odd marks. The number of reduced marks which can be formed with 2p integers is evidently 2'^. Let Ej, be the number of even marks, i^ the number of odd marks. The first column is (0, 0), (1, 0), (0, 1), or (1, 1). The first three, when placed before a given mark, leave its character (as even or odd) unaltered ; the fourth alters the character. Hence Therefore F, = 3F,_, + E, 'p-i- E,-F^= 2{E,_,- F,_,) =■■■= 2'{E^_^ -F^)=-= 2^-\E,-F,) ; that is, E^-Fj, = 2". But E^ + F^=^ 2-" ; hence E^ = 2-f-'- + 2"-', F^='Jr^-^ -2^-\ In this chapter we consider the case j5 = 2. The number of reduced marks is 16, and of these 6 are odd. These 16 marks may be arranged in the order 00 00 10 00 01 00 00 10 10 10 01 10 (00) (12) (5 6) (3 4) (2 3) (13) (14) (2 4) 00 10 01 11 01 01 01 01 (4 5) (3 6) (4 6) (3 5) 00 10 01 11 11 11 11 11 (A), (16) (2 6) (15) (2 5) and, for this arrangement of marks, B "0 0" 10 (vi, I'o) can be replaced. in what is called the current-number notation, by 65(1)1, v.^, and simi- larly for the other functions. It will be observed that df,, ^j, ^u, Q^ ^ui ^15 are the 6 odd functions. The meaning of the third row in the table will be seen in the next paragraph. § 229. The sum or difference of two marks is found by adding or subtracting the corresponding constituents. Thus ri3n + ro9-| = r6i2-| = rooi. L24J L62J LseJ LooJ 346 DOUBLE THETA-FDNCTIONS. by Let us denote the reduced odd marks, namely, 11 01 01 10 10 11 01 01 11 11 10 10' (1) (2) (3) (4) (5) (G), 00 and the even mark „ „ by (0). From the definitions (l) + (3) + (o) = (0), (2) + (4) + (6)^(0), and therefore (1) + (2) + (3) + (4) + (5) + (()) = (0) • This shows that the sum of any odd marks = the sum of the remaining odd marks. Further, (1) + (3) =(5), (2) + (4) = (6), etc. Excluding (0), the other 15 marks are congruent to the sums of the 6 odd marks two at a time. AVe have just proved this for the odd marks, and have only to show that, of the 15 duads formed from the 6 marks, no two can give congruent marks. First (1) + (2) can- not be congruent to (1) + (3), since (2)^ (3); and, secondly, (1) + (2) cannot be congruent to (3) + (4); for and therefore (5)5^(6), (1) + (2) + (5)= (3) + (4) + (6); (l) + (2)5^(3) + (4). Example. Prove that any even mark is congruent to the sum of three odd marks, in exactly two ways. AYe shall name each of the fifteen reduced marks, (0) excluded, by that duad to which it is congruent. For example, since 11' .0 LiiJ + 10" Lii. = (3) + (4), we shall denote "11" 0. by (3 4). The corresponding theta-function will then be 63 4 (v,.); and the excluded function will be written df,^{vi). It may be remarked that for an even mark, (0 0) excepted, the sum of the numbers in the duad is odd, and for an odd mark the sum is even. The table (A) of the preceding paragraph gives the connexion between the marks and the duads. As in the case of the single thetas, the notations used for the double thetas are numerous. Tables for comparison are given in Cayley's memoir (Phil. Trans., t. 171, 1880), in Forsyth's memoir (ib., t. 173, 1882), and in Krause, Die Transformation der Hyper- DOUBLE THETA-FUNCTIONS. 347 elliptischen Funktionen erster Ordnung (1886). Cayley's current number notation is from to 15, instead of from 1 to 10; and he uses the letters A, B, C, D, E, F instead of 1, 2, ■••, 6 in forming the duads. The notation of duads employed here is based on Cayley's, and agrees with that of Staude if we replace 6 by in the duads (Staude, Math. Ann., t. xxiv.). § 230. The periods. Eeferring to (1), the change of -y, into i\ + ij.^ is equivalent to the change of 7i, into /i, + 2/x,; and, therefore, if 2 fi^ be an integer, 6.9r iv, + ^,) = e 9r ■ (IV) (3). From (2) and (3), if /n,. be an integer. k\ M- Let /ii = 1, ;uo = 0. When g^sO (mod 2), (I'l, v^j, 6,9r \k {i\ +1, v.2)=d g. and we say that 1, are a pair of periods of the function ; but, when gi = l (mod 2), 2 and are a pair of periods. Another pair of j)eriods is 0, 1 when g., = (mod 2) ; 0, 2 when g.2=l (mod 2) . Write for shortness <^ (Xi, Xo) = TiiXi^ + 2 Tj^iX, + T2^2'- Then the typical term in (1) is exp ni [<#,(«! + g,/2, iH + g,/2) + 2 2(«, + g,/2) {v, + V2) ]. Consider the effect of changing n, + g^/2 into ji, + g^/2 + X,.. The expression in square brackets is increased by 2(n,4-J7,/2) ^-^(^ + ^(A,, X,) + 22A,(iv + V2), and therefore it becomes 4>{n, + g,/2, n, + g,/2) + 2 2(«, + g,/2)U + \ ^-^^^^ + ^/2) 348 DOUBLE THETA-FUNCTIONS. The theta-f unction accordingly becomes H t i ('"' + 1 It) '""p "' ^''' "^ ^ ^^'^''' "*■ '^'Z^^^' where = (f) (Xi, X2) • But, as the change contemplated was merely the change of g^ intc 9', + 2 K„ the function also becomes e 9r + 2Xr (Vr), on the supposition that 2 X^ is an integer. Therefore ,19. ' 2 8A J ^. + 2X, exp — 7rj[(^(Xi, Aj) + 2 2A,r, + 2X^iJ (4). If X, be an integer, the function reproduces itself except as to an exponential factor. When Xi = l, X, = 0, -— ^ becomes tj^ and 2 8X, we have one pair of quasi-periods, t^ and tj, ; when Xi = 0, X2 = 1, - — becomes t^, and we have another pair, tw and t,,. All other 2 8X, I , 1- _ periods and quasi-periods are compounded from the two pairs of quasi-periods and the two pairs of periods by addition and sub- traction, the general form being where «i, and 0, £2 are pairs of periods. Where there is no danger of misapprehension, we shall use the term periods for both the true periods and the quasi-periods. § 231. The theta-function satisfies the relations (V„ 1'2 + 1) = {-iy^o g. (Vr), 0\9r \k 69. \k 6 9r = {-iy.t (Vi + Tu, l'2-f Ti2) = (-1)\( 9r\{Vr) (w,)-exp — 7ri(2Vi-|-Tii\ (wi + T12, V.2 + T22) = { — l)\e. (7,!(t),)- exp— iri(2i'2+T22), DOUBLE THETA-FUNCTIONS. 349 and also satisfies three differential equations. Conversely, everj' function which satisfies these conditions differs from 6 "^ («,) only h, by a factor, which is independent of the variables and moduli. The following is the proof generally given for this important theorem (see, for instance, Krause, Die Transformation der Hyperelliptischen Funktionen, p. 5): — Assume, for simplicity, gi = g.2 = hj = lu = ; and let f{Vi, Vo) be a function which obeys the above equations. Because /{v^, v.i) is simply periodic as regards Vj, V2 separately, it must be expressible as a sum of exponentials CO CO 2 2 a„„ exp2^^i(?i^v^ + rl2V2)• The third condition, /(■t'l + Tu, V2 + T12) =/(i'„ 1)2) exp -7riX2 vi + Tu), gives 2 2a„^„^ exp 2 tti [?ii {v^ + th) + n.iv.^ + Tn) ] = 2 2a„^„^ exp 2 ni{niVi + n^v^ -v^ — ^th). By equating the corresponding terms on the two sides, we get «»,-!, n, exp ni [ (2 )!i - l)i-u + 2 ihTjo] = a„^„^, and repeated use of this relation gives «„,n, = "O., exp 7rl(«rVii + 2 HirijTis). By similar reasoning based on the fourth condition, it is easy to show that tttaj = «oo exp -iriniT^ Therefore /(vj, v^ = aoo 2 2 exp iri(TuTii^ + 2 T-anin^ + T22n2^ + 2 tii^i + 2 ngVa) = aoo^oo(yr)- The quantity ago is independent of the parameters, in virtue of the differential equations. For ^ = 4 Tt -^ ; 8u,2 8tii that IS, aoo g^ = 4 TTi Uoog;^ + «o( therefore 5-^ = 0, which shows th larly, aoo is independent of r^, r^i- 0^00 therefore ^ = 0, which shows that aoo is independent of tu. Simi- 350 DOUBLE THETA-FUNCTIONS. § 232. General formulce for the addition of periods and half- periods. From (3) and (-4), "'+'^'+*8x;r''i/^+2;.A If X, and /i, be integers, this is ^ ,(iv)-exp-,r/[<^+22M,+2X,(7i,+2/.,)]. & (IV) • exp - ,r/[<^ + 2 2A,r, + 2((7,^t, + hX)'\ (5), the formula for the addition of jjeriods. Let m = m 1 77in is used. 9r , W?o = K Ik f^r ; then for 2(g'rMr + Mr) tlie symbol If X, and /i, be multiples of 1/2, write Xr/2, Atr/2, in place of A^, /u,,. Then e \9r = 6 (iV + ,a,/2 + i8/2-\. J/r + Ar K + Mr From formula (7) it is clear that each of the 16 theta-functions can be changed into any of the other 15, save as to an exponential factor, by the addition of some pair of half-periods ; the number of pairs of values of \„ fi„ given by X, = 0, 1 and /n^ = 0, 1 being just 15, when the case X] = X, = ^i = jn, = is excluded. In the current number notation the result of adding two marks is not immediately evident, but the case is otherwise with the nota- tion by means of duads. After the addition of half-periods the suffix of a theta-function is altered. To obtain the new suffix from the old, when any half-periods are added, let (a), (/3), (y), (8), (t), (^), be the six odd marks (1), (2), (3), (4), (5), (6), in any order, and observe that we may regard ' as a mark. Let it be Ml 1^ expressed in the form («) -1- (/8) or («/3). Four cases arise: accord- ^' is of the form (yS), («y), {a^), (00), ing as h. fif.+X, isoftheform («) + (/3) + (y)-f (S), K + l^r («) + (/3) + («)+(y), («) + (/3) + («)-f(^), («) + (;8); and therefore = (e^), (fiy), (00), (a(i). We can express this shortly by the symbolic equations No attention is here paid to the exponential factor, which is readily filled in from formula (6). As an illustration of the above, 9 let "10" .01. "01" .0 These two marks are (36), (56); hence the new theta-function is ^35 ; also the multiplier is exp .^l(IP + V, + ^ Hence ■■5 Osb(vi + T12/2, V, + T,^/2) = e„(i'„ n) exp - 7riir.„/i + v, +1/2). 552 DOUBLE THETA-FUNCTIONS. 01' 11' 10' 01 10 11 01' 11' 10' § 233. It is well to notice that the symbol m | mo of § 232 = 1 or (mod 2), according as m and m^, do or do not contain a common letter. To prove this, we write the odd marks as follows : — 1,3,5 '' '' '' 2,4,6 and observe that if m and wio be different marks, selected from the same row, m | mo = 1 ; if m, m^ be selected from different rows, m I wio = 0, while m\m = 0. Now a/3 I ay = « I « + « ! ^ + « I y + ^ I y either a, /3, y are all from the same row, or two of them are from the same row. In either case, a|/3 + a|y + y81ysl. Again, a;8|y8 = a|y4-a|8 + /3|y + ^|S; if y, 8 be from the same row, « | y + « | 8 = 0, and /? | y + ^8 | 8 = ; but if y, 8 be from different rows, a | y + a ] 8 = 1, and I3\y + I3\S = 1. In either case a/? I y8 = 0. Lastly, it is evident that 00 | a/3 = 0. It will be useful in the sequel to remark that, given mo = («/3), there are eight marks m such that m | mo=0; and that, since m | mo=0 implies (m + mo) | mo s 0, these eight arrange themselves in four pairs: — m m+»„=mm„ -, 00 «/3 y8 c^ 7« ^8 y^ Se J (B); and that, similarly, when m [ mo = 1, there are eight marks m which may be arranged in the four pairs ay Py aS p& «£ /Se ui /3^ (B'). DOUBLE THETA-FUNCTIONS. 353 Since the mark m^ can be chosen in 15 ways, (00) being ex- chided, there are 30 such sets of four pairs, into which fall the 16 • 15/2, or 120, pairs which can be formed from the 16 marks. § 234. Tlie Rosenhain hexads. The six odd marks form a Eosenhain hexad, or, simply, a hexad ; and any six marks obtained from the odd ones, by the addition of the same pair of half-periods, also form a hexad. Thus there are 16 hexads, as follows : — ^13 ^35 ^15 02. 0^ 02. (12) 0.^ e^ 0-2. 0u 0^ Om (13) ^00 e,, 035 0^ 6^5 0^ (14) ^34 e^ 0^ 0]i 0iO 035 (15) ^35 0u ^00 03& 0-a 03i (16) ^30 6j, 0,0 03S 0u 0]2 (23) 6i2 e-^ 0^ 0-^ 0^ 03S (24) 6-^ 0m 036 ^00 0^ e« (25) ^46 0-^ 0U 0^ em 0^ (26) 0^ 0u 03i 0« 02i ^00 (34) Ou 0^ 0x 0-^ 036 015 (35) ^13 0\» 0^3 0m 0^ 0u (36) ^10 0^ d.2\ 0u 03i 023 (45) 0-^ 0^ Bu 0-25 05, ^13 (46) e-^ 0n 6.^ 0^ 0va 02. (56) ^24 0SS 0m ^13 0^ 0-25 (C). The hexads will be named by the pairs in the column to the left. For example, the hexad [15, 00, 13, 16, 12, 14] will be called the hexad (35). In each hexad, except the first, there are two odd func- tions ; any two hexads have a common pair of functions ; and any function occurs in six hexads. The hexads are of two types, which may be called pentagonal and triangular. The pentagonal ones are of the form 00, «/8, ay, «8, at, a^, and are obtained from the original hexad by the addition of an odd mark. The triangular ones are of the form «A /8y, ya, Se, «(;, (;8, 354 DOUBLE THETA-FUNCTIONS. and are obtained from the original hexad by the addition of an even mark. Any pair of functions belongs to two hexads. Thus d^„, 6^1, belong to the two triangular hexads {a^e, y&O and {a(3^, ySc). The functions 6^^, 6^^ belong to the pentagonal system (a) and to the triangular system («/3y, Se^). The functions .^ooi ^a/s' belong to the two pentagonal systems a and /J. The name of each hexad is inferred from the marks in it very simply. In the triangular hexad («/8y, Se^) the figures in each triad are all like (all odd or all even), or two are like and one unlike. If all are like, the hexad is (00); if not, the two unlike ones name the hexad. For instance, the two unlike ones in (12C, 3-45), which are 1 and 4, name the hexad (14). The hexad is, in point of fact, derived from (246, 135) by the interchange of 1 and 4. In the j^entagonal hexad («) the hexad is named by the two figures congruent to a; for instance, we speak of the hexad (35) instead of the hexad (1). § 235. The Oopel tetrads. Any two marks belong to two hexads ; a third mark can be chosen in six ways, so that the three do not belong to any hexad. If the two be, for instance, 00 and a/S, the third is chosen from y8 • c^, ye ■ 8^, y^- Be.; for it must belong to neither of the pentagonal hexads («), (13). Taking any of these six marks, say yS, the marks 00, «/3, yS do not belong to any hexad, and further, no three of the four marks 00, a/3, yS, ef belong to any hexad. Such a system of four marks or functions is called a Gopel tetrad. It is evident that a Gopel tetrad may be constructed by taking any two of a set of four pairs (§ 2337. From the 30 sets of pairs, two of a set can be chosen in 30 x 6 ways ; and since a tetrad is compounded of pairs in three ways, as 00 • «/?, y8 • e^ ; 00 • yS, a,8-e^; 00 -e^, a/3 -yS, the total number of Gopel tetrads is 30 x 6/3 or 60. There are 15 of the type 00 • ayS • y8 • t^, and 45 of the type ay Py a8- /3S. § 236. The zeros. It is evident that an odd function vanishes when v, and V2 vanish. We ex])ress this by saying that 0, are a common pair of zeros of the first hexad. The second hexad is obtained by the addition of the half-periods, which have the mark (12), to Vi, v^; that is, by writing i\ + ti,/2 for v,. Therefore th/2, TyJI are a common pair of zeros for the second hexad ; and so on. Thus six DOUBLE THETA-FUSCTIONS. 355 pairs of zeros, which are incongruent as regards the periods, are determined for each function. For instance, referring to the posi- tion of ^00 in the cohimns of the table of hexads, we see that ^oo vanishes, when Vi, v.,, are equal to the half-periods (13), (35), (15), (24), (46), (26) ; that is, when l\ = (1 + Tn)/2, (ra + T^)/2, (1 + Ti,)/2, (1 + rn + r,,)A t,,/2, (l+Tn)A U, = r^/2, (l + ri, + T.,)/2, (l+r,,)/2, (t,, -f r,^) /2, (l + ro,)/2, (l + r:.)/2. § 237. Approximate values of the functions. In the double series (1) consider the lowest powers of expirjVn, expTriris, exp iriVoj. If gi = 0, jTo = 0, the lowest power occurs when ni = no = 0, and is then merely 1. If j/i = 1, ^o = 0, the lowest power occurs when ni = nj = and when ni = — 1, no = ; and, combining these, it is 2cosir(ri -I- Ai/2) exp jTrjTu. Similarly, if (/i = 0, jto = 1, the lowest power is 2 cos ir (uo -I- ^2/2) exp J- iraVoo. li gi = l, g2 = 1, the lowest powers are contributed by ni, no = 0, 0; - 1, ; 0, - 1 ; -1,-1, and are 2 cos^r^Bi + v« + hjLb\ exp ^ (t„ -I- 2ti2 -I- T22) -I- 2 cos ir ^ui - uo -f !hJZ^\ exp — (Til - 2 T12 -f T22) . These values are sometimes useful for the verification of formulae. If we put Bi = ■uo = 0, we obtain approximate values of 0a^(O, 0) for small values of expTTlTll, eXpTTtTlo, exp 7rtT22. When (ftyS) is any even mark, the value of d^^(0, 0) will be denoted by c^^. § 238. Eiemann's theta-formula. Let a, + b, + c, + (l = 2ay a, — b, + c, — (l = 2c,' a,-6,-c, + c^ = 2d/ J ('• = 1,2). (i.). 356 DOUBLE THETA-FUNCTIONS. Solving these equations for a„ 6„ c„ d^ we find that the equar tions (i.) are still true when unaccented letters are replaced by accented, and accented by unaccented. Hence accented and unac- cented letters may be interchanged in any equations we may deduce. "When a, becomes a, + ju,, + - -^, X, = 0, 1 ; //., = 0, 1, «/ becomes "' 2^ isT' and the same change is made in 6,', cj,.dj. By such additions of periods and half periods, Bji^ becomes, from formula (5), 6„a, exp — iri [<^ + 2 2X,a, + m | mo], and 6,^aje^lhje^cje„d,' becomes, from formula (7), e^-aj . e^..h; ■ e^..c; . 6^..d; ■ exp - irt [<^ + 2\,(a/ + 6/ + cj + d/) + an even integer]. 9r m' = K' , m" = where .,„__,,.„_ or e„..a/ . 6„..b; ■ 6„..c; ■ e„..dj ■ exp - :ri(<^ + 22X,«,). Now when m' runs through all the 16 possible values, in" does the same. Hence :s ( - 1 ) " ""'Q^oJ ■ 6„hJ ■ e^-cj ■ O^-dJ, m' summed for all reduced marks, obeys the fundamental relations of § 231, when regarded as a function of a^; that is, it reproduces itself, save as to the exponential factor exp — tti [<^ + 2 2X,a, + m \ m^']. Therefore it differs from 6„a^ only by a factor independent of a,. Similar reasoning holds for b^, e^, c?„ so that 2(- i)""°'-e'„.a; . o„.b/ ■ o„.cj . e^.d;=Tce^a, . ejb, ■ e„c^ ■ e^ ■ • (8), m where h is independent of «„ b^. c,, d„ and therefore either a numeri- cal constant or a function of the moduli th, tj,, t^i- There is a more general form of the theta-formula which has been used by Prym and Krazer as a basis for their researches on the theta-functions. This extended formula, in the case p = 2, can be found by changing b„ c^, c?, into , ,1 ,,18<^' .1 „ , 18<^" , . 1 ,„.18<^"' DOUBLE THETA-FUNCTIONS. 357 where K'+K"+K"'=H^r'+l^r" + Hir"'=-0, and <^(«) means <^(Ai<«), X^'"*). For then a/ is unchanged, by (i.), and b^, c„ d, are changed by pre- cisely the same amounts as bj, cj, dj. By these additions of half- periods any mark m is changed into mnii, mnio, mm^, where mi = Wl, = ! V'l and the marks mmi, m7n.,, mm^ are unreduced. The right-hand side of the formula (8), already obtained, changes by a three-fold use of formula (6) into ^^A ■ 0„n,br ■ O^m^Cr ■ 6„„d, • exp - TTtJi (,^' -I- " + ^.3 + K,") + i >^i.' = A-(>^,. + X,,") + 4 X,,'. The same is true for any interchange of accents, so that HK, + K'!) + 4 >^,/' = k{X,, + X,,') + 4 x^" Therefore bj^ subtraction (^. _ 4) (X,/ - X,,") = (A- _ 4) (X,; _ X,J'), an equation which cannot be true for all values of a,.', a,.", etc., except when A; = 4. For instance, if o/ = rt/' = the half period (15), Xjg' and X13" vanish, while tlie other terms need not. Therefore for certain values of the variables k = 4, and since A: is independent of the variables, its value is always 4. Writing i//„ for X„ + X„' +XJ', we have 2 ^„ = Ks' + ^..' + ^'j - >(-2: - X,,' - ^^', and therefore ^„, + (-l)'""'V™=0 (10), where ??i and m' are different odd marks. § 240. Dealing similarly with the general formula (9), we have 4 e„,a, . e„„,&, . &„,„,/, . e^„,ci= 2 ( - 1)" '"■e„,a; . e,,„,&/ ■ e„.^Sr' ■ o^„.,{-dj), 4 e„a/' . ^_,V' . ^_/;' . e„„J - fZ/') = 2( - l)"""e„,a/ . ^„.„,6/ . e„.„// . e^.^^d'. Case i. Let mm.^ be an odd mark. Then, by subtraction, 2(X„ + X„.") = -2(-l)"''"X„,', where XJ') now denotes ^„o/«' • 6„,„,6,<-> • e„,„,,cM . ^I^^^rf,*"', and vi'ni is any odd mark. Let, for example, 7)13 = (12). Then, because mnii and ni'm^ are odd marks, m and m' belong to the hexad 14, 16, 23, 25, 35, 46, derived from the hexad of odd marks by the operation (12), as in § 234. 3 DOUBLE THETA-FUNCTIONS. 359 The symbol vi m' = 1, when m and m' have one common figure. To fix ideas, suppose m = (14 ) ; then 2 (^14 + ^14") = — ^14' + ^iG + ^4o' ~ ^ii' ~ '^li' — ^35' ; or, if ^„=5V"\ 2 ^„ = X,/ + X,,' + X,„' - X^' - X^' _ X.^', and therefore xj;„ + {—l)"""il/„. = (11), where m and m' belong to the hexad (12). And similarly for any hexad. § 241. Case ii. Adding the same equations, we have 2(X„-X,„")=2(-1)-"»V, where m'm^ is any even mark, wihij any odd mark. Interchanging the accents, we have in all three such equations, which give on addition To reduce this equation, which contains ten terms, let us take two different values of m. First let these be of the form («/3), (yS), as for instance (12) and (34). These belong to two hexads, (23) and (14). If both values of m make iJiMJa odd, m^ must be (23) or (14). Selecting the value ?)i,.5 = (23), m' is any mark not contained in the hexad (23), since m'm.i is even. That is, m' is obtained from any even mark 17 by the addition of (23). The even marks are 00, 12, 14, 16, 23, 25, 34, 36, 45, 56, and therefore mi' is one of 23, 13, 56, 45, 00, 35, 24, 26, 16, 14. "When m takes in succession the values (12) and (34), 5 ( — 1 )"" "''i/'„' becomes respectively — Ip-S — i'\3 + lA^C + '/'45 + "AoO + fe — «/'i4 — i'X — "AlO — in = 0, and — i/'ij — ij/is + fe — i/'4o + "Aoo — fe — '/'24 + fe + "Alii — "Au = 0. Therefore, by subtraction, fe + "A-B = "Aic + fpx- But this equation may be obtained directly from the considera- tion of the ten even marks ; for m' = ,, + (23); therefore m | m' = ?)i 1 77 + m | (23), 360 DOUBLE THETA-FUNCTIONS. and consequently 2( — l)"''''i/',„- = 0. Writing out 2(— l)"'''i/', when m= (12) and (34), we may combine the expressions, and replace each r] tliat remains by the corresponding in'. Thus, from the even marks 00, 12, 14, 16, 23, 25, 34, 36, 45, 56, m = (12) gives the sequence xj/m + ^u — •/'u — "Aie — fe — fe + iAm + fa + "A-io + fe J m = (34) gives •Aoo + ii2 — ^u + fa — fa + fa + fa — fa — fa + fa ; the half-difference is — fa — fa + fa + fa (12), and, when (23) is added to each mark, there results as before the equation fa + fa = fa + "Aie- The other value for m^ is (14). This, in the second method of reduction, has only to be added to each mark in formula (12), giving fa + fa = fa + ^PiS- The argument is clearly independent of the choice of (a/3), (yS), when both are even ; and it is left to the reader to verify that, whatever be the marks m, provided their form is (w/S), (yS), the reduced equation is fa + fa?= fa + fa- Next let the marks m be (afi), (ay). For the sake of definiteness suppose them to be (12) and (13). The values of m^ are the names of the hexads in which (12) and (13) occur, namely (35) and (25). When m= (12), 2(— 1)"" '>«/', is, as before, fa + i^ia — fa — i^ie — fa — fa + fa + fa + fa + fa ; when m= (13) it is fa — ^12 — fa — fa — fa + fa — fa — fa + fa + fa- The half-difference is >/'i2 — fa + fa + fa- Therefore, when m, = {3o), fa = fa + fa + fa, and, when mi = (25), fa = l/'l5 + 'Pk + «/'lC- DOUBLE THETA-FUNCTIONS. 361 This proves that for two marks of the form («,8), (ay), of which one is even and the other odd, we have results of the forms and it is left to the reader to show, by working out a typical case, that the results hold for any pair of the form («/3), («y). Lastly let the marks m be («/3), (00) ; say (12), (00). Then ma = (35) or (46) and 2(— l)"'''i/'r, is '/'OO + "Aia + "All + "/'lO + ^'23 + >P-25 + 'pM + 'Pie + ^iS + "As© or i/foo + "Ais — "Am — "Aie — fe — fe + i/'34 + fe + "Au + '/'se- The half-difference is ■Au + lAie + fe + fe' Therefore, when ?)i3=(3o), •Aai + "Aai + fe + >/'23 = 0) and, when m3 = (46), "Aid + lAu + >/'i5 + "Ais = 0- This proves that for the form («/3), (00), when («/3) is even, we have results of the form lAoy + faS + l/'ae + "/"a? = J and the same result holds when («^) is odd. Combining these facts, we have the result : — Of the ten marks which are not in the hexad rris, four are in any other hexad. If this other hexad be triangular, then according as the four marks are of the form a^, /3^, yt, Se or /3y, e^, (8, 8e, there is the relation or l/'p-), = l/'ef + l/'^6 + „ (13), where m is any one of the four marks, m' is in turn the other three. 362 DOUBLE TH ETA-FUNCTIONS. When the four marks are given, m.^ is determined as follows. To the hexad belong two other marks, which appear in the hexad ()?i3). For instance, let the four be 12 • 34 ■ 45 • 53, belonging to the hexad (li). The remaining marks 16-26 also belong to (24). And (24), when combined with any of the four, gives an even mark. § 242. Fornuda for the reduction of the marks in i/'„. Let mi and m., be reduced marks ; we shall determine the formula which will enable us to reduce the marks in such a product as e e ^ e ^ e i^y§232, e„^„^or is reduced by the factor exp7ri2/t,/x/(r/, + A;). Similarly for wi +m,; and for m — 9)i] — m,^ we have e g^-X;-Xj' =6 Sf. + V + A;'!exp,ri2(,x/ + iLc/')(^, + V + V')- \-^J-lj,J< h^ + ^J + ^y^ The last mark will be reduced by a factor of the form exp wi'S.K,{g, + K' + K"), where k,=0 when /(, 4- i^r + iJ-r" = or 1, and =1 when K + f'r + m/' = 2 or 3 ; these are all the permissible values of h^ + fxj + fx,", inasmuch as m, nil, m^ are reduced. The conditions are satisfied by K, = h^{ixJ + p.;')-\- fijfij' ; the reducing factor for m — mi — m^ is exp ^i l.ig^ + A/ + X<') (k, + ;x/ + ^/'), and the whole reducing factor is exp ni 2 [2 gXil^r + m/') + /*.(A,V; + Kyj< + A/'V/") + ((/.-A/") (;.>/' +;x/ + /x;')], or, omitting a factor which is independent of m, exp7ri[2/t,(A,V + A,V') + ?.(M>/' + M; + M/')] • ■ (14). This result is not altered by writing A,'", /./" for A/, /x/ or for \ " ,1 " DOUBLE THETA-FUNCTIONS. 363 § 243. Addition theorems. I. When vi'm^ is odd, let us jjut c'rt &r, c„ d, = IV + vj, 0, 0, i\ - 1)/. Then a/, b/, c/, d/ = iv, t\', i\', -v„ and a/', &/', c/', d/' = r/, i\, i\, i\'. Let m = (13), m' = {lo). "We seek to express the relations between Oi/^v ± v'), 6u(u ± c') and 6-functions of v, v\ using for that purpose formula (11), ^„+ (-!)""">„. = 0, in which mv^, m'm^ are odd marks. Since m??i3 and m'm^ are odd, ms={00) or (35). Taking 7)13= (35), we have ?)ii + m^s (35). Now in order that the functions 6„, &„., with the arguments V, ± v/, may not disappear, it is necessary that the marks mi + (13), mi + (15), m2 + (13), m2 + (15) be even. Therefore ?)ii and m, must be selected from (12), (14), (IG), (24), (26), (46). The condition nii + m2= (35) further restricts m^ m.^ to the three pairs 12, 46 ; 14, 26 ; 16, 24. Take mi = (12) = ! 1 loo , m, = (46) = 01 01 The reducing factor of § 242 is (-1)'=; that is, + 1 for m = (13), — 1 for m' = (15). Therefore, from formula (11), since m, mnii, mm,, ni7rt3 = (13), (23), (25), (15), and m', m'mi, m'm.2, m'Hi3 = (15), (25), (23), (13), c^c^\e,^{v + v')e^{v - v') + 6^{v + v')e,,{v - v') \ — 2 6ii6^6.2^di5 + 2 6ii 02362^615 = 0, where the arguments of are Vj, v^, and those of 6' are Vi, v^'. 364 DOUBLE THETA-FUNCTIONS. II. In the second case, when m'm^ is even, we use formula (13), Let a„ b„ e„ d, =%\ + v,', v, — %\',(i, 0; then aj, bj, c,', OJ =v„ v„ v,', —v/; and a/', &/', c/', d," = v„ v„ v,', i\'. Let two of the four marks be (13), (lu), and ?)ii = (35). That 6is{v±v') and 615(1; ±1;') may not disappear, we may choose m., = (12), mj = (46). From either of the two hexads which contain (13) and (15) two other marks must now be selected. But, in order that for these 6„(w±v') may not enter, we impose the condition that mwis be odd. This restricts the choice to (14), (16) from the hexad (35), or to (35), (46) from the hexad (00). Choosing m = (13), (15), (14), (IG), we have m»ii=(15), (13), (26), (24), «im,s(23), (25), (24), (26), 9nm3=(25), (23), (16), (14), and the reducing factor = 1,-1,-1, 1 ; the reducing factor of § 242 being, as before, (— V)^. The equation "Ais + •Au + I^U + l/'ju = becomes -2e,A.e.^o.r; -2ej.j,:e,^ + 2e,,%,d.^%,' = o, or c^c^\e^,{v + v')e^{v - v') - e,,{v + v')e,,iv - v') i - 2 e,fi.^6,:e,^ + 2 6,aa:o.^ = 0. Similarly, if we replace «„ h„ c„ d, by 0, v, + vj, v, - vj, 0, the rest of the work being unaltered, we have cuciiio^iiv + v')e^{v - v') - e.^{v + v')e,,{v - v') \ + 2 0M6-^'d._, - 2 e,,'e,,e.J-^' = o. From the two equations already obtained for ^^(y + v')6^!,{v — v') and 6ii{v + v')di:i{i; — ^•'), it is evident that each of these quantities can be expressed in terms of 6-functions of v and v'. DOUBLE THETA-FUNCTIONS. 365 Example. Prove CooCiol^a^Ci; + v')e^{v-v') + e^^{v + v')es, {v - v') } - 2 o^$^eoo'6,o' + 2 e^e,s^6j = o. § 244. Relations betiveen the squares of four theta-functions of u hexad. Let thea and a„ b„ c„ d, = v„ v„ 0, ; a/, bj, c/, d,' = IV, V,, 0, 0, o,", V, e/', d/' = t;„^;„0, 0. Further let wij = (00). Then since ttij + mj + Tn^ = 0, nij = — wij ; and ij/^ takes the form Here the reducing factor is (— l)^^'-'"''", when mnij is reduced. Hence, dropping the factor 3, we may put f^ = {-l)^s-:''"-ej{v^)c. 2 in formula (13). We thus have a linear relation between the squares of any four functions of a hexad. For example, let wij = (00). Then we have the six independent relations <-(» ^00 ^ Ci2"C'i2 + Cn C'i4 + C]6 Pie' ^ C23 Pog + Cji'Pji + C^fO^ and nine others which are readily deduced from these by subtrac- , and (15), tion. Again, if wig = (13) = 10 m = (13), (12), (14), (16), 101 10 ' 10 1 00!' 01 10 00 11 366 DOUBLE THETA-FUNCTIONS. (16). we have, attending to the factor (- l)^^'-'"-"' or (—!)''■, Similarly, -ejc^'-ejc-J+e.JcJ+e.JcJ=(i, 9»3= (35), e,:-c^-+e-j-cj-ejc,.?-6:^%:-=o, m.,= (i5), 6,ic^+6,fc,]:-ejc,^ + e^c,^=Qi, nu= (24), ejc,j^+e,icj+ejc,,f-ejcj=o, m,= (46), The total number of such relations is the number of sets of four which can be chosen from the hexads ; namely, i • 6 • 5 • 16 or 240. Example. Prove the relation § 245. Relations between the c's. If in the square relation we put i\ = 0, we derive six relations of the form Co,/ = Ci.^ + V + Ci6* (l*"). and six relations — CuCj + C^-cJ + Cio'Cj = ~ eucj + Ch'V — Ci/cv = Ci3"Ci;- - Cjcj + Cs,rcJ = Ci/fi/ - CifCM- + c,,.-\v = — c,.-C|„- + c.jcj + cjcj = Cu-Cj + C:i^C^ — Cjcj = . This may be stated as follows : The ratios of Cy?, c,/, Cio" ; — c.J, c^-, c~/ ; — &,,,-, — c^-;, CrJ' to c^ are equal to the coefficients in an orthogonal transformation of co- ordinates. (See Caspary, Crelle, t. xciv.) For if we write for these ratios 'ill 'l2i 'l3 ) '2I) '221 '23 ) '3I1 '32; '33> we have the relations (s = l, 2, orS), (r=l, 2, or3), (s = 1, 2, or 3; s' = 1, 2, or 3 ; s ^ s'), ()•=!, 2, or 3; r=l, 2, or 3; r^r'). (18). 2?,r r=l 2 Ur. r=l = 8 2 Ur: •^1 = DOUBLE THETA-FUNCTIONS. 367 Interpreted in this way, the table ^■12" C14- Ciij- Aoo= (D) affords a compendium of the relations in question. It may be observed that the sign is minus when the figures in the suffix are displaced from their natural order. § 246. Relations between the squares of five tJieta-fiinctlons. Excluding e^, take any five functions 6^, 6.^, 0.^, 6^, 6-,, no four of which belong to a hexad. Eeferring to the table of hexads (§ 234), we see that in any case two triads, 6„ e,, 6, and 6,, 6„ $„ can be chosen, which belong to different hexads. The two hexads have $1 and some other function 6^ in common. Therefore 6^-, 6./, 6i, (9/ and (9/, i9/, 6,% d^-, are linearly related. Eliminating 6^', we have a linear relation between 61', O.j', O-i, 6i, Oi- There cannot be two such independent relations. For if there be a second linear relation between 6^, 6.?, 6/, 6/, 6/, there is a linear relation between $1'. O.f, Of, ^/, of which three belong to a hexad, and therefore have a common pair of half-periods as zeros. If then 4 2M/=0, ^1, 6.,, 0, vanish for values which do not make 6^ vanish, and conse- quently k^ = 0. The relation reduces to 2 JcJJ' = ; but 61 and 62 belong to some other hexad, and therefore have a common pair of half-periods as zeros, which are not zeros of 63. Therefore k., = 0, and kiOi- + k.jO.f = 0, a relation which is impossible, since all the pairs of zeros of 0{-, 6-j- are not the same. If one of the originally selected functions be 6,„. the addition of suitable half-periods gives five functions not including 6,x,; and from the relation between these five follows a similar relation between the original five. There is then a linear relation between the squares of any five functions, no four of which belong to a hexad. The proof supposes that there is no linear relation between four squares which do not belong to a hexad. This is evident if any 368 DOUBLE THETA-FUNCTIONS. three belong to a liexacl, and the only outstanding case is that of a Gopel tetrad. Suppose that then, when v^ = 0, "■'12^12 I "'34^34' ^^ ^ j and when r, = /ir/2 + \ ^'P/^Kj and it follows from (7) that Therefore, either k-^ = k^ = 0, or The former supposition is impossible, because it would imply a linear relation between two squared theta- functions ; the latter-suppo- sition is untenable, because there are three independent parameters, I'll, T12, T22, of which the c's are functions, and the approximate values for the c's in terms of the t's disprove the equations ^12=^34 ) <-12 ^ — ^u The total number of sets of five is 16 • 15^- 14 • 13 j 12 . ^-^^ ^^^_ ber of sets of which all are in a hexad is 6 • 16, and the number of 6 • 5 sets of which four are in a hexad is • 10 • 16. Hence the num- ber of linear relations between five squares is 1872. The general theorem at which we have arrived is the following: Four squared thetarfunctions are connected by linear equations when, and only when, their marks belong to a Rosenhain hexad ; but there is always a linear relation between five squared theta- functions, no four of which belong to a hexad. Hence every 6*^ is linearly expressible in terms of four selected squares, if these four do not belong to a hexad. For instance, the four theta-functions may form a Gopel tetrad. § 247. Product theorems. Let a„ h„ c„ d, = v„ v„ 0, ; then a/, 6/, c/, d/ = v„ v„ 0, 0, and a,", hj', c/', c?/' = v„ v„ 0, 0. DOUBLE TH ETA-FUNCTIONS. 369 Therefore Ym — ^ - m " mmi^ mmi*- mm^^ and, from § 241, M.™.c,„„,c_3=2(-l)'"i'»M».- mfim-m^Cm-ms! the summation being for any three marks m' which, like m itself, give an even mark when combined witli m.j. Now (/(,„ exists only when mm., and tmiis are even, and since, regarding in., and m^ as given, there are six values of 7)i for which vim.2 is odd, six values for which mm,, is odd, and two values for which both mm,.^ and mim; are odd, there must be six different values for wliich both tool, and mw^ are even. But mm^m.i^mmg, mmim..=mm.,, since mi + m., + m^ = 0. Therefore, if m be one of the values, ))i»ii is another; and corresponding to any two marks m.j, m^, there are three products of pairs of functions, 6„,6„„.^, between which there is a linear relation. The number of such relations is the number of ways in which ?)72 and m^ can be chosen ; that is, .V • IG • 15 or 120. Since mvu and in m- are even, :L{g^ + K"){K + f^r")=0, and, by addition, m \ in., + m \ in., + 5X/'/x/' + 2X/'Vr"'=0, therefore in \ m, is given. This shows that the three linearly related pairs belong to one of the 30 sets of four pairs of § 233. Four relations can be formed from each set by leaving out each pair in turn, the total number of relations being 4 x 30 = 120. Taking the set determined by m, = (12), m | mi s 0, we see that. since mi = 10 00 , it must be 00,12; 34,56; 35,46; 36,45 The determination of mz, m, is effected by the consideration that mo + W3 + (12) = 0, 2VV + 2V'V;"=0, .therefore m^, m^-OO, 12; 34, 56; 35, 46; 36, 45. The reducing factors are respectively 1,1, (-l)^ (-l)v 370 DOUBLE THETA-FUXCTIONS. Therefore, taking m = (00), to' = (34), (35), (36), and observing that g-, = 0, 1, 1, respectively, we have from ^m = i/'.u + fe + - tischen Funktionen erster Ordnung ; Krazer, Theorie der zweifaeh unendlicheu Thetareihen auf Grund der Riemann'schen Thetaformel ■; Krazer and Prym, Neue Grundlagen einer Theorie der allgemeinen Thetafunctionen ; Prym, Un- tersuchungen ueber die Riemann'sche Thetaformel. Ail these works are pub- hshed by Teubiier. The theta-functions form an essential part of Riemann's theory of Abelian Integrals, and therein lies their present importance. 'Works on Riemann's theory contain more or less of the Algebra of the theta-functions ; accordingly to the above references can be added those at the end of Chapter X. The founders of the theory of this chapter were Gopel (Theorife transcenden- tium Abelianarum prinii ordinis adumbratio levis. Crelle, t. xxxv.) and Rosen- hain (.Memnire sur les fonctions de deux variables et a quatre p^riodes. JlOm. pr6s. a I'Acaddmie des Sciences de Paris, t. xi.). A bibliography will be found in Krause's book ; but special mention ought to be made here of a memoir by AVeber (Math. Ann., t. xiv.) and of Cayley's memoir in the Phil. Trans. (1880). The method used in § 2.30 and following paragraphs for the discussion of formula (9), which is due to Prym, appears to be new. CHAPTER IX. Dikichlet's Problem. § 252. Hitherto the j)arts of a function /(z), = m + iv, have been considered in the combination xt, -{■ iv, and x, y in the combination X + iy. Let the functions u, v be one-valued and continuous, as also their first differential quotients, throughout a connected region of finite dimensions which includes the point z; in other words, throughout some neighbourhood of z. By § 20 of Chapter 1., f{z) has at this point z a differential quotient /'(z) which is independent of dz. Conversely, in order that u + iv may admit a continuous differential quotient f'{z) at a point z, within whose neighbourhood M, V are continuous, the real functions it{x, y), v(x, y) must give rise to continuous partial differential quotients of the first order. For, by supposition A(m -f iu)/^z= A(m + ill) /Ax, A(m + iu)/Az = A(?( 4- iv)/i\y; and, when Az tends to zero, the left-hand side tends to/'(z); hence the right-hand sides must tend to hu/hx -\- i8v/&x and — iSu/8y + hv/hy ; i.e. Bu/Sx=&v/Sy, Bu/8y = — Bv/Sx (1). But /'(z) is continuous by supposition, therefore the components of the right-hand sides must be continuous ; hence the condition imposed upon/(z) — that it is to have a continuous differential quo- tient — implies the continuity of &u/&x, Su/By, Sv/8x, &v/&y in the neighbourhood of z. The differential equations (1) might have been selected as a starting-point for a theory of functions of a complex variable.* From them it is possible to deduce Cauchy's theorem | f{z)dz = 0, where C is a closed curve contained within a region r, throughout which u, V, and the first partial differential quotients are one-valued * This 18 the plan adopted by Picard in his TraitiS d' Analyse. 374 dirichlet's problem. 375 and continuous. As a consequence of this theorem it follows that all the higher differential quotients of u, v exist within r, and are finite and continuous ; also that/(«) at a point Zi of the region can be expressed as an integral series P{z\zi), the circle of convergence extending at least to the rim of T. Since the higher partial differential quotients of u, v exist and are one-valued and continuous in r, it follows that S'tt/&x^ = B-v/&ySx, S2«/S2/2 = _8\./3a;S(/, h^u/dx- + l,-u/d>/ = • (2). The discussion of the properties of an analytic function might be carried out, ab initio, with the initial assumptions that in the neigh- bourhood of a non-singular point {x, y) the functions u, 8u/Sx, 8ii/Sy, B^u/Sx% i^u/Sy" are continuous, and that &^u/8x% h^u/hy^ satisfy the equation hhi/hx"" + h''u/hy- = 0. The equation (2) is a special case of Laplace's equation Pu/Si^ + ?,'u/Sf- + S'u/&z'- = (3), which plays so important a part in Mathematical Physics. In many branches of Mathematical Physics a problem which is constantly arising is to determine u throughout a region of space, given the values of u on the boundary of the region, and given that u satisfies the equation (3) throughout the region in question. The problem which we propose to discuss in this chapter belongs to this class. On the general subject of the jjartial differential equations which occur in Mathematical Physics we refer the reader to an im- portant memoir by Poincar^, Amer. Jour, of Math., t. xii., p. 211 ; this memoir contains among other things a most original discussion of Dirichlet's problem for three dimensions. Eiemann's special mode of treatment of functions, with Laplace's equation as a basis for the theory, was imdoubtedly suggested by physical considerations immediately connected with the Potential theory ; although his theorems are couched in the language of pure analysis.* The most novel feature of Eiemann's work was pre- cisely his attempt to define a function u -f iv of a; -f- iy within a region by its discontinuities in that region and by certain boundary conditions, in preference to the use of an arithmetic expression as a basis. The difficulties presented by such a method must of necessity » Attention is called to tho physical aspects of Riemnnn's work by Klein in his valuable pamphlet Ueber Rieinann's Theoiie der Algebraischen Functiunen und ihrerlntegrale. 376 dieichlet's pkoblem. be formidable, for little is known as yet of the nature of the proper- ties of functions in general; so that when certain properties are postulated of a function it is essential that some proof be given of the existence of a function with these properties. Despite the diffi- culties associated with it, Riemann's method has proved of immense service in the study of Abelian integrals and automorphic functions. In view of the yearly increasing importance of tlie mathematical investiga- tions which have been carried out on Riemann's lines, rigorous proofs of exist- ence theorems are of the highest value, even though the analysis be too indirect and laborious to serve as a method for the calculation of the functions whose existence it establishes. To a student of applied mathematics these proofs may appear unnecessarily long and tedious ; the more so as in certain applications the existence of the potential function is indisputable. For his benefit we may be permitted to quote a passage from the memoir of I'oincare's, to which reference was made above : " Xeanmoins toutes les fois que je le pourrai, je viserai a la rigueur absolue et cela pour deux raisins ; en premier lieu, il est toujours dur pour uii gc'cimetre d'aborder un probleme sans le re^oudre completemeut ; en second lieu, les Equations que j'etudierai sont susceptibles, non seuleraent d'ap- plicalions physiques, mais encore d'applications analytiques. C'est sur la possi- bility du probleme de Dirichlet que Kiemann a fondu sa magnifique thfeorie des fonctious abeliennes. ])epuis d'autres gijoraetres out fait d'importantes appli- cations de ce meuie principe aux parties les plus fondanientales de I'Analyse pure. Est il encore permis dc se contenter d'une demi-rigueur ? Et qui nous dit que les autres problemes de la Physique Mathfimatiijue ne seront pas un jour, conmie I'a d6ja etc le plus simple d'entre eux, appeles a jouer en Analyse un role considerable ? " The basis for Riemann's work is a famous proposition known among continental mathematicians as Dirichlet's Principle, or Troblem. § 253. DiricJilet's jvoblem for two dimensions. To find a function u{x, y) which, together with its differential quotients of the first two orders, shall be one-valued and continuous in a region r, which shall satisfy Laplace's equation and shall take assigned values upon the boundary of the region. In the problem just enunciated we must pay attention to the order of connexion of the region, to the character of the rim or rims, and to the manner in which the values are assigned along the rim or rims. In the simplest form of the problem, the region T is simply connected and bounded by a simple contour such as the circle or ellipse ; also the values on the rim are continuous. In the most general form of the problem, the region T is n-ply connected and bounded by an exterior simple contour Co and by ?* — 1 interior simple contours Ci, C.,, •••, C„ i, which are exterior to one another. dirichlet's problem. 377 "We shall denote the collective boundary by C, and shall speak of points within the region as points in T, and of points on the boundary as points on C. The curves which constitute C are sup- posed to have tangents whose directions vary continuously, except at a finite number of points at which there is an abrupt change of direction; a case which would arise, for instance, were the rim a polygon composed of a finite number of circular arcs. The values assigned to the rim will be continuous except at a finite number of places. Any real function n {x, y) which, together with its differ- ential quotients of the first two orders, is one-valued and continuous in r, and satisfies Laplace's equation for two dimensions, is called harmonic (Neumann, Abel'sche Integrale, 2d ed., p. 390). Dirichlet's problem can be re-stated as follows : to find a function u{x, y) which shall be harmonic in T, and which shall take assigned values on G, these values being supposed to be continuous along C except pos- sibly at a finite number of points. It should be remarked that the boundary value at a point s on C is to be identical with the limit to which u{x,y) tends along a path in V which leads to s, this limit being assumed to be the same, however the path may he chosen* Green was the first mathematician who enunciated the general problem (" An Essay on Electricity and Magnetism," 1828, § 5), hut his proof was based on purely physical considerations. For a treatment of the problem from the physical point of view, the student is referred to Gauss, Ges. Werke, t. v., Allgemeine Lehrsatze in Beziehung auf die im verkehrten Verhaltnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungskrafte, Art. 31 to 34 ; Lord Kelvin (Thomson), Reprint of Papers on Electrostatics and Magnetism, Art. xxviii., and Natural Philosojihy, t. i., Appendix A (d); Riemann, Schwere, Elektricitiit und JIagnetismus, p. 144 ; Lamb's Hydro- dynamics, pp. ;i8 et seq. Weierstrass has pointed out that Riemann's proof, and ill fact all those which make use of the Calculation of Variations, are open to grave objections. On this point the reader should consult Picard, Traitfi d' Analyse, t. ii., fasc. i., p. 38. § 2o4. If the problem be solvable, the assigned values on C must be, in general, continuous. The following two theorems which relate to the continuity of the rim values were published by M. Painleve in his thesis, Sur les Lignes Singulieres des Ponctions Analytiques, Paris, 1887. They involve the conception of a passage * Scbwarz hjia diecussed in a very lucid and thorough manner the suppoeitions which must be made in Rolving Dirichlet's problem, and the coneequences which follow from tbem. See bis memoir Zur Integration der partlellen Differentialglelchung .iM = 0, Ges. AVeike, t. ii. For instince, he i>oinl8 out that an assumption of the continuitj- of 6it/&x,&ri/&y on C Is not legitimate, since Bucli an asi-unipiion would imply that the rlin values are not merely continuous, but also have a differential quotient with respect to the arc. 378 dikichlet's problem. with uniform continuity to the values on the rim. By this it is implied, that to an arbitrarily small number c, given in advance, there corresponds, for every point s on C, a region of finite dimensions which penetrates into r, and is bounded partly by an arc of C which contains s, partly by a line interior to F; the distinctive property of this region being that the difference of any two values of u, in the interior or on the rim of this region, is equal to or less than £ in absolute value. In what follows, we shall use the letter s for a point on C, and also for the variable which determines the position of the point on C. Theorem I. Let a point s describe a part D of C, and let there be associated with s a path cs for the function u{x, y), cs varying continuously with s and leading from the interior point c to s. Then if, along D, u tend uniformly to the value U{s) which is assigned to s, the function U{s) must be a continuous function of s, and u{x, y) tends to U{s), ichatever be the interior j^ath cs which leads to s. Theorem II. On the other hand, if xi{x, y) tend to U{s), what- ever be the path cs, U{s) must be a continuous function of s, and u{x, y) must tend uniformly to U{s) along D. The following remarks may help to show the raison (Vetre of these theorems. To fix ideas let the region r be a circle of centre and radius B, and let the approach of a variable interior point (.r, y) to a point s of the rim C take place along the radius from to s. Given c in advance, a positive quantity 5, can be assigned such that, for all points {x, y) on the radius from fo s at a distance from s less than S,, \u{x, y)- l\s) j<£. Let 5, be the largest value consistent with this inequality. As s moves round the circle C, 5, varies ; suppose that the lower limit of the quantities 5, is 5, where 5 is different from zero. Then, far the ansir/iied method nf approach to the points of C, the values ti(x. y) tend uniformly to the values C/(.s-). A ques- tion which suggests itself is whether the values n(x, ?/) would tend to the same limiting values f/(s) when there is a change in the curves of approach to the points s ; and whether, if there be no change in the limiting values, the passage to the rim values is still effected with uniform continuity. There are, so to speak, two degrees of freedom in the question ; the infinitely many paths to an assigned point of O, and the infinitely many points of D. That a variety of paths to an assigned point of D may lead to one and the same limit can be illus- trated, by analogy, from Chapter IIL Suppose that in § 00 the approach to the point 1 is along a straight line Xol which does not coincide with a radius ; then the limiting value is the same for this path as for the radial path to 1. (Stolz, Schlomilch's Zeitschrift, t. xxix., p. 127.) dirichlet's problem. 379 I. To an arbitrarily small positive number e, given in advance, tliere corresponds, by the conditions of Theorem I., a length -q such that when an arc sc' of length ij is measured off on sc, we have u{x, y)- U{s)\ <£, for all positions of {x. y) on sc'. Let s^ be the poiut s + 8s ; and let the path at Sj which corresponds to the path sc\ be SiC,', the length of this arc being rj. The points c', Cj', ■•• trace out a continuous line L. Since i((x, y) is continuous along L, and since the path sc varies continuously as s moves along D, there must exist an interval (s — /» to s + h) such that for all points s + 8s in this interval, But, by the conditions of Theorem I., \L\s)-n,.\<(, I U{s + Ss) - «,^. I < £. Hence | U{s + hs) - U{s) \<3e. This proves the continuity of f/(s). Next, let (x, y) be any point interior to the curvilinear quadri- lateral bounded by D, L, and those two curves of the set sc which pass through s — h, s + h. By the conditions imposed initially, {x, y) must lie on some curve of the system sc; suppose that it lies on the curve through s + &s. Hence, \u(x,y)- C7(s + Ss)|<£. But I u{x, y) - U{s) I < I u{x, y) - U{s + 8s) | + j U{s + 8s) - U{s) \ , i.e. \uix,y)-U{s)\<4:e. This shows that a circle can be drawn with s as centre, and with a radius sufftciently small to ensure that for every point {x, y) inte- rior to the circle, the above inequality shall be satisfied. Hence u{x, y) tends to U{s) at s, whatever be the path of approach to that point. II. In the second theorem the data show that each point s of the arc D has a domain, say of radius p, such that for those points (x, y) of r which lie in the domain of s, \u(x,y)-U(s)\ ^€; 380 dirichlet's problem. for otherwise there would be paths sc' such that | u(x, y) — U{s) \ > e for points of sc' arbitrarily close to s, contrary to supposition. Measure off two arcs ss', ss" of lengths p, and let a point {x, y) of the domain of s tend to a point s + Ss of s's". By supposition u{x, y) tends to U{s + Ss). But throughout the domain of s, \u{x,y)-U{s)\^.. Hence | U{s + Ss) - U{s) \ ^ £. This establishes the continuity of 'U{s) . If s/, s/' denote the middle points of ss', ss", there corresponds to every point s + Ss of the arc SiS^" a circular region A, of centre s + Ss and radius >p/2, such that for every point (x, y) common to r and A, \uix,y)-U(s + Bs)\<2c. By reasoning about s/, s/' in the same way as about s, and by a continuation of the process, we must arrive finally at the extremities of D, unless the radius p tends to a limit 0, when the centres s tend to some point Sq. To see that this is an impossible case, observe that there corresponds to Sq a circular region Aq, say of radius po, such that when (x, y) lies inside Aq, \u{x,y)- U(s)\ >£/2, and therefore to the points s near s„ there correspond domains of radii at least equal to po/2. These considerations show that we can find a length ^ such that for points inside a circle of radius t, and centre s, \u{x,y)-U{s) >2e, whatever he the position of s on D. Thus u{x, y) passes with uniform continuity to the rim values U{s). We shall prove, in the next paragraph, sonre theorems which •will be of use at a later stage of the discussion. These theo- rems embody well-known results in the ordinary theory of the potential. § 255. Greenes theorem for functions of two variables. Let Ml, M2 be two functions which are harmonic within V, and let r be a region which is completely enclosed by r'- The functions Ui, Mj, together with their partial differential quotients of the first two orders, are continuous in T and on its boundary C. From the remark of Schwarz quoted in the foot-note to p. 377 it appears that dirichlet's problem. 381 it is not permissible to assume in all cases the existence of partial differential quotients on the boundary of T' ; this is why r is chosen. Green's theorem is Here the differentiation is with regard to a normal drawn into r and the direction of integration over a rim is positive when tlir.t rim is regarded as an independent curve. Since dx/dn = — dy/ds, dy/dn = dx/ds, we have Smj _ _ S"i f/^ . 8m, dx . 8n Sx ds S.y ds' and there is a similar formula for M2. Integration by parts shows that where the first integral is a curvilinear integral taken positively over C. Similarly, By addition we have =X'-(l-*-|-)-XX«■ Similarly rf|?!i>, 838«,)^,^^ r 8u,^^ . . (6). J Jr\bx6x^ t,y &y ) ^ Jc on 382 diuichlet's problem. An important result can be derived from (0) by writing n^ = iu=u ■ namely, JJ | (g J + g^ J } d.d^ = _ J. gj rfs • • (7). This formula shows that there cannot be two harmonic functions ■which take an assigned system of values on C. For suppose that Ui, «,' are two such functions. Replacing »/ — »,' ^y "> '^^'e have M = on C. Hence, if we assume x- to be finite along C, the for- ou mula (7) becomes // ;g^(I)>-»- That is, a sum whose elements cannot be negative vanishes. This can only happen when every element vanishes. Hence &II/&X = 0, 8u/By = 0, and «, — ?(; = a constant throughont r. This constant must be zero ; for otherwise there would be a discontinuity in jiassing to the rim. Thus, if the problem of Diriclilet admit a solution, that solution is unique. Picarcl has pointed out the limitatioiLS to this proof. Besides the necessity of assuming, in general, that " does not become infinite, there is the further Sn limitation that the integrals, as also their differential quotients of the first order, are to remain continuous in r and on C. (Traite d' Analyse, t. ii., fasc. i.,p. 23.) Theorem. | — ds = 0, when u is harmonic within a region r' J con. which encloses V. The formula which results from the combination of (5) and (6) is "■.■^-"■s; *-» (8)- i 'c\ da '&n Replacing Wj, u.j by u, 1, we have •Sm fc&H X — f'« = (9). Theorem. Let r be enclosed as before in T', and let r be the dis- tance of a point {x, y) from a fixed internal point (a, 6), then «(«,.) = iJ(...,|?-»»^').. . . . (10). DIEICHLET S PROBLEM. 383 "With centre (a, b) describe a small circle Ci of radius p. Let t(j = log 7-, Wj = u : then by (8), if the normal to Ci be drawn towards = log p ) — ds + - 1 uds = - r«c7s by (9). P Jci Since the left-hand side is independent of p, the expression J uds must be independent of p. When p tends to zero, we know that an approximate value of - j uds is u{a, ^) • - | cJs, i.e., 2int{a, b). Since this value is inde- p.yc, p«^Pi pendent of p, it must be the exact value, and the theorem is estab- lished. The result thus obtained is of high importance, because it expresses the value of u at an internal point in term of values along a certain boundary. In the course of the proof, we have established incidentally a theorem which has important consequences in the theory of the potential. This theorem will now be stated explicitly. Theorem. A harmonic function u cannot have a maximum or minimum in r. If p be the radius of a circle C with centre (a, b), 2Trv(a,b) = - Cnds= r"u(p,e)dd ■ ■ ■ (11), pjc Jo in polar co-ordinates. Let p be made sufficiently small; then if u{a, b) be greater than all the values within, or on the circum- ference of the circle (p). !77m(o, h)<~ { M(a, b)ds, pJc p- i.e., 2 ttw (a, 6) < 2 ttm (a, 6) . This shows that u{a, b) cannot be a maximum. Similarly it can be shown that it cannot be a minimum. The same result might be deduced from (9). This theorem extended to three dimensions amounts to the statement that the gravitation potential of an attracting mass has neither a maximum nor a minimum value in empty space. It was discovered by Gauss. 384 dirichlet's problem. Theorem. If u{x, y) be a function which is harmonic in any region r, and continuous on the boundary C, the value of u at every point in T is intermediate in value between the greatest and least values on C. Here we assume that there are no points on C at which the function u is discontinuous, and that u is not a constant. The values of u on C liave a finite upper limit G, and a finite lower limit K; and the values in r and on C have a finite upper limit G' and a finite lower limit K'. At no points of T can the limits G', K' be attained. Let us assume, if possible, that /( attains the value G' at an interior point (a, b). The formula (11) shows that on the rim of every circle interior to C, with (a, h) as centre, the value of u= & ; for if some of the values be less than G', others must be greater than this quantity, and this is contrary to suppo- sition. Hence at all points of a circular region of centre (a, b), which lies wholly inside r and extends to C, u is constant. By choosing a new point (o, b) in this circular region and drawing a new circle, it follows that ?t is constant throughout the new region. In this way it can be proved that the value of u in T is everywhere equal to G'. This is contrary to hypothesis ; hence there is no point of r at which u attains the value G'. But we know that the continuous function u attains its upper limit at some point (§ 63). This point must lie on C. As the upper limit on C is G, G' must be equal to G ; and similar reasoning shows that K' must be equal to K. Corollary. A harmonic function which is constant on C has the same constant value in r. § 256. Schivai-z's existence-theorem for a circular region T of radius 1. [Schwarz, Werke, t. ii., p. 185.] Let /(«) be a real function of a, which is continuous and one- valued upon the circumference, and which resumes its value when a increases by 27r; no other restrictions are imposed upon /(«). There exists always one and only one function u which possesses the following properties : — (1) M is continuous in T and on C; (2) Su/8x, Su/By, Sht/Bx-, Wu/hy'^ are one-valued and continuous in r and on C, and 8-m/8x- -f S'w/Si/^ = ; (3) the values on the rim are equal to/(«) at all points of G. The proof is based upon a consideration of the integral ^ = hr^^''^ l-2r costal g) + >- '^"' dirichlet's problem. 385 where r. 6 are the co-ordinates of a point on the rim of a circle whose centre is and radius ?•(< 1). This integral has a perfectly determinate meaning for all points within the circle, and equals the series ~ f'^f {6)0.6 + - i Cy{6) cos n{a -e)de- r". This latter series is convergent in r. On C it assumes the form of Fourier's series. Scliwarz proves that (i.) i, is harmonic in V ; (ii.) ^ takes the contour value /(^). The proof of (i.) is eifected in a very natural and simple manner. We know by the properties of monogenic functions that when <^{z) is holomorphic in r, the real part of <^(j) is harmonic in T. Hence the real part of is hariaonic in r. But this real part is precisely Poisson's integral u(x, y) = t = — I /(«) da. To prove (ii.), Schwarz examines the difference t,—f{6), where 6 is arbitrary, and shows that as (x, y) tends along radii to the various points (1, 6) of C, u(x, y) tends uniformly to the limit /(^). To simplify the denominator of the integral write u + 6 for a; this involves no change of limits, since /(«) is periodic. At any stage of the following proof the limits can be replaced by new limits y, 2Tr + y, where y is any positive or negative quantity. The new form of the integral is J_ r-'' (l-i'')f(n + 6)da 2 IT Jo 1 — 2 r cos « + r^ ' or -1 r*- {l-r')f{a + e)da 2TrJ-. l-2rcos« + 9-2 This may be written J_ r+'T (1 - 1^) If (a + 6) -fjO) \du ^ f{d) r^" {l-r'-)da 27rJ-.r 1— 2rcos« + r* 2Tr J-n l — 2r cos a + r^ 386 dirichlet's problem. For all values of r less than 1, 1^ r+- (l-?-')da ^^ 27rJ-'w 1 — 2 )• COS a + 1-^ Hence ^ = /(^) + f C' ^ -';)?{(" + ^)-/(/) J'^" - ■^^ ^ 2-n-J-n 1-iircosa + i- We have now to show that when e is an arbitrarily small quantity given in advance, it is possible to find a number i; such that for all values of 1 — r L and for all values of 0, IX +>r (l-r-)j / (r4 + g)-/ (g)ic?« 1 — 2 ;• cos « + ;•- <£. This inequality is required in order that the passage to the contour values may be effected with uniform continuity. It will be observed that the paths of approach to the contour are along radii, but this limitation may be removed by means of Painleve's theorems. After replacing the limits by — 8, 2 tt — S, let the integral be sepa- rated into the two parts -L r' + l- r'~' a-^^)lfUc + e)-f(e)\da 2 7rJ-6 27rJ+5 l-2rcos« + r'' where 8 is a small positive quantity. Let g be the upper limit of /(« + 6)—f{0) in the interval (— 8, 8). The absolute value of the former integral is <9- |27rJ-5 1 — 2)-COS« + 7- 27rjJ-4 1 — 2?-C0S« + The absolute value of the latter TT Js 1 — 2 ?• COS a + r^' where G is the upper limit of the absolute values of /(a) in the interval (8, 2)r-8). But 1 p-5 (l-7^)da ^ 1- ttJs 1 — 2?- cos a + 9-^ rCl — + 9-2 r(l-cos8)' for since the expression cos a — cos 8 is negative in the interval in question, the denominator 1 — 2 ?• cos a + r- > 2 r(l — cos 8) . "What- dirichlet's problem. 387 ever be the value of the small quantity B{=f=0), it is possible to choose 1 — r so small that 1 — i-^ G -X^^-) 2i.Ti-^,- ^^' where | c — s | is the distance from c to a point s of C, and U{s) is the value of u at s. When c coincides with 0, this formula becomes '^=1 "-vXs^de or the mean of the values U(s) on the rim of is equal to the value of u at the centre. By a known formula iu Trigonometry R'- 2 Ep cos {a -e) + p' '^AbJ Hence when x = p cos 9, y = p sin 6, ui^, y), =f{p, ^)> =^ rV(«)f^«+- 2 { r7(«)cos nada-cosnd + f "/(«) sin nwdu. ■ sin nB \{-jA = ^ + 2 (a„ cos nB + 6„ sin n 9) (^, 2 n=l V '' / o8» DIRICHLET S PROBLEM. Tliis expansion has been proved on the supposition p R. Is the expansion true whenp=2?? We know that the limit to which Poisson's integral tends, when p tends to R, is u[x, y), or f{R, 0), where x = RcosB, y = BsinO; but it is not evident that the integral series in -£- tends to the limit " R -" + 5 (a,, cos nO + b„ sin n0). Z n=l That this is what actually happens has been proved by Paraf. As soon as it can be shown that the series just written down is convergent, Abel's theorem on integral series (§ 90) enables us to say that this series is the limit of the series in pjR. Paraf s proof of convergence. [Thesis, sur le Probleme de Dirichlet et son extension au cas de I'equation lineaire generale du second ordre, p. 56.] The coefficients a„, 6„ are continuous functions of R, which satisfy the relations | a, |, | 6„ [ < h/n'^, where h is a constant which depends only on the values of f{R, a) and not on i?,. For, after a double integration by parts, a„ = - I f{R, a) cos nada, b„ = - 1 f(R, a) sin nada, TT^a ttJo . C 7"(-K, «) cos nada, b„ = ---~ Cf'iR, «) sin nada. give a =--.- IT TV Hence | a„ | and ! 6„ j < 7i/n^ where h is twice the upper limit of the values | f"(R, a) |. This shows that the series -° + 2 (a„ cos nS + &„ sin nB) is absolutely and uniformly convergent. Hence when p tends to the limit R, f(p, 0) = I + j/;|)"(«n cos ne + b„ sin nO) gives « (a;, 2/) = /( iJ, 6) = ^ + 2 (n, cos nO + b„ sin nO) . dlrichlet's problem. 389 § 258. We have arrived at a series for u{x, y), namely w(2!, y) = ^-\- 2 (a„ cos ne + 6„ sin nO), in which the coefficients are definite integrals. Let us now make a fresh start with the equations 8m _ 8v S?( _ Sw hx 8y By Sa;' or with the corresponding equations in polar co-ordinates 8M_18i' &v__lSu Sr~r60' Br~ rS8' where the pole is at a point (a, b), and (r, 6) are the co-ordinates of an arbitrary point of that circular region of centre (a, b) and radius M which extends to tlie boundary of T. Here r is the region of the z-plane for which the functions u, v and their first differential quo- tients are one-valued and continuous, these differential quotients being subject to the above equations. Since v^* + ^^) an(j {u + iv) ^^^ finite when r = 0, we must 8x By have for all values of 6, lim t=0 86 = 0, lim Sv :0. On a circle of radius r { = - n vGosnede = 0. Ll &6 Jo Since — is a continuous function of both r and 6, Br ("" -sin nede= - C ''u Bin nOdd. Jn Sr SrJo Hence, when w > 1, r^ + nc„ = 0. dr Similarly, multiplication by cos nO gives S«n 7 A r— 2 - nd„ = 0. Sr From equation (ii) we can derive, in a similar manner, the two differential equations dr Sr The equations Jo Sr rJo Se Jo Sr rJo se give ^^'=0, 1^ = 0; or Sr dikichlet's peoblem. 391 therefore cto, Co are constants. To find b„ we must integrate the equation »-^^r + '--"-«^6„=o. bl- br Hence b^ = A„j-" + A„';--" ; but knowing that Su/8r is to be finite for r = 0, it is evident that A„' must vanish. Corresponding to 6„=A„>-'' we have the three equations Hence 00 u+iv = 4-(a„ + ic„) + 5(;u„ - a J (cos nd + i sin n^))-" = P{z-z,), where x,, = a + ib, z — z„= re^'. This is the Cauchy-Taylor theorem. [Harnack, Fundamentalsatze der Functionentheorie, Math. Ami., t. xxi., p. 305.] § 259. In the generalized problem for a simply connected region r the values U(s) on C are in general continuous, but suffer discon- tinuities at a finite number of points a,, ««, ■••, a^, •••, a„. The dis- continuity at a point a is of a special kind. One condition is that for the region common to V and to a small circle of centre a all the values of u are to be finite ; but further the values U{s) are to tend to limits [/"(Oj + 0) or U{a^ — 0), when s tends to a, in the positive or negative sense, where U{(2o) which maps Tj upon T,. Let z, = .r, -j- i'^';, <^(z.?) = 'Xi{x.2, yi)+ i'>^-2(x-2, y-i)- Let u{Xy, y{) be a harmonic function with assigned rim values C/i on d; let the values K assigned to the rim C. be equal to the values Ui at the corresponding points of Cj. The function uQ<-i, >^i),= u'ix.^, yo), 22 ' 5;2 f satisfies the equation - — • + ^-r, = ; dx.r 02/2' Also when (x.,, y^) tends to a point 3-2 of C., u', = u, tends to the assigned rim value C/i(si),= C/.Cs.)) where Si corresponds to s.^. § 261. TJie conform representation of a simply connected region T, bounded by an n-sided rectilinear polygon, tipon the pyositive half-plane. Arrange w + 1 real numbers x„, a^, a.^, ■■■, a„ in ascending order of magnitude, and let z' be defined by z'= C\a,-zyr\a,-zy^-^-- {a„-zyn-'dz, 396 dibichlet's pkoblem. where the X's are positive, and 2 X, = )i - 2. The function z'(z) is K=l holomorphic throughout the positive half-plane, since the critical points Oi, tto, ••-, a„ are all situated on the real axis. Let z move in the positive half-plane, and let b^, b.,,---,K Ije the points in the z'-plane which correspond to cii, a.,, •••, a„. The differential equation dz'/dz = (oi - z)h-\a2 - z) V ... (a„ - z)^""' defines a holomorphic function z{z') at all points in whose neigh- bourhoods the expression on the right-hand side is holomorphic (Chapter V., § 156). Let z describe the real axis except near the points Oi, o,, • • •, o„, round which it describes infinitely small semicircles situated in the positive half-plane, and let it start from a point a to the left of a^. As z moves towards ai, z' is real ; near z = a^ d:'/dzcc(ai-z)^i-^ so that after the negative description of the semicircle round a^, the amplitude of z' becomes 7r(A.i— 1); and as z continues its path from Oi to a^ the point z' moves along a straight line bfii, which makes with the direction bfi an angle ^T\■^. When z describes the small semicircle round a, and moves along the straight line ciMs, the z'-path changes from the straight line 6,62 to the straight line bJD-i, where bjj^^ makes with b-p^ an angle ttX^; and so on for the remaining sides. Since 2X^ = 11 — 2, the point z' describes the closed polygon b-fii--- 6„&i, when z describes the complete axis of x. Further- more, as z moves upwards from the real axis, z' moves into the interior of the polygonal region ; and when z' is once inside, it remains inside as long as z remains above the real axis, since an exit from the region would mean a passage of z' over one of the sides, and therefore a passage of z over the real axis into the negative half-plane. Conversely, while z' remains in the polygonal region, z must remain in the positive half -plane. By the transformation z" = az' + p, one of the corners 6, can be made to coincide with an angular point c of an assigned polygon c,Co---e„C], so that one of the sides shall coincide in direction with 6,&^+i ; and the constants o, X can be chosen so as to make all the sides of the two polygons coincide. The determination of these constants is a matter of considerable difficulty. dirichlet's problem. 397 The form of this solution is due to Christoffel and Schwarz. Schlafli has elaborated a method for solving the equations which involve the constants in a memoir Zur Theorie der conformen xVbbiklung, Crelle, t. Ixxviii. Objections have been urged by J. Riemann against Schlafli's proof of the possibility of the conform representation of any rectilinear polygon whatsoever upon a circle, while the validity of Schlafli's work has been upheld by Phragmfin (Acta Mathematica, t. xiv.). However this may be, there can be no question as to the fact that every rectilinear polygon can be mapped on a circle ; and when this is granted, Schlafli's method provides a definite process for the determination of the con- stants. On the problem of the conform representation, the reader should consult Riemann's Werke, The Inaugural Dissertation ; the many important memoirs on this subject contained in the second volume of Schwarz's Werke ; Darboux's Lemons sur la Theorie Gfinferale des Surfaces, t. i., chapter iv. ; Harnack, Die Grundlagen der Theorie des logarithmischen Potentiales, 1887 ; Picard's Traits d' Analyse, t. ii.. Fascicule 1 ; Laurent's Traitfi d' Analyse, t. vi., chapter vii. He should also consult M. Jules Riemann's valuable commentary on Schwarz's memoirs, Sur le Probl6me de Dirichlet, Ann. Sc. de I'Ecole Xorm. Sup., Ser. u, t. v., 1888 ; Schottky, Ueber die conforme Abbildung mehrfach zusammen- hangender ebener Flachen, Crelle, t. Ixxxiii. ; Christoffel, Sul problema delle temperature stazionarie e la rappresentazione di una data superficie, Annali di Matematica, Ser. 2, t. i., Sopra un problema proposte da Dirichlet, ib., t. iv., Ueber die Integration von zwei partiellen Differentialgleichungen, Gottingen Nachrichten, nr. 18, 1871. By the use of inversion (a special case of orthomorphic transformation), Dirichlet's problem for a region which encloses the point ao can be made to depend upon the simpler problem for a region in the finite part of the plane. Suppose that two regions F, Fi can be represented conformly on each other by X + i>j = n C and throughout the exterior plane, or some values of u on C are negative. In the latter case the value at oo is neither a maximum nor a minimum, and the extreme values for the region lie on the boundary C. [See Harnack, Grundlagen, p. 48. Further theorems relating to the point ao are to be found in § 27 of the same work.] § 262. Schwarz's method for solving Dirichlet's Problem employs theorems from the theory of conform representation relative to regions bounded by regular arcs of analytic lines. After showing that a simply connected region bounded by regular arcs of analytic lines can be mapped upon a circle of assigned radius, Schwarz employs his alternating process and proves the existence theorem of the harmonic function for simply and multiply connected regions with rims of highly arbitrary shajDCS.* Regular arcs of analytic lines. Let P(? — ?„), Q{t — io), where to is real, be two integral series with real coefficients. These series can be continued along the real axis, by the method explained in Chapter III., until further progress is debarred by the presence of singular points. In this way t\vo functions /](<), /^(t) are defined for two portions of the real axis which contain t(,. Let these por- tions overlap along a stroke tit.J, and let this stroke be shortened by making fj.), for every interior point {x, y); hence, when the point (a;, y) tends to a point s of C, |m„+i + «„+2H 1- «„+, I tends to a limit which cannot be greater than t. We proceed to the proof of two important theorems due to Harnack. Theorem I. Let the component terms of the series 2 w, be har- monic in T and continuous on C, and let u^{x, y) tend to U,{s) when the point {x, y) in r tends to a point s of C by any route dirichlet's problem. 401 whatsoever in r. Then if the series m = 2 w, be uniformly conver- gent for the points of C, it must also 'he uniformly convergent throughout r, and u{x, y) tends to U{s) = ^ U,(s), whatever be the route in r from {x, y) to s. This theorem is the converse of that just established. To prove it we must show that ('•) 2 i( is continuous in r, (ii-) 2 «, is harmonic in r. K=l (i.) Let S„= L\+U,+ - + L\, 0-n = "l + «2H !-!(„, pPn = ■"„+! + M„+2 H h "„+p, P,. = W«+l + '<»+2H to 00. If c be given in advance, we can find a number /u such that |,i?„!/x), for all points of C. But j,p„ is harmonic in r, and tends to p^„, when the interior point (x, y) tends to a point s of C. Hence | pp„ | < e throughout T. Thus the series 1u^ is uniformly convergent in r ; and the sum u of this series must consequently be continuous in V. (ii.) Next we have to prove that u is harmonic in r. To prove this, select any point c of r, and describe round c as centre a circle Ci of radius R, such that the enclosed region Tj lies wholly within r. Let a function v.' be constructed which has the same values as u upon Ci and is harmonic throughout T^. For the points Sj of Ci the series 1<'(S,) = Wi(Si) + U,{Si) + «3(Si) + ••• is uniformly convergent ; let it be integrated term by term after multiplication by {R- - r)dtt/(R^ - 2 Rr cos (« - 6) + !^). Hence where r, tf are the polar co-ordinates of a point (x, y) in Tj, and ii„(jB, «) is the value of o.(x. v\ at {R, a). 402 dirichlet's problem. Since j p„ j < e upon Ci, it follows that lim , ■■■, Cj be constructed in r, such that Co cuts Ci, Cj cuts Cj. •••, Ci cuts CVi, the circles being so selected that c', c lie in Ci, C\. Since 2?<„ is convergent in C\, it must be convergent in C.>, Cs, •", Cj; also the convergence is uniform. By Harnack's first theorem, the function defined by the series is harmonic. § 264. Schwarz's solution of the problem of Dirichlet is effected by the employment partly of the method of conform representation, partly of a process known as the alternating process. The object of the latter process is to effect the solution for a composite region T] + To — r', supposing that the solution is known for tico component parts T], r,, which overlap along a region V Schwarz first establishes an important lemma. Suppose that the contours of T,. r, (which are taken to be simple) intersect at two points a ; and that T, is divided into arcs Cj, C^ and To into arcs C., Ci; the arcs C3, d bounding V. (i.) Let ?/i be a function which is harmonic in Fj and which takes values 1, on C], CV The values of u^ in T, must be all 404 dikichlet's pkoblem. positive and less than 1. Upon Ci there must be an upper limit for the values of Ui, say 0. This number cannot be greater than 1, for all the values of it, in r, lie in the interval (0, 1). It cannot be 1 inside the region T, ; nor can it be 1 at a point a where the line C4 meets the rim of Tj, for this would imply that the line Cj touches d- Hence 6 is a proper fraction. (ii.) Let values with an upper limit G be assigned along C,, sucli that there exists in Tj a harmonic function m, with these values on Ci and the value on C3. Then Gui + n., is upon C3, is nowhere negative on Ci, and therefore nowhere negative inside r,. Its values along Ci are consequently positive. Writing Gu, + u, = G9+ v., + G(wi - e), and noticing that the second term is nowhere positive on C,, it follows that G9 + it, is nowhere negative on C4. Similarly, from the function Gu^ — lu, it follows that GO — % is nowhere negative on C4. Hence, at each point of C4, I u, , > GB. With the aid of this preliminary proposition, Schwarz establishes his method of combination. Let values f/be assigned for the lines Ci, C-,, and let (?, K be the upper and lower limits of these rim values. For the surface F, there can be constructed a function it, which takes the prescribed values {/on C, and the value K on C3. This can be done since the generalized j)roblem can always be reduced to the ordinary problem. As the line C, lies within F,, the values ? alon;:: Cn, must be > /r. Hence «2 — "1 must be positive in r'; for on C^ it vanishes, and on C3 it equals nP' — u{'''> = (a positive quantity which is equal to or greater than K) — K. Thus lu — m, is positive throughout V. Next, two new functions ?<„ v, are constructed for T,, V«. Along C„ W3 takes the values U; along C3 it takes the values u.P^- By reasoning similar to that already used, u^ ^ m,. Let the values of W3 along Ci be Mj**'. Along Co, u^ takes the values U; along C4, it4 takes the values Wj'^' ; therefore u^ ^ «„. Along C3 call the values of Ui, M4<". I5y proceeding in this way, two series arise, Wl+(M3-Wl) + («5-'0+---+(«2n+l-W2„ 1)+ • ' (^), Mj + {Ui - M2) + ( !(„ - Ui) H h («2„ - il2„_2) + • ( JS) , dirichlet's phoblem. 405 whose terms are all positive. Along C, two terms of the same rank in {A), (B) are equal, while along C, a term of rank k in (A) is equal to the term of rank k - lin {B). The convergence of the series can be established by jjroving the existence of lim m.,„+i, lim n.,. The existence of the former limit for r, follows from the fart that «„ u.., u., ... are in ascending order of magnitude and all less than G; hence, if lim (^,„^, = ,t', the sum of the series {A) is v' Similarly, the sum of (B) is u", where u" = lim u.„. From the fact that all the terms of {A) and (B) are positive, it follows that «', m" are harmonic (§ 263). The series under consideration are uniformly convergent. For consider, in the first place, the term >,,-u,. As this term vanishes on C\ and is less than G~K{=D) on C„ it must be less than D throughout T^. In particular where 6^ is a proper fraction. Hence in r. In particular w^'^ _ „,(3) < e^Q^D_ ^yhere 6. is a proper fraction. The term u^ — u^ vanishes 011 C^, and takes along C3 the value M4''' — M2*''. Hence in Ti, A continuation of this process of reasoning leads to the results «2„+i- "2,-1 < ^(^1^2)"-' ill Ti, «2„+2 - «2» < .0-^1 (^1^0) "-Mn r,. These results prove that the series u', n" are uniformly con- vergent. Since the above inequalities apply also to rim values, it follows that the two series converge to sums U', U", where C7'' = lim U2„+i and C7" = lim U.,„. The functions u', u" are equal throughout r' and are both harmonic. The function V takes the values U along Ci ; and the function U" the values U along G,. Along Cy and Ci the values of U', U" are equal, since at a point in these lines lim U'^,,+2 = li"i U^n+st by hypothesis. 406 dieichlet's pkobleji. The following reasoning shows that the limiting values to which u', u" tend, when they approach a point a at which C^, C, meet by some path interior to r', are equal. Let the path in question make angles (^i, <^3 with C,. Cj, and angles .,, <^4 with Cj, C,. Then the limiting values to which u', u" tend at a are F'=(n)/i-^V(t^)/i-- But ( cr,). = ( u,).(i - ^i^yi u,)/i - ^^^ ( f/-3)„ = ( ro<.^i - ^^)+ ( c^;)/i - ^^y Hence, using (^j + <^. = tt, <^o + c^^ = tt, 93 + ) + («g — 't4) + "-. By the use of Harnack's theorem these series can be shown to be uniformly convergent. For if the maximum value of I )(, — n-j \ on Q, be G, the maximum value of the same expression on Qo < Gd.,. On both banks of Q.,, Ui — u. = ru — ?(<. Therefore | xu — u^ ; is a function which equals on the rim of r, and ^ GQi on Q^. Its value on Q,, a line interior to r,, must therefore ^ G6id-,. Similarly, I «3 — "5 ' ^ G61O.,, I !<4 — «6 ' ^ GdiOi', etc. The uniform convergence of the series can be established by exactly the same method as before, and the sums be proved equal to lim ?(2„+i. lim Uo„. On the rim Qr + Ci + Q/ + Co of r', u' = u" + A ; hence in the interior of r' the snme equation holds. On the rim of F — V, u' = w" ; there- fore in r — r'. )(' = 7(" The function u' is precisely the solution required by the con- ditions of the problem for Tj. For let u = n' throughout T-^; the analytic continuation of %i over the line Qi is u", which equals u' — A. The function u defined by u' and its continuation w" is the required solution. § 267. Greenes funi'tion. L^t {x, y) be a point in a region T, or on the contour C, whose distance from an interior point (a, b), dirichlet's problem. 409 or c, is r. "We shall prove presently that there exists a function g(x, y) which is equal to log l/c for all points (x, y) on C, and which is harmonic in T. This function is often called Green's function, but Kleiu has pointed out that this name is more suitably- given to that one-valued potential function which is infinite at (a, b) like log r for r = 0, and is continiaous elsewhere in r, the rim values being zero. Green's function G(x, y) is therefore ij{x, y) — log 1/r. To avoid confusion we shall speak of g as the (/-function ; also for the sake of simplicity we shall sujipose the region r to be simply connected. Assuming the existence of g{x, y) for the region r, we proceed to the proof of some of the properties which make it specially suitable as an auxiliary potential function in the investigation of Dirichlet's problem. Since 6r(x, y) is upon C, and a very large negative quantity near c, the function must be negative throughout r. Formally the jiroof runs as follows : Describe a small circle C round c ; then the function G is zero on C, negative on C, and har- monic throughout the region between C and C. The function must therefore be constantly negative within this region. Changing from Green's function to the (/-function, tlie theorem states that througliout V the value of g is less than that of log l/c. Tlieorem. Let Cj, c, be two points in r, and let (/„ g.^ be the cor- responding (/-functions ; then the value of gf, at c> is equal to the value of (/o at Cj. Let C" be a curve, interior to C and arbitrarily close to C, upon which Green's function G^ (relative to fj) takes the constant nega- tive value «. Ultimately, « will be made to tend to zero. Apply Green's theorem J J ( hx hx^ 8y 8y ) -^ J^ ' ' Bn ' where G<, is Green's function relative to c.,. to that region of T which is bounded by C", and two small circles of radii pi, p^ round Cj, Co as centres. The simple integral is taken positively over the outer rim and negatively over the inner rims with respect to the region, and the normal is drawn inwards. The part relative to C vanishes; the remaining two parts contribute = _ J((?, _ «) ^-^p.cW -f{G, - a) ^^p/W -J(G, - a)cW. 410 Now let pi, p2 tend to ; the expression reduces to - (\Q,-u)de, or -2w{G,-a),^. Hence, when « becomes zero, Symmetry shows that the light-haud side can also be written 27r((?,),,. Hence, (G,)c,= (G.)v and {gi)c={g-^c,, a result of great importance. The next property to be proved is that when &, tends to a point s on the rim, g., at C] tends to log 1/r, where r is the distance of a point Ci of r from s. Let c, lie on a curve for which j/i— logl/)'i=«, where Vi is the distance of a point of the curve from q, and « is a negative constant which is arbitrarily small. Since the value of g., at Ci is equal to the value of g", at c.,-, it follows that the value of g., at Ci is logl/|C,— e^j + «, where \ci—c., rejiresents the distance ai)art of Cj and C2. When Cj tends to s, a tends to and the function g., tends to log 1/ I Ci—s I, i.e. log 1/r. Let V be the region which remains after the removal of a small region which contains s and c,; the passage of g., to the rim values log 1/r for the region r' is uniformly contin- uous. For (72 — logl/)-2, where r^ is the distance from c, of anj'' point in r', or on its rim, is equal to the small negative quantity a at Cj, is harmonic in r', and negative or zero on the rim of r'- Hence, at any interior point g, — log l/j-j is equal to the product of a by a finite quantity (see § 2G3). When Cj tends to s, a tends to and the function g^ in r' tends with uniform continuity to log 1/r. Further, all the partial derivatives at Cj of the harmonic func- tion G = gt — log l/)-2 tend to when Co tends to s. For at each point p, ^ of a circle with Cj as centre and R as radius, we have (§ 258) 8p TrRn=l \R ^^(^^±in('> cosnd ]'' G' cos ned6+ sin n$ C^^G'sinnOdO — s'mne f'^'' G'cosnOdO+cosnO \''G'sinnedO p 86 itR,^\ \R_ where G' stands for the values on the rim of the circle. [Harnack, Existenzbeweise zur Theorie des Potentiales, Math. Ann., t. xxxv.J dirichlet's problem. 411 The absolute value of each integral in the brackets is increased by the suppression of the factors cos nO, sin nd in the integrands. But the integrals then become | ' G'(W = 2TrG^, vrhere 6r„ is the value of G at the centre of the circle ; and the new series converge. Hence, throughout the circle {li), \SG/dp , '^8G/86 are equal to Go multiplied by tinite quantities. Therefore, when G,, tends to 0, the expressions S(?/Sp and SG/p&O tend with uniform continuity to for all points of the circle (It). l!y selecting a point in (li) as centre for a new circle, it can be shown that the same theorem holds for the new circle, and so on until the whole region T has been accounted for. Similarly, every partial derivative of G tends to under the above conditions. JIarnack's Method. § 26S. Suppose that the solution can be found for Dirichlet's problem in the case of any simply connected region bounded by a rectilinear polygon. Harnack has shown that the f/-function can then be found for any region whatsoever in the plane of x, y, and thence that Dirichlet's problem can be solved with complete gener- ality. The theorem may be proved for simply connected regions and the passage to multiply connected regions eifected by the alter- nating process in the manner described above.* Regard the region r as the limit of a series of regions T], Tj, •••, bounded by rectilinear polj"gons which are subject to the following conditions : — (i.) Each polygon is supposed to be interior to V; (ii.) The rim of r„. , lies wholly within the rim of r„, whatever be the value of n ; (iii.) As n tends to cc, the rim of r„ tends to coincidence with C. One method for satisfying (iii.) is to select a sequence of de- scending real positive numbers (f4i, «,, •••) which deiines (see § .jO), and then let C„ lie entirely within that region of T which is covered by a circle of radius «„, when its centre describes the com- plete boundary C. Construct for each of these polygons the g- f unction with respect to a point c of T; and let these gr-f unctions be * Harnack defines a simply connected continnum in the finite part of the (x, »/) plane by raeana of masses of points. See Die Grundlagen der Theorie dcs Logarithmischen Potenliales, §39. 412 dirichlet's problem. 9i, g.,, ■••. Upon the rim of r„ ^r, = log 1/r and g., < log 1/r, where r is tlie distance of a point of the rim from c. Therefore throughout Ti 0i>U-2>g3>---- The same reasoning shows that within the larger region Fj,, whatever be the value of the positive integer p. Harnack's theo- rems with regard to series of harmonic functions show that g = lim g,^ = gi + {g-, - (/i) + (g^ -g-2)+-; is convergent. For each of the quantities g„ is greater than log l/fx, where ju, is tlie maximum distance of c from C, therefore lim f/„ exists and is finite ; and further the terms in brackets are harmonic and all negative. Thus by Theorem II. of § 2C3, the function g is harmonic in T. It must be proved that g takes on C the proper rim values, viz. log 1/r. To simplify the proof Harnack introduced in the place of g tlie new function <^ defined by (^ = re". The harmonic function <^ is equal to 1 on C and has the value at c; its values at points inte- rior to r and other than c are all included between and 1. Con- sider the curve 4> = ^'i where A; is a proper fraction. This curve cannot be a closed curve which excludes c, for it would follow that log is constant on the perimeter of a closed curve and harmonic inside, and therefore, also, constant throughout T. That the curve must be a closed curve can be proved from the properties of harmonic functions ; for a point at which the curve stops abruptly would be a point at which log <^ is a maximum or minimum. As the value fc changes from to 1, the curves of level expand from an infinitely small curve round c to the complete boundary. Let T be contained wholly within a region T' bounded by a rectilinear polygon C", and let one of the sides of C pass through a point of C. Suppose that g' is the [/-function for V relative to the point c, and that i//= re"'. We have to prove that <^ = 1 at each point s of C. Let T' be so selected that s is the point common to C" and C. Then, when {x, y) approaches s along any path (interior to T) which ends at s, the values of 41, „, for all orders of n however great. But when the point (x, y) tends to .1, the function 1// tends, by the definition of the gr-function, to 1. Hence the limit of is also 1.* * This moditicatiou of Harnack's proof is due to M. .lules Riemann. dieichlet's problem. 413 The existence of the gi-function has been established for a simply connected region whose rim is subject to the sole restrictions that it is not to intersect itself and that it is to be a continuous line. The rim may suffer inhnitely many abrupt changes of direction ; it is not even in fact necessary that it should have a determinate direction at any of its points, or an element of length. The existence of the (/-function for a multiply connected region in the plane of x, y, follows by the use of Schwarz's method for creating a multiply connected surface by eojnbinations of simply connected surfaces. § 269. "With the help of Green's function it is possible to repre- sent any simply connected region V in the finite part of the plane, conformly upon a circle of radius unity. Let Green's function G{x. y) = (j{x, y) — log !/?• be constructed with respect to an interior point (a, b) of T. Draw a line from (a, b) to the rim C, and let the surface r be called r' when this line is treated as a barrier. Let H(x, y) be conjugate to G{x, y), so that ^<^'')-£:(£*-f"^' ,\hx 8y The values of H{x, y) are determined (to a constant term) in r' and differ on opposite banks of the barrier by 2-. When (x, y) describs^s a line G = constant, the equation BH/8^ = — &G/&ti shows that H increases constantly ; therefore G + iH takes an assigned value only once in r. Consider now the function The values of this function in V are all less than 1, and on C the absolute value is precisely 1. The point (a, b) passes into the origin 0' of the z'-plane and the points of T correspond one-to-one with the points of a region T' in the z'-plane, bounded by a circle C of centre 0' and radius unity. It will be remembered that H con- tained an undetermined constant ; hence, assumintj that the points of C, C are related one-to-one by the above transformation, the simply connected region T can be mapped upon the circular region r' so that the arbitrarily selected interior point (a, b) becomes 0', and an arbitrarily selected point ef C becomes an assigned point of C". Harnack has called attention to this assumption. Writing z' = ze ,g+m when 2 tends to a point Zo of C, z' tends to C", but it is not evident that it tends to a definite point of C. Painleve has removed this 414 dirichlet's problem. obstacle to a complete proof for the case which we have been con- sidering, namely, that of a rim with continuous curvature except at a finite number of points at which there is an abrupt change of direction. [Sur la theorie de la representation conforme, Comptes Itendus, t. cxii., p. Goo, 1891. J In this proof we have had occasion to make use of the function II(x, y) conjugate to (?(.v, y). It is worth while to call the reader's attention to the fact tliat when u is harmonic in an «-ply connected surface, the conjugate function has periods at some of ?; — 1 barriers (§ 1C8). § 270. It is now possible to comjilete the solution of Dirichlet's problem for a simply connected region r, whose rim C has an inte- grable element ds and suffers only a finite number of abrupt changes of direction. Here, as elsewhere, we assume that at a point s where there is such a change of direction the two parts of the curve C which meet at s do not touch. Harnack assumes the existence of a function u wliich is harmonic in T and has assigned rim values U{s), and arrives at a formiila which gives the value at any interior point c whatsoever. The values u in T are to pass with uniform continuity into the values U{i^) on the rim. First, then, we assume the existence of a function u with the given values U{s) on the rim. In order to arrive at the formula referred to above, let us suppose that F is contained in a slightly larger region V This supposition is made in order to avoid difficul- ties as to integration over a natural boundary (§ 104). At a point c of r whose co-ordinates are x, y, u is given by uix,y) = A Cu'-^^as-l-C'Jllo,l/r.cIs • (1), 1-Jc bn 1 T Jc bn where r is the distance of a point s of C from c. Xow, although we have proved the existence in T of the (/-function relative to c, we know nothing about the nature of E(j/6n along C. For this reason let us consider, in place of C, a curve C" which is interior and very close to C; and let us form, relatively to C", the equation '^"l-'',-'/f^ys'=o .... (2), bn bn J where accents denote that the values are those taken by U, 8f//S)i, •.., on C", and where Sg'/6n' is constructed with respect to an inner fJ( dirichlet's problem. 415 normal. When C" approaches indefinitely near to C, rj' tends to log 1/r and SU'/&n' to SU/Sn (for u is supposed to exist in a region which extends beyond T). Hence liin fcj' SU'/8n' ■ ch' exists and equals \ (joU/dn ■ ds ; and therefore also lim j U' Sc/' /Sn' ■ ds' exists 1 11 c=cj and equals the same quantity. Subtracting (!') from (1), we have «=i Cu'J^ds-±lunfu'^^':ds' (3). This is the formula to which reference was made above. "We shall now consider the expression ZttJ on 'J,Trc=cJ Sit' on its own merits, without reference to what precedes. The prob- lem is to show that this exjyression defines a function u u-hich is har- monic in r and tal:es the rim values U{s). Consider separately the two parts of which it is composed. I. The integral fu ^ ^°" V'' fe The form of this integral shows that some restriction must be placed upon the rim C, since an integration is to be effected with regard to an arc ds. The rim must possess an integrable arc ds. Harnack, to simplify the work as far as possible, imposes the further restrictions that the rim C is to have, in general, a definite tangent whose direc- tion alters continuously along C; and that cos i/f, where ij/ is the inclination of the tangent to the axis of x (an arbitrary line), is not to take the same value at infinitely many points of C, unless it be along a rectilinear portion of C The geometric significance of these restrictions can be found without difficulty. The curve C is assumed to have only a finite number of points at which the tangent alters abruptly. Suppose, if possible, that the axis of x meets the curve in infinitely many points ; between two consecutive pioints there must be a jjoint at which the tangent is parallel to the axis of X, or else a point at which the tangent changes abruptly. But the latter points are finite in number ; hence the number of points at which cos i/' = is infinite, contrary to hypothesis. Thus no straight line (not included in C) can intersect C in infinitely many points. A simple meaning can be assigned to the expression "| ' ds. Whatever be the position of the interior point c, let $ stand for the 416 dirichlet's problem. angle between the normal dn (drawn inwards) and tlie line from ds to c. Then SIoq;!//- „, and the expression ^—^ds = the angle subtended by ds at c Sn = {ds),. in Xeumanu's notation. (Xeumann, Abel'sche Integrale, 2d ed., p. 403.) The element (ds), (i.e. the apparent magnitude of ds as viewed from c) is to be treated as positive or negative according as the inner or outer side of ds is turned towards c. Suppose that the point c tends to a point t of the rim C. The rim C can be separated into two parts Cj, C-,, the former of which contains t; we suppose that Ci is small. Call ?(i, n., the two parts of the integral which correspond to Cj, Co respectively. When c moves to t, the integrand of jt, remains iinite and continuous. In the limit, when the extremities of Ca tend to t, let u., tend to | U{ds)„ whatever be the position of the interior point c. Since the length of Ci is at our disposal, and since the values U{s) are continuous near t, Ci can be chosen so short that for every point of its arc we shall have, in virtue of the continuity of U, I U- U, I < c, where £ is arbitrarily small and fjis any value of the function V{s) upon Ci. Then «i- U,f{dSi),'<.J\{dSi)/.. When c moves up to t, | (ds,)^ becomes that angle between the two lines from t to the extremities of Cj which does not contain c. Let Cj shrink indefinitely; then | (dsi), tends to a limit 2-77 — a, where a is the angle between the two tangents at t measured ^cithin r, this angle being ordinarily equal to tt. In the limit when Ci shrinks up to t and c moves to t, lim «, = ttU,, or (2:7 — «) U„ and ^ lim («i + v,), or — lim Cu^^2SlIlds, JiTT 2 IT J hn dieichlet's pkoblem. 417 according as the point t is, or is not, a point at which the tangent changes its direction abrupth-. II. The integral — lim i'u'^ds'. We have to prove that the value of this integral is *- w 77*/ In the first place the value of the integral must be defined more precisely, for the values U' at the points of C" are not known. Let the values U' assigned to the jjoints of C" be equal to the values U at corresponding points of C, the correspondence between the points of C and C" being in general one-to-one. The method bj- which Harnack effects this is to separate the curve C into parts which are concave and into parts which are convex to the axis of x. The former parts are translated through an arbitrarily small distance S parallel to the axis of y downwards, and the other parts through the same distance upwards. It is clear that a point s' of the new curve C" will correspond to that point s of the original curve C, wliich is at a distance 8 from s'. As the curve C" is to be wholly comprised within C, the above method has to be modified near points of C, at which the tangent is vertical or suffers an abrupt change of direc- tion. Consider the case in which a tangent, parallel to the axis of y, touches C externally ; near the point of contact there must be a vertical chord which passes through two points of C at a distance 2S apart. The translation-process brings these two points together, at a point p say ; these two points and all the points between the chord and the tangent are to be represented by p. It is true that there is a discontinuity in the values of U at p, but this discontinuity can be made arbitrarily small by making S tend to 0. The case in which a vertical tangent touches C internally differs very slightly from the preceding. The point of contact changes into two points, and there is a gap in the curve C. This gap can be filled up by connecting the two extremities of the separate arcs by an arbitrarily short arc interior to C. Let all the points of this arc correspond to the single point of contact on C; thus the value of U' on this short arc is constant. 418 dirichlet's problem. § 271. Having assigned in this way a definite meaning to the U' of — \ U' • -ch', let us now consider the function 2 ttJ n' 2 77 J 8u' ' where g' is the value on C" of the [/-function for r relative to a point c, and '-- is the derivative with regard to a normal of C drawn inwards, C/" replacing V. Let (.r', y') be a point on C, and let (a, b) be the point c. Con- struct the functions g{x, y; a, b) and g{x, y ; x', y') relative to (a, b) and (x', y'). Then g{x', y' ; a, b) = g{a, b; x', y'). But in the neighbourhood of (a, b) the function on the right- hand side can be expressed by a Fourier series, when (a, b) is chosen as the origin for polar co-ordinates. The coefficients in this expan- sion can be expressed, in the neighbourhood of (o, b), as definite integrals, which can in turn be expressed as integral series in x', y' throughout the neighbourhood of (a;', ?/'). In this way it can be proved that all the composite derivatives of g with regard to a, b, x', y' are harmonic in the neighbourhoods of (a, b), {x', y'). Therefore p. + ^^ = ^ Cu^^ + ^cls' = 0. Sa- 86= 2 77 J Sn\Sd- 8b-J This proves that v is harmonic in T. Now when c tends to a point t of C, g' and -f- pass with uniform continuity into log 1/r and — log 1/r, where )• stands for the distance of t from the points of C. Hence when c tends to the point t on C, v tends to a value V which is the same as the value of — j U— log 1/r ■ ds'. As C" 2 77*/ 0)1 tends to C, the values of the harmonic function v and the rim values V alter. AYe have to determine the exact value of the limit V to which V tends. As before, separate C into two parts, one being an arbitrarily short arc s, on which t is situated. On s, the values of the continuous function U differ by less than e. Through the extremities of s, draw vertical lines to cut C" in an arc s/ parallel to Sj ; denote the remainders of C, C by s.,, sJ. Xow let the parts of the integral which are associated Avith s/ and s.,' be ^=fJ^al.^°^"V'--'^^- dirichlet's peoblem. 419 and ^=hf^t'^°°^^''"^''- When 8 tends to 0, the curve C tends to C and the values of r in B pass -^-ith uniform continuity into tlie lengths of the radii from t to points of So. Therefore lim B = -}- fc/A log 1/r ■ ds, = A Cu{ds.,)„ - TTty OH J TTtJ the passage to the limit being accomplished with uniform continuity for all points t of the rim. To find lim A, denote as before the value of U at t by [",. The values of U on the arc s/ can be made to differ from d by an arbitrarily small amount by diminishing the length of the arc. Thus the equation yields the inequality A-;^Lrf{ds,'),\<,.± f\{ds,'),. -TT J I lizj I But I (cZs/), = — (the angle which s/ subtends at f ) = — /8', where the negative sign is used because it is the outer side of ds/ which is turned towards t. As C moves up to its limiting position C, y8' tends to the value /3, where /? is the angle between the two lines from t to the extremities of Sy. When the arc S; is made to shrink indefinitely, /3 tends to the limit ir or a, where a has the same meaning as before, and we have \xmA = -\v,ox-^U„ lim B=^ lim frCds,), = ^ ^V(ds), ; and F; = lim.-l + limS = -^U;+^ ^Vids),, or V, = -^l\+^ fu{ds),. It remains to be shown that the passage of F]' to the limit F, is effected with uniform continuity for all positions of t\ in other words, that there is always a position of C such that the values of v 420 dirichlet's problem. relative to C" differ from V, by less than an arbitrarily small posi- tive number tj. To prove this, observe that TV = - 11 L' + ,f f C7' A log 1/r'dsJ + c„ _ r J 7r»/»'2 on V,=-fu, + ^ f U^ log 1/rds, + c,. Here r, r' are the distances of t from corresponding points of C, C". In considering T^' — V,. we have to examine tj — to, /3' — j8, and i- r6^'A]ogl/;-'rf,sV-J^ frllogl/rds,. The absolute value of the first of these can be made less than 77/3 by choosing Si sufficiently small ; the difference between /S and /3' can be made less than ry/3 by making C approach sufficiently near to C, i.e. bj- making S less than some number 81 ; finally the third expression can be made less than t;/3 by making 8 less than some number Sj. for r' can be made to differ arbitrarily little from r by making 8 sufficiently small. Here 81, &> are determined for all points t of C. Now let (8', 8", 8'", •••) be a regular sequence which defines 0, the members of the sequence being positive numbers arranged in descending order of magnitude. Let the harmonic functions v which correspond to the values 8', 8", 8'", •••, of 8 be r', v'\ v'", •••, and let the values of these functions at t be Fi „ V-,,,, Vs,,, ••■■ ^Ve have just seen that the sequence (Fi,„ V-,,„ T^,,,, •••) defines a value V, ; that is, F,,, +(Fo,, - Fj,,) + (F3,, - F,,)+ ••• =V,. By § 263 the v' + {v"-v') + {v"'-v") + --- defines a function v which is harmonic in T and continuous on C; and the value of this function at the point t is V,. The combination of the two integrals 27rJ hi 2rr=cJ 8)^' is a function u which is harmonic in V and passes with uniform continuity into the assigned rim values U. Thus Dirichlet's problem has been solved and solved uniquely. § 272. Before passing on to the discussion of potential functions with discontinuities we shall establish a proposition which relates to Dirichlet's problem for the circle. Suppose that u is the harmonic DIRICHLET S PROBLEM. 421 function which takes assigned values (7(ij) on a circle of radius R, and let a concentric circle be described whose radius p is less than It. The diiference between a value u on the circle (p) and the value w„ at the centre is — p{p — R cos a )da + e) E- — 2Rp cos a + p- 7r«/o=0 \ p sm a \/R'- — 1' Rp cos a +p-. The expression sin~ value sin~'p/^. p sm n Vi2' — - Rp cos « + p'. has for its maximum Suppose that as a increases from to 2 tt this maximum is first attained when a = =3lT— /• is less than G sirr'p/R, and p sm« f{a + e)d[ sin->- "=2"-* \ ~\/R'— 2 Rp COS u+p-. —/ is harmonic throughout T). [Harnack, Grundlagen der Theovie des Logarithmischen Potentiales, § 4.'); Neumann, Abel'sche Integrale, 2d ed., pp. 44o, 4G1 ; Schwarz, Werke, t. ii., pp. 163, 166.] § 274. Take a circular region ; call this region Ti, and let its rim be Ci. Let C-j be a concentric circle of smaller radius, such that all the places of discontinuity of / are situated within the circular region bounded by C,. Let the region exterior to Co be called Tj. Along the rim of any circle with centre and radius intermediate to the radii of C\, C^, — I fdO = a constant A ; '2tJo or, replacing/— A by /, :0. 1 r-" JttJo Apply Schwarz's alternating process to the two regions T„ T^. Construct for Tj a harmonic function ?(, with the values F.2 on C.,, where F2 stands for the values of / on C-j ; for Ti a harmonic func- tion M2 with the values «i"'— i^j on Ci, where Wi<'* stands for the values of ? — F„ and ^ is a proper fraction which depends solely upon the radii of C'j, C.,. Since u-^ takes on Co the values u.P + F,, we have Since «,,,<-' — m/^) _ „ un^ ^he absolute value and the oscillation of Its — ? 1, it is also a branch-place at which r sheets hang together. Since, near s, R = {z-cy^"Po{z-cy', the value of j Rdz is Jiz - c)V'jao -I- a,(2 - c)'/' + a.iz - cY" + - \dz, = bo{z - c)".""-"- + h,{z- c)<«.+'-+»/' + ■■: Case I. If qj + r>0, | Rdz = P, ^,(2 — c)'/'. Here s is a branch- place, or an ordinary place, according as r^ 1. tione of Kronecker are one-to-one transforraationfl of a curve in the plane. The former trane- formationa separate coincident tans^cnts, but do not resolve a multiple point with distinct tan- gents into its component nodes. ABELIAN INTEGRALS. 429 Case IT. If q^ + r^ 0, \etqi + r = -t; then jRdz = bo{z -c)-"-+b,{z - cy'-')/'+... +j)^ log (z-c) + P{z - c)'/'. Here s is both a branch-place and an infinity. The nature of the infinity depends on the values of the coefficients. When bo = b, = b,= --- = b,_, = 0, the infinity is purely logarithmic. On the other hand, if b, = 0, the infinity is purely algebraic; but if bo=bi = b.,= ■■■ = b, =0, we fall back on Case I. For every other system of values the infinity is logarithmic-algebraic. Case II. shows that the integral is infinite at an infinity of E. except when that infinity is a branch-place on T such that 9, + ?■ > (>. We have established the fact that the integral has only a finite num- ber of infinities which are algebraic, logarithmic, or logarithmic- algebraic; and that no other infinities can occur. An integral is said to be of the first kind when it has no infinity on T, of the second kind when its infinities are algebraic solely, and of the third kind when it has logarithmic or logarithmic-algebraic infinities. The simplest integral of the second kind is that which has only one infinity on T, that infinity being a pole s of the first order; the integral of the second kind is then said to be elementary. The simplest integral of the third kind is that which is infinite at two places of T like + log (z — c,) and — log (z — c.,), and is elsewhere finite ; the integral of the third kind is then said to be elementary. We shall denote elementary integrals of the second and third kinds by Z and Q, and shall indicate by suflBxes the places at which the integrals are infinite. Thus iu the present notation Z, is an ele- mentary integral of the second kind, Q,^,^ one of the third.* AVe have had instances in § 219. The integrals of the three kinds are continuous functions of the place on T, but are not one-valued. When the upper limit of an Abelian integral describes a closed path on T, the final value of the integral may differ from the initial value by multiples of periods f due to the cross-cuts or to the logarithmic discontinuities. It should therefore be borne in mind that the integral of the first kind may become infinite by the addition of infinitely many periods. • In cases where no ambiguity will arise we shall say that an integral is on at a place s like (j-s)-' or log («-»), where what is meant is that it is to like (« — 2(s))-i or Iog(«-«(s)), s(») being the point of the »-plane at which a lies. t The word period is used for shortness. It would be more correct to use modulus of periodicity and reserve period for the Abelian functions which arise from the inversion problem. 430 ABELIAN IXTEGRALS. In Case II., writing z — c = z{, Ave have ^Rdz = h^C + h,z,'-' + ■■■ + rb, log «i + P(zi). P If the path of integration be a small closed curve round the place s, this gives f Rdz = 2Tti)%. The coefficient rb, is called the residue at the point in question (§ 180). Eound an ordinary point, and round a critical point such that in the expansion of R no term (z — c)"' enters, r Rdz = 0. Let Si, 8-2, ■•■, s, be the logarithmic infinities of I Rdz on the surface T. The surface T can be reduced by the ordinary cross- cuts A, B, C to T' As in § 180, all the logarithmic infinities must be cut out; the surface is now (K-|-l)-ply connected. Let it be reduced to a simply connected surface 1" by k further cuts. On T" the integral has no logarithmic infinities; hence taken positively round tlie complete boundary of T" it vanishes. The parts of this boundary which are traversed twice in opposite directions contribute nothing to the sum-total, since each element Rdz is cancelled by an element — Rdz. The only jjarts which are not of this category are the small closed cxirves round the places s; these contribute 2Tri (the sum of the residues). Hence the important theorem that the sum of the residues of | Rdz on T is zero. In the proof of this theorem we have integrated round the boundary of a dissected surface. This method is used constantly in Kiemann's theory. It seems desirable therefore to point out explicitly that elements on opposite banks of a cross-cut are com- bined with each other,* and that the cross-cuts C play a subsidiary part, the period across a C being zero (see § 180). The theorem with regard to the residues shows why an integral with tico logarithmic infinities is chosen as the elementary integral ; for an integral with only one logarithmic infinity is non-existent. In practice use is made of a special equation, and the results are open to objections on the score of incomplete generality. To meet these objections it is necessary to place the Kiemann sur- face T in the fore-front of the theory, and to prove that Abelian ♦Integration round the rim of a dissected Burface is nnnlogous to iiuegr.ition round the rim of s\ parallelograni of periods in the caee of a doubiy periodic function. See Chapter VIT., § 197. ABELIAX IXTEGIIALS. 431 integrals of the three kinds exist upon it. It is assumed that this surface is spread over the z-phme iu a finite number of sheets (say n), that its order of connexion is L';j + 1, and that the sum of the orders of its branchings is 2}) + -n— '2. To avoid interrupting tlie discussion of properties, we shall assume the truth of certain exist- ence theorems, and give the proofs at a later stage. Strictly we ought not to speak of Abelian integrals of the first three kinds for such a surface, but rather of functions of the place of the first three kinds, since there is, at first, no basis-equation. But no con- fusion will be caused inasmuch as the functions prove to be of the form J B{w, z)dz, where u' is an algebraic function of T which is connected with z by an algebraic equation. § 277. The existence theorem for integrals of the first kind establishes the fact that there exists upon T a function W with the following properties : (1) TFis continuous on T ; (2) TFhas constant periods «, + i/3,, «,+y+ i/3,+p, at the cross- cuts A^, B^, where the «'s are assigned arbitrarily. That the name Abelian integral is bestowed appropriately on the function can be shown by the following considerations. On oppo- site banks of a cross-cut the difference between TI+ and TF_ is con- stant. Hence d]VJclz=dW_/dz, and the function dW/dz is continuous except at the branch-points, where it may have algebraic discontinuities. Such a function is necessarily a rational function R{u: z) of w and z, where F(w", z) =0 is an irreducible equation associated Avith the surface. Thus W is an Abelian integral. The property (1) shows that it is an Abelian integral of the first kind. Suppose, to take a more general case, that there exists on T a function I which has the following three properties : (1) It is continuous on T except at isolated places ; (2) Its discontinuity at each place of discontinuity is algebraic, logarithmic, or logarithmic-algebraic ; (3) The value at any place of T is altered merely by an addi- tive constant after the description of a closed path on T. The derivative dl/dz is continuous except at certain places where the discontinuities are algebraic ; for the logarithmic infinities are converted by differentiation into algebraic infinities, and dl/dz 432 ABELIAN INTEGRALS. has no periods at the cvoss-cuts which change T into T". Thus the function dJJdz is of the form R{iv, z), and I is an Abeliau integral. A special case of Green's theorem. Let TF be an integral of the first kind ; let W= U+ iV, where U, Fare real. The only places on T' at which the first derivatives of U, V can become infinite are the branch-places of T. Let these be cut out from T ' by small closed curves. By an application of Green's theorem CuclV^ C(- U^-fdx + U^-^dy J J \ oy Sx ' =/J linw'"'' since 5-{7/8a^ -f i-U/Sy- = 0. The single integral is taken round the boundary of T ' and round the small curves which enclose the branch- points. The integrals round the branch-points become infinitely small when the curves become infinitely small; therefore we have the theorem that X--J/Kf)'-(f)'}"- ■ <"• where the single integral is taken round the rim of T'. [See Ch. IX., § 255, and Ch. VI., § 176.] By integrating round the lines A, B, C, we get, since U+ — U.. is equal to «^ along A^ and to a^^^ along B^, X^''^^=A{"'P^^+«-X^^}' for each cross-cut is traversed twice in opposite directions and dV+ = dV_. The integrals on the right-hand side are equal to ;8,+p, — )8, ; hence the expression : a positive quantity, or • (2). In particular, W reduces to a constant when all the «'s or all the yS's vanish. That is, an integral of the first kind with jyeriods all real, or an integral icith jyeriods all jmrely imaginary, is a constant. Re- ferring to what was said at the beginning of this paragraph, we can now see that the conditions imposed upon W determine it to a constant term. For let W be an integral of the first kind whose «'s are the same as those of W. The function W— W is an inte- gral of the first kind which has purely imaginary periods at the ABELIAN INTEGRALS. 433 cross-cuts ; therefore, by the theorem which we have just proved, W— W is a constant. If the ratios of the periods be all real, the periods themselves are of the form ka^, ka^^_^, where A; may be com- plex ; ir/fc is now a function whose periods are all real, and there- fore ir/fc and W are constant. Example. An integral whose periods at the curves ^, (or B^) are all zero, is a constant. § 278. It is of great importance to determine how many integrals Wean be linearly independent. Let there be q linearly independent integrals TF"^, (X= 1, 2, •••, q) ; and consider the expression A-0+ 2A-,F„ where the coefficients ^\ are constants. The expression is continuous on T and has constant differences 2 A:^(D;^,, 2 A.\a);^ ,+ at the cross- cuts A^, B^, where o);^,, w^, ,+, are the periods of W/^ at these cross-cuts. It is therefore an integral of the first kind. Denote it by IT and its periods by ^„ k^ be resolved into their real and imaginary parts «. + 1/3,, a;^, + «/8a.,, Ma + 'V^^. We have the system of 2p equations 2 (Ma + ^■•'a)(«a. + «^A,) = «« + *'A A=l '«5 2 (ma + i>'A)(«A, K+p + %, .+?)= ««+p + «/S« (3), K = l, 2, •■•,p- By equating the real parts, we get 2p equations, from which the 2q quantities /u.^, va can be eliminated, provided q /3a«i "a k+ ' Px «+p' "«' "«+p- Choosing a set of values for a, which do not obey these relations, we have an integral of the first kind which is not linearly related to the T^^'s. If 5 -f 1<;7, we can by the same argument obtain q + 2 linearly independent TFs, and so on until p is reached. Hence there are not less than p linearly independent integrals of the first kind. Suppose that W^, W.,, •••, W^ are p linearly independent integrals of the first kind; then every other integral W of the first kind is expressible in the form W=k,+ ik,W^ (4), A=l a theorem of fundamental importance. In proof of this theorem observe that whatever be the real parts of the periods of W it is 434 ABELIAN INTEGRALS. always possible to assign such values to /x^, v^ as will make the real p parts of the periods of the linear combination 2 ^'JV^^ identical with the real parts of the periods of W. And when this has been done, p W— 'S.hJV-^ is an integral of the first kind whose periods are .all purely imaginarj', and is therefore a constant. The theorem which has just been enunciated cannot be regarded as proved until it i.s shown tliat the determinant formed by tlie coefficients of tlie quantities a^„, etc., in the equations for a^ and a^^p, does not vaiiisli. Neumann sliows by considerations drawn from determinants tliat the vanishinjc of this determinant is inconsistent with the linear independence of ll'j, ll'j, •••, 11'^,. Neumann, Abel'sche Integrale, 2d ed., p. 243. Corollary. When p = 0, there is no integral of the first kind. The theorems of this and the preceding paragraph are the ones referred to at the end of § 220. § 279. The bilinear relations for integrals of the first kind. The integral | TF^cZTF"^-, taken over the complete boundary of 1', is zero. The cross-cut A^ is described twice in opposite directions, and = ( {W--W,-)dW^\ = <^A«'"A-,,+«; see Fig. 74, § 180. Similarly, W^dW^. = - o);,.,a,x,r+.- Also, r W^dW^. = 0. Therefore, after addition. P (U 2 A« '"\,P+K «=1| ,, = 0, (k = 1, 2, ..■,p). The integral SA^TF^^ has now no periods at the cross-cuts, and is therefore a constant. Hence the ir/s are not linearly independent, contrary to hypothesis. § 280. Normal integrals of the first kind. An integral W may have the period 1 at the cross-cut A^, and the period at all the other cross-cuts A^. For the equations 2i\ 2A-. :0, 1, (k = 0, determine the p quantities A\ without ambiguity. Hence W is com- pletely determinate, save as to an added constant. Such an integral is called a normal integral of the first kind. "We shall denote it by ?t, and its period at B^ by t^,. Corresiwnding to thep cuts A^ there are }} such integrals, whose periods form the scheme A, A- ■A A B,- ■5. Ml 1 • ■ T]l 1-12- • -^Ip Ms 1 • • T21 1-22- ■ T2, U,\ ■•■ 1 \t,: * Other relations bcBides the bilinear one connect the periods of two Abelian integrals of tlie nral kind. See Clebecb, t. lii., p. 185. 436 ABELIAN INTEGRALS. The p normal integrals are linearly independent. For, if possible, let ito + 2^\w, = 0. Then at A, ^\ • 1 = 0, and the vanishing of the coefficients implies that there is no linear relation. The bilinear relation becomes, in the case of the normal inte- grals, U^, M„ I 1 T„ r. + T., 1 T, = 0, (Ct t,K In virtue of this relation the array formed by the t's is sym- metric. The simple character of the properties of the p normal integrals ?(„ m,, ••■, v^ makes them specially suitable for use as the elements in terms of which all integrals W are expressed linearly. In § 277 it was proved that "■p+K p «, a. >0. If we apply this result to the integral 7liUi + nM2-\ hWpMp, where rij, n2, •••, «, are real, we have, if t,, = «„ + 1;3,^ «. = n„ P. =0, p 1=1 1=1 and therefore 2 2n.n,/3„>0; «=i .=1 that is, Pn^i + 2 /8,2n,n2 The sign of equality obtains only when the integral 2 nju^ is con- 1=1 stant, a case which implies the vanishing of rii, %, •••, n^. § 281. Tlie functions *. The derivative, dW/dz, of an integral of the first kind belongs to a system of algebraic functions on the surface which is of paramount importance. We shall denote such ABELIAN INTEGRALS. 437 a function by *. Thus j Mz is always an integral of the first kind. These functions are (2p +2n- 2)-placed. For near a place s at which z = c let = W{s) + ai{z - cyi' + a.,{z - cY"- + .... Ifoi^O, $ = 6,(2-c)<'-"/'+ ..., and $ is x'"' at s. Therefore is /3-placed, where ^=2(H-l) = 2^9 + 2(n-l), (§ 170). Of the 2p-\-2{n—l) zeros of we can at once account for 2n. For by supposition there is no branching at oc, and accordingly, when c = oc, W=W{s) + P,{l/z), * = P,(l/z). Therefore each of the n places at oo is a zero of the second order. There remain 2p — 2 zeros, which are said to be moveable, inasmuch as they are different for different *'s. That this is the case will appear in § 288. It has been assumed in the proof that «i, and therefore hi, does not vanish. When r — 1, the vanishing of Oj evidently means that the place s is itself a zero of , and in this way also we account for the vanishing of Oj when r > 1, by saying that then a zero of the $ in question happens to be at a branch-point. And similarly when in the expansions at co the coefficient of 1/z in the series for W is missing, one of the moveable zeros happens to coincide with a fixed zero. This proof is from Klein-Fricke, t. i., p. 544. The ratio of two functions * is evidently, at most, (2p — 2)-placed. There can be chosen p functions 4> between which there is no homogeneous linear relation ; but between these and every other * there must be a homogeneous linear relation. This is proved at once by diiferentiating (4). From this again it follows that half the moveable zeros of a * can be assigned arbitrarily. For, if s„ «2, •••, Vi ^® *^® assigned zeros, the constants A; can be chosen so as to satisfy the p — 1 equations. 2fcA(s,) = 0, where «= 1, 2, •••,71 — 1 ; 438 ABELIAN INTEGRALS. p that is, we can make 2^\4'J^, which is the general expression of a $, vanish at the p — l places. It appears at first sight as if the p — 1 assigned zeros determine the other p — 1 zeros ; but this is not necessarily the case, for tlie above p — 1 equations need not be independent. For example, when the surface is two-sheeted the basis-equation can be written IV- = n (z~ «,), we can take ( § 182) for the p independent *'s z'^/w, A = 0, 1, • • •, p — 1, and for any , 2A\«*/iu. The 2p — 2 moveable zeros form p — 1 pairs of places, say ss, each pair having the same z. Xow pro- vided that among the assigned zeros there are no pairs, the remain- ing zeros are all determined, for they are S), s.j, •••, Sj,_i; but when among the assigned zeros there are i pairs, and therefore jj — 1 — 2i unpaired places, of the remaining zeros p — 1— 2t are determined, and 2t form pairs unspecified in position. The case of a two-sheeted lUemann surface, or surface which can be rendered two-sheeted, is called the hyperelliptic case. The surface depends on the 2p -I- 2 branch-points, which we can regard as arbitrary points of the 2-plane. By a bilinear transforma- tion, 3 of these can be made to take assigned positions. Therefore the number of independent co-ordinates of such a surface is 2p — 1. All important property of the functions * is that the ratio of two such functions is an invariant witli regard to birational transformations. This arises from the fact tliat in such transformations an integral of the first kind remains an integral of the firet kind. See Weber, Abelsche Functionen vom Gesohlecht 3: and Klein-Fricke, t. i., p. 540. § 282. Integrals of the second kind. It is a consequence of the existence-theorems that on the arbitrary Eiemann surface T there is a function Z„ which has the following properties : — (1) It is continuous on T except at a single place s; (2) It is infinite at s like l/{z-c);* (3) It has constant periods at the cuts A^, B^, whose real parts are assigned arbitrarily. Such a function must be an Abelian integral. For dZJdz is everywhere continuous except at s, and possibly certain branch- places. At the places at which it is discontinuous, it is algebraically *If s be a brancli-pliu'c at which )■ eheete haug together, Z, is infinite lilie 1/(2 — c)Vr. ABELIAN INTEGRALS. 439 infinite. It is therefore an algebraic function of the surface, and is of the form E{iv. z). The conditions (1), (2), (.3) determine the integral to a constant term. For if Z„ Z,' be two functions with these conditions, Z, — ZJ is nowhere infinite, and has at the cross- cuts i^urely imaginary periods. Accordingly Z, — ZJ is a constant. "We shall denote the period of Z, at A^ by -q^, and that at B^ by j?p+,. The expression Z;»' = Z, + A-o + 2^-,Ma A=l satisfies the conditions (1), ('2), (3); and since the 7) constants k/^ can be so chosen as to make the real parts of the periods of Z,'"' take assigned real values, while c is at our disposal, Z}"' is the most general elementary integral of the second kind. The normal integrals of the second kind. Let — A\ = the period of Zj-"' at the cross-cut A^, {\ = 1, 2, •■•,p). This choice of the constants makes all the periods of Z,<°' at the curves A^ equal to 0. Denote the integral so defined by t.„ and call it the normcd integral of the second kind icith an infinity at s. To obtain the periods of 4 at tlie cross-cut B,, observe that the integral i t„duy must be Jr equal to i i„du^ (§ 178). Jit) Now C ^,du^ = i r^/K + 2 f La :.du^ = 0+2 ( Vp-^,du^, the integral being taken once along B^ and in the positive direction, But = ^Vp+k\ rf"A- Jdu^ =0 or — 1 according as k #= X or = X. Hence J , t,,du^ = — Vp+k- On the other hand, we have in the neighbourhood of s (which we assume to be an ordinary place) 4=l/(2-c) + P(z-c), and therefore, fdn. _ 9 Zirl dz Is 440 ABELIAN IMTEGEALS. Hence .,^^^ = -2 .i(^^). Let dujdz be denoted by 4)^. Then the scheme of periods for 4 is Ai A., •■• A^ ■•• A B, ... B^ § 283. Algebraic functions on a Rinmann surface. We are now in a position to prove tliat upon an arbitrary Rieniann surface T there exist algebraic functions wliich have simple poles at /x distinct but otherwise arbitrary places, whenever fi >p. Let Si, s.,, ..., s^ be any y. places on T which are subject to the restrictions that no two coincide in the same sheet or lie in the same vertical ; and assume that there is an algebraic function lo of z, which is infinite like k/^/{z — c^) at the place s^, (A = 1, 2, •••, ;u.). Construct the integrals ^ and call them, for shortness, i^^. The expression ^ is a function of z which is continuous on T. It is therefore an inte- gral of the first kind. As w and the ^^^'s experience no discontinu- ities at the cross-cuts A^, such a function, if existent, must reduce to a constant. Hence, the form of w is w = fco + 2 k^^^. A=l It remains for us to find out when the last expression is an algebraic function on the surface. The necessary and sufiicient con- dition is that it has no periods at the cross-cuts B^. This condition is expressed by the p equations E^ = -2ni^k,%{s,)=0. A=l These equations may or may not be linearly independent. When the former is the case, p of the constants k^ are determined as linear functions of the other fj.—p. Thus there remain, including fc„, ix—p + 1 arbitrary constants in iv. But if the equations be not all linearly independent, there will be a certain number t of linearly independent relations 2 «,<*'£. = 0, (h = l,2,...,r); ABELIAN INTEGRALS. 441 in this case the places s„ s.,, ■••, s^ are said to be T-ply tied, aud there remain ^x — p -t- t + 1 arbitrary constants which enter linearly. The T equations in E show that t is the number of linearly independent combinations 2 6^, which vanish at the places s. This theorem was given by Koch (Crelle, t. Ixiv.) as the extension of a result of Eieuiann's, and is known as the Riemunn-Eoch theorem. It is clear that if iJi.>p, there is a solution of the p equations which is distinct from 'k^ = k,= ■■■ =k^ = 0; that is, there is an algebraic function of z which is infinite only at the ju, places or (if some of the A,\'s be zero) at some of these places. When to is infinite at yu,i of the /i places, it is a /xj-placed function of z and satisfies an algebraic equation F{iv''. zi^^) = 0. In this way the exist- ence of an algebraic function of the arbitrary Eiemann surface T follows from the existence theorems for the Abelian integrals. It is especially worthy of note that in Riemann's theory the integrals come first and algebraic functions second. From the Riemann-Eoch theorem we can infer that the poles of algebraic functions are necessarily tied when ix 2^9 — 2, the function cannot be special, as there are only 2p — 2 moveable zeros of a 4> (§ 281). Let the /x poles be moveable; then there are on the whole fi + fi—p + l independent constants, or oc-f"'"^^ functions with /x poles. Each of these maps the surface T into a /x-sheeted surface T^, so that the correspond- ence of T and T^ is (1, 1). Let there be ooP transformations of T (and therefore of T^) into itself. Then the number of surfaces T is crJi'-''+^-i'. These are all in a (1, 1) correspondence. Let B = 2p + 2^ — 2, then (§ 128) ^8 is the number of branch-places 442 ABELIAN INTEGRALS. (supposed simple). When tlie branch-points are assigned, there is only a finite number of possible surtaces, for the number of ways in which the sheets can hang together at these points is finite. Thus there are cc^ ^-sheeted surfaces; and on each of them algebraic functions exist. These considerations show that the number of essentially distinct surfaces is cc^"-''"^-'"'"^'' or cc*"^"'"''. Hence the number of characteristic constants of a surface is 3j) — 3 + p. These must all be equal for two surfaces which can be mapped on each other. These characteristic constants are called by Eiemann the class- moduli. When ^5 = 0, we can take for the surface the a;-plane itself, and the birational transformation is bilinear, say iv = {az + b)/{cz + d). There are therefore three quantities at our disposal when we seek to transform the plane into itself, and p = 3. Hence there is no modulus. It can be proved that when p = 1, p = 1 ; and that when ^) > 1, p = 0. Hence when p = l, there is one modulus, and this can be taken as the absolute invariant g? jgi of the quartic which occurs when the surface is reduced to the standard form (§ 218). When ■p > 1, there are 2>p — 3 moduli. In the hj'perelliptic case we found that there are 2^5 — 1 ch.aracteristic constants (§ 281). Now 3 p — 3 > 2j> — 1 when p > 2. Hence when p > 2, there must be relations between the moduli when the surface can be brought to the two-sheeted form. In this paragraph we have followed Klein's tract Ueber Eiemann's Theorie, to which (pp. 64-72) the reader is referred for more information. The proof is similar to the one given by Eiemann (Werke, p. 113). Proofs drawn from the theory of curves are given in Clebsch, t. iii., ch. 1, and by Cayley, Math. Ann., t. viii. On the relation of the hyperelliptic case to the gen- eral one, when p > 2, see Klein-Fricke, t. i., p. 571. § 285. Tlic integrals of the third kind. The most general Abelian integral has logarithmic infinities which may be Cj, e,, •••, c,. For simplicity, assume Cj, c,, •■■, c, to be ordinary places of the surface. After cutting out these points we must render the surface simply connected by a further cut D which extends from Ci to c.,, from Cj to Cj, •■•, and finally from c^ to some point of the boundary, as for instance the point where Ay, B^ meet. When this cut has been made, the surface may be named T". Let a^ be the residue at c^ (A.= l, 2, ■•■, k). The part Z>, which extends from c, to the boundary furnishes no period, for the integral along a path on T" ABELIAN INTEGRALS. 443 which encloses all the points c„ c,, ■■■, c„ and leads from one side of L», to the other, is equal to 'Iwi^a^, = 0. Hence there is no discon- tinuity at Z),, which behaves in this respect like the cuts C,. That part D^_, of D which leads from c,_, to c ^ furnishes a period 2 wia^, the pai-t D^_« a^ period 27ri(«, + a^_,), and so on. This may be illustrated by J d log R, where E is a /^.-placed function of the sur- face. If c^ be the zeros, c^ the poles of R, where A= 1, 2, •.., /x, the cut D may be drawn from c/ to Cj, from Cj to cJ, from c/ to Cj, and so on. Across a cut from C;^' to c^ there is a period 2-Ki, and across a cut from c\ to c\^i there is no period. Let us assume that there exists on the arbitrary Eiemann surface T a function which (1) is continuous on T except at two arbitrary places c,, c,; (2) is infinite at c^ like log {z — Cj) and at Cj like — log {z — Cj) ; (3) has constant periods at the cross-cuts of T ", with arbitrary real jjarts. By the same reasoning as before we can prove that the function in question is an Abelian integral. Denote it by Q, ^^. If there be a second function with the same properties and whose periods have the same real parts as those of Q. reasoning similar to that already used shows that the two func- tions differ merely by a constant. As an elementary integral of the third kind remains of the same kind after the addition of any inte- gral of the first kind, the most general elementary integral of the third kind is Qc,c, + A'o -f 2 Ic^u^ ; for the /t\'s can be so chosen as to ensure that the real parts of the periods of this expression take values which are assigned arbitrarily. Bv a proper choice of the Ar^'s we can make the periods at the cross-cuts A^ all vanish. The resulting integral is the normal integral of the third kind with the prescribed logarithmic infinities, and may be denoted by 11,^,^. The remaining periods of n,_,^. In the general bilinear relation write I=W, I'=Q.,.,, and denote the periods by the scheme W I 0)1 (02 • B, B., -B, p f/x., 1 C,/l + Ci/o + C3/3 by dco.f *Clebscb and Gordan allow f=0 to have cusps, but this is an unnecessary complication (see § 189). It may be added that the work of Chapter IV. shows that multiple points with separated tangents cause no difficulties as regards algebraic expansions. t This diflerential expression rf». Is called by Cayley the displacement of the current point on the curve/=0. See Cayley's memoir on the Abeli.in and Theta Functions, Am, Jour., t. »., p. 138. 448 ABELIAN INTEGRALS. The general Abelian integral is now | p^cZu), where x is a homo- geneous form in x^, x,, x^, the degree of whose numerator must exceed that of its denominator by n — 3, in order that the integral may not be altered when x^, x.,, x^ become kx^, kx,, kx^. Clebsch and Gordan assume that iv and z in F(iv, z) = are the parameters of two pencils w 1 cfi^.i I + I ai&2a-3 1=0, 2 I &1C2X3 ! + \a^c^^ \ = 0, where the constants are the coefficients in the homographic trans- formation defined by the equations Xxi = aj, + hiZ + CjW Xajj = a^t + h.z + c.,vi X.Xs = cist + bsZ + Cgw which connect x^, X2, Xj, with t, z, w, and which change F{io, z, t) into X'/iXi, X2, Xj). To pass to the non-homogeneous form of an Abelian integral, use the relations X"/(a;i, Xj, X.,) = F{iv, z, t), X"- \Xi, X2, X3)„_3 = (iv, z, <)„_3, Ci/i + cj, + cj, = W/ho = A.SF/SW, A >? I Cj.TjcZxj I = I ai&jCj j {tdz — zdt), then the Abelian integral CiX^doCs I /' (Xi, X2, Xs)„. ■3- C1/1 + C2/2 + C3J3 passes, when t = 1, into dz f (W, 2, 1)„_3- ^ ' ' ''» ^SF/8w Here it should be observed that F=0 is of order n in the quantities w, 2 combined. This is Clebsch and Gordan's F, and not Riemann's. The process used by Clebsch and Gordan for the classification of Abelian integrals is purely algebraic. (See their treatise, pp. 5 to 8.) The result at which they arrive is that every Abelian integral is expressible linearly in terms of the following kinds of integrals : — (1) Integrals of rational functions of z ; (2) Integrals of the form ) -^yj-< where BU an integral polynomial of order n — 3 in w, a ; ' ' ABELIAX INTEGRALS. 449 (3) Integrals of the fom I- -f=-^, where His an integral polynomial of order n - 2 in ir, .- J {.z - i,)iF/iw (4) Integrals which are derived from (3) by differentiations with regard to a. They then proceed to an examination of the nature of the discontinuities of these integrals, and to a discussion of the integrals of the first, second, and third kinds. § 288. The adjoint curves <^„_,,. The integral T-^— , where H J SF/dw is a polynomial of order ?i — 3 in v.; z, can be finite for all values of 2, 10. To see that this is the case, observe that the integral can become infinite (if at all) only when hF/aic = ; that is, at the branch-points and nodes of i^ = 0. (1) Xear a branch-point (a, h) and the integrand is hence the integral is P^{z — ay'^, where k ^ 1. This proves that the integral is finite at a branch-point. (2) Near a node (a, 6), ic-h = P,{z-a), and the integrand is Hdz- P„{z-a) . z — a hence the integral is infinite, like log {z — a), except when H= at the node. /' Hdz - — — cannot be everywhere finite unless H=0 passes through all the S double points. When this hapi^ens, the number of arbitrary constants in H is |(n_l)(n-2)-8, =p. That is, the mtmber of linearly independent integrals which are finite for all values of z is equal to the deficiency of the basis-curve. These p integrals are Abelian integrals of the first kind. As H is of order « — 3, n is at least equal to 3. If ^5 = 0, there is no curve H= which passes through all the double points. But in this case vj and z can be expressed as rational functions of a parameter t ; and every Abelian integral becomes the integral of a rational function of ^ 450 ABELIAN INTEGRALS. In the homogeneous form the integral becomes S^ where <^„_3 = is an adjoint curve of order n — 3, i.e. a curve of order ?i — 3 which passes tlirough all the double points of /= 0. If <^„_3 = were not an adjoint curve, the nodes would make /„ f^, /a vanish, biit not <^„_3; they would therefore be infinities of the integral. The only other points of /=0 at which the integral could possibly become infinite are the points at which tangents from (ci, c.,, Cj) touch /=0. These points are points of intersections of /= U with the first polar of c. The integral must be finite at these points, for its value does not depend on c,, c.,,C3, and consequently the position of its infinities cannot be dependent in any Way on the position of the point c. The curve <^„_3 = is to pass through the 8 double points of /=0; the number of the remaining iutersections with /= is ,j (,i — 3) — 2 8, or 2 J) — 2 ; i.e. the curve <^ = has 2j? — 2 moveable • points of intersection with /= 0. Since there are jj arbitrary con- stants in a curve <^„_3= 0, it can be made to pass through jJ — 1 arbitrary points ; in general jJ — 1 of the intersections of a curve <^ = with/= determine the remaining points of intersection. 87** A function H/ — whose integral is always finite is evidently Sw identical with a function of § 281. And the 2p — 2 moveable zeros of a ^ are, on the curve, the intersections of an adjoint curve <#>n-3 with /= 0. ]\Iore generally the points at which 0/$' takes a given value A are, on the curve, the moveable points at which the adjoint curve <^ — X<^' = cuts /= 0. There are 2p — 2 such points unless <^ = and (^' = intersect on the curve. Let any rational function E(iv, z) become, in homogeneous co-ordinates, M(Xi, x^, x^)IN(x„ x.,, x.^). The points at which the function takes the value X are the points in which the curve 3f— XN= cuts the curve /= 0, exclusive of the common inter- sections of M= 0, N= 0, /= 0. At such a point the value of 3I/N is easily seen to be A.', if M—X'X=0 be that curve of the pencil which touches /= at the point (§ 217). Of the moveable points how many are arbitrary ? The question is answered by the theorem of Riemann and Roch. ' Namely, if there be fi moveable points, and if through them t independent adjoint curves are conies through the nodes. Let ju, = 2. Through the two points and the two nodes we can draw oc' conies, of which two are linearly independent. Therefore t = 2 ; /i— ^J-l-T-|-l = l; and the function reduces to a constant, which need not be oc at the two points. This does not stultify the theorem, which refers to functions which are not infinite at other than the fj. i^oints without asserting that they are infinite at the ^ points. Xext let ju.= 3. The 3 points and the 2 nodes determine a conic; therefore t=1, fi— 2) + t + 1 = 1. There is no algebraic function which is 00 at the 3 points. But if the 3 points be in a line L with a node, the conic becomes this line and any line L' through the other node. Xow t = 2, fi—p + T + l = '2; and there exists a function of the form {(if,N+ UiM^/N, which is oo at the three points, and there only. "We can take as JVthe line L itself, and as Mi any other line through the same node. Lastly, let /x = 4. Here t = 0, lx.—2} + T + l — 2, and the function required is still of the form (f^o^-1- aiMi)/N. Let JVbe an adjoint cubic through the 4 points; the cubic cuts /= again at 7 points, and we take for M^ an adjoint cubic through the 7 points. The reader is referred for an exhaustive account of Eiemann and Roch's theorem to Clebsch, t. iii.. Chapter 1, § 2. The system of points on the curve or on the Eiemann surface, at which an algebraic function takes a given value, is said to be coresidual * or equivalent with the system of points at which the function takes any other given value (§ 217). Before leaving the subject of the adjoint curves <^„_3, it should be mentioned that they are' employed by Clebsch (t. iii., ch. 1, § 3) in the problem of selecting, from the class of curves which arise from a given curve by birational transformations, that of the lowest order. The result is that when p > 2 a curve of deficiency p can be trans- formed into a normal curve of order p— tt + 2, where 73 = 37r, Stt + 1, or 37r + 2. For ^3 = 0, 1, 2 the normal curves are respectively a ♦The word coreBidual has the same meaning as in the theory of re«iduatlon, explained in Salmon'^ Higher Plane Curves, Chapter V. The theorem of Rieraann and Roch is fundamental in thia theory. 452 ABELIAN INTEGRALS. line, a non-singular cubic, and a nodal quartic. [Clebsch and Gordan, § 3 ; Salmon's Higher Plane Curves, § 365 ; Brill and Xcither, Math. Ann., t. vii., p. 287; Poincare, Acta Math., t. vii.J § 289. Homogeneous form of Q* Consider the integral S-. da Its possible infinities are those points of /= at which (1) /i, /,, /j vanish simultaneously. These are double points of /==0, and make the integral infinite unless i//=0 passes through them ; (2) Ci/i + c.,/2 + C3/3 = 0, without fi,f.,,f being all separately equal to 0. These are branch-points and do not contribute infinities to the integral ; (3) «i.Ti + a.:ir,., + K^x^ = 0. The number of intersections of this straight line with/= is n ; at each of these n points the integral will be infinite unless 1/^ = happens to pass through the point. To get rid of the infinities due to double points, make i/?„_2 = pass through the double points. The curve i//„_, = cannot be made to pass through more than n — 2 points of intersection of a^Xi -\- (1.2X2 + ct^x^ = with /= 0, except when \f/„_2 = degenerates into the straight line and an adjoint curve <^„_3=0 of order n—3. In this exceptional case = dw ^ I „_',d(o', that is, the integral is of the first kind. Euling out this case, the number of infinities of | '^^^^ du> is two, and these are due to those two points of intersections of a^Xi -f (ux^ + a-^x,, = with /= 0, through which i/f„_2 = does not pass. To determine shortly the nature of these two infinities, we have in the non-homogeneous form ^^^, where v,+.j, is some integer. Collecting these facts, we have 2 \W{x,)-W{y^\= 5(v.«., + v.+,a,,^,). The places x, y have been defined as zeros and poles of a func- tion R(w, z) ; it might therefore appear, at first sight, that a dis- ABELIAN INTEGRALS. 455 tinction is to be made between 0, «:,aiKl ordinary constant values k,k'. That no such distinction is inherent in the nature of the case will be apparent from the consideration that we havej if Il = {R'-k)/{R'-k% E' = k, k' when R = 0,oo. The places x„ y, are characterized adequately by affirming that they are the points of T at which the /.-placed function R' takes given values k, k'. Two such systems of places have been called equivalent systems (§ 217). As R' changes continuously from k' to k, we may regard the system y, as changing continuously into the system a-,, and may say that the f. paths are equivalent. The paths are supposed subject to the restrictions that no path passes through a branch-place, and that no two paths intersect. Suppose that in the above equation the system x, shifts into the equivalent system y, + dy,. On the supposition that none of the places con- cerned he near a cross-cut, the equation holds equally for the new system ; and therefore, by subtraction, ^jmy. + d'J,)-W{y.)l = 0, or 2fnr(2/,) = 0. Integrating from the system y, to the system x„ we have 2 I d W{x,) = a constant, where the paths of integration are equivalent paths. So long as no path crosses a cross-cut, the constant is zero ; but each time that the path crosses a cross-cut, the constant must include the period at that cross-cut. Generally the constant may be written p where f,, 1',+^ are the numbers of positive crossings of A^, B^, less the number of negative crossings. Comparing this result with the equation of § 285, we infer that the value of log R taken round a cross-cut is to be obtained by considering the points at which that cross-cut is crossed by a system of equivalent paths which lead from the poles to the zeros of R ; in fact, - — ; I d log R is the excess of the number of these positive crossings over the number of negative crossings. [Neumann, ch. xi.J 456 ABELIAN INTEGRALS. § 292. Abel's theorem for normal integrals of the third kind. Let Ilf, be a normal integral of the third kind. Cut out from the sur- face a dumb-bell-shaped double loop formed by two small circles (^), (r/) round $ and rj and by a connecting line Z>» whose direction is from ^ to -r; ; to reduce the resulting surface to simple connexion a second line D' must be drawn from Cj to the boundary of T'. Let Jt have the same zeros and infinities as before and let these 2^ places be cut out from the surface by 2/^ small circles and by connecting lines L,.,, L.^, ••■, ij^-i, 2^ ; the last circle can be connected with the boundary of T' by a line D". On the simply connected sur- face T" which results from these various cuts, D' and D" play a part similar to that played by the C"s ; they may therefore be neglected.* By applying the extension of Cauchy's theorem to the integral f Ui„dB/R, we find that f U(„dR/R = 2 r ntAR/R -|- 2 f m„dR/R + f U^4R/R. The last integral can be transformed into - f log iJ . dHf,, »/ normal integrals of the first kind are left undetermined, these p quantities are determined on T' save as to additive constants. By making a suitable choice of these lower limits it is possible to simplify the work. Let definite lower limits be assigned to u,, m,, •••, u^, and, in accordance with Riemaun's notation, let j du^ have the same lower limit, for a definite A, what- ever be the upjjer limit x. The theorem which has been enunciated above can be proved by means of the integral J-. fu.dlogO (\ = 1, 2, ...,p), taken over the boundary of T'. Knowing that u^ is everywhere finite on T' and that log 6 is finite except at the p places Xi, x^, •■•,Xj, at which it is simply infinite with residues 1, we derive at once the equation - — ; I u^d log 6 = the sum of the residues of the integrand, 2 -nil/ = Wa (^i) + Ma (a;a) + ■ ■ ■ + Ma (»p) ■ The integral on the left-hand side of the equation can be treated as the sum of 2p integrals over the 2p lines A^, B^, each line being described twice in opposite directions. Along A,, B^ we have, as before, d log {Q^fBJ) = 0, «/ — u^- = or 1, according as A =^ k or = K ; d log {e^/e.) = -2 iridu^, V = Ma" + t.a- A line A^ contributes the part — _ | (u^"^ — u^~)d log 6+ ; the value of this part is when A. ^fc k, and - — , j d log ^ when X = k. 2 TTt J ill But I dlog^ = log (^+/^_), where the signs refer to jB;^, and we know from the values of 6 along B^ that log {eje.) = 2 ^i\u,(c,).-e^ + rjl + w, 462 ABELIAi^' INTEGRALS. w^(cj)_ being the value of u^ at the point Cj of T' (see Fig. 74, p. 254). Thus the p lines A contribute a part «a-«a(ci)-+/»^-W2. The sum of the j^ integrals over the lines B^ = 7^.^ f ("/^^ log ^+ - «rd log e_) = 77-.^ f [^.A^^ loge_ + 2,riV,/Z«, + 2rtl4,-dMj. To find the value of this sum, consider separately the integrals C dloge., C du^, f u^du^. %/B^ i/^^ ^S^ The first of these is equal to log {6-/6+) along A^^ ; that is, its value is g^ • 2Tri. The second integral is equal to t(,~ — it,"*" along A^ ; that is, its value is — 1. The third integral may be ■written dw. Thus the sum of the 2jj integrals over the lines A^, B^ is 1 '' e^ - u^{Ci) - + K- txa/2 + 7^. 2 [2 nig.r,^ - 2 tjt.^ + ^ir^^ Hence, ^, f «,d log fl = e, - M,(cO- + /i. - rjl + 2 [fl-.T., - tJ2 and 2u^(a;,)=e;, + A:^ + /t^ + 9'iTu + 9'2T2x + ---+9',^A, • C^)) where A/2+ir(M/ + «x-)dw. ABELIAN INTEGRALS. 463 Theorem III. The lower limits of u^, u,, ■■■, u^ can be so chosen as to make all the quantities k^ vanish, and then «A (a^i) + "a (a-'iO + • ■ • + n^{x,?) = e^ + K + giriK + (J-ir.^ + • • ■ + p^r^,,. Eiemann proves this by the following argument : — The fact that u^^ possesses at A^ the period 1 and at the other cross-cuts A the periods 0, determines »^ only to an additive con- stant. Let definite values be assigned to the quantities u^. u.,, ••■, u^,, and let us still use Eiemann's lower limits for these integrals. A simultaneous increase of u^, e^{\ = 1, 2, ■■■,p), by a,^ leaves the func- tion 0{ui — Ci, 11., — 62, •■■, u^ — e^) unaffected and produces, accord- ingly, no change in the positions of x,, a\,, ••-, .r^, or in the values of f/^, h^. The left-hand side of the equation (A) increases bypcc^^, the right-hand side by a^ + the increment of Z,\. Hence the increment of t;^ is (p — l)a;j. Choose a;^ so as to make the new k^ = 0. This will be effected if the increment of fc^ be — A,\, that is, if aA = - V(P-l)- "When the lower limits of the integrals are changed by the inclu- sion of the constants a^ just defined, the quantities Ci, e,, ■••, e, must be congruent to Ui{xi) + tiiix.) -\ h «i(a:,),- u-,iXi) + u^ix,) -\ \- u^{x^), ■••, U^{Xi)+U^{X,) -^ h Uj,{x^) with respect to the array of periods 1 • • nl Tl2- •T„ 1- .0 Tu T22 • ••^2, 0-"1!t]j, T2, •••Tjy. § 296. The function 6(t<, — Cj, u, — e^, •■■, v^ — e,) differs from 0(..., u^{x) - u^{x,) - u^{Xi) ^Ki^p), •••) by an exponential factor. Owing to the length of the p arguments it is usual to print only one of them, and in order to show more clearly the exact character of these arguments and the position of the p zeros, Clebsch and Gordan, Weber, and others write the theta- function in the form is>^-£:'''-s>--s:'-^ ^A I' 464 ABELIAN INTEGRALS. where c, c,, Cj, •••, c, are fixed places which are assigned arbitrarily on T'. Owing to the arbitrariness of the lower limits, an expression fc;^ must be reintroduced. Denoting by x^, x^, •••,Xj, arbitrary places on T', it is possible to determine the quantities &„ k.2, ••■, k^,, inde- pendently of X, Xi, .T2, •••, x^, so that the theta-function shall vanish at the p places Xy, x.,, ■•■,Xj,. It should be observed expressly that we have assumed that the theta-function, considered as a function of the co-ordinates z, w of x does not vanish identically. See § 300. § 297. Having established those properties of the theta-function which will be needed in the sequel, we shall now prove the converse of Abel's theorem for integrals of the first kind. Suppose that Xi, x^, •■•,x a.ndi yi, y.,, •••, 2/^ 3-re two systejns of places on T at which a /n-placed function R{w, z) takes two assigned values; then by Abel's theorem r'du,+ pdu,+...-f- r''dM,=o {x=i,2,...,p) where the sign of congruence means that the right-hand sides are of the form '»A + (/iTu + gir^K -I 1- 9p-^^P> the ^'s and 7i's being integers. Tlie converse of Abel's theorem. Suppose that the p congruences ?(^'.Vi -I- M/2"= -\ h ?i/l^V = are satisfied, and that a function R{w, z) takes one and the same value iJo ^t the ix places 2/1, 2/2, ••■. y^. Then R must take one and the same value Ri at the fx places Xj. x.2, ■■■, x . To prove the theorem we shall construct a function R{w, z) which is 00^ at the places y,. y.,, -■•, 1/^ and 0' at the places x^, x.j, •■•, x . The method for constructing such a function has been described by Weber, Zur Theorie der Umkehrung der Abelschen Integrale, Crelle, t. Ixx. The building up of algebraic functions on T, with assigned zeros and assigned infinities, can be effected, whenever the problem admits of solution, by the use of methods closely analogous to those employed in the construction of the general elliptic function with the (7-function as element. In the latter case aii is one-valued and continuous throughout the u-plane, has one zero and no infinity within the parallelogram of periods, and is quasi-periodic (see Ch. VII., § 206). When the quotient of suitably chosen a-products was ABELIAN IKTEGEALS. 465 multiplied by a suitably chosen exponential, the one-valuedness remained, the quasi-periodicity was converted into the periodicity of the whole function, and the zeros and infinities of the function were the zeros of the elements in the numerator and denominator respectively. In the present case the j)-tuple thetii-fiuietion and the surface T' take the place of the tr-function and the parallelogram of periods, while the single zero of a o--funetion is replaced by the ^j zeros of a theta-function. These p zeros are denoted by the upper lim.its x-^ Xj, •••, Xj, of the arguments Cdu^- C'du^- pd», f^K + ^A (^ = l,2,-.-,p), or, (as they are often written for shortness) •yc ^Cj ^/Cj mJCp ^K> here it is to be understood that /c^ is supposed to be related to the lower limits in such a way that the zeros shall be x^, x^, •••, x^, and also that these p zeros are not so situated as to make the theta- function vanish identically. Suppose that ju, is an exact multiple of p, say fx. = sp. If the number of integrals be not an exact multiple of p, add on the required number of integrals with coincident limits. It is required to construct a quotient Q of theta-functions which shall be 0' at fsp places a;,, a:,, •••, a-,^, and =c' at sp places ^i, y^, ■■■, y.^ Separate the sp places Xi, X.,, •••, x,^ into s sets of p, (Xj, x.,, ■■-, x^), (.r^+i, a-^^,, •••, a;,,), •••, (a;,._i),^i, a;„_i,p+2. ••■, ^,p); and the sp places y^, y.,, ■■-, y,^ into s sets of p, (z/i> 2/2, •••)2/p)> (2/p+i. yp+2, •■■,y2p), —, (2/(.-ii,.ti. y^,-l)p*2, ■•■,y.p)- The factors of the numerator of the quotient Q are where the places c, Cj, Cj, •••, c^ are arbitrary but iixed. The factors of the denominator are obtained from those of the numerator by replacing a;'s by y's. The theta-functions in question are discontinuous at the cross- cuts B and continuous at the cross-cuts A. At B^, e( V - ^a) = <9("r - «a) • exp - 2 ,ri(M.- - e. -f T„/2), [,p ,-"1, ip /»Vr "1 where r' is the remainder after r is divided by p. 466 ABELIAN INTEGRALS. Thus Q^ = Q_ • exp 2Tri 2 I du^. Now, by supposition, : ?)l, + Kiri, + noT,, H 1- W T, 'P* KP^ therefore Q+ = Q_ • exp 2 Tri (jiixi, + iut.^^ -\ h ?ij,T,y) . Let P=exp 27ri(?!] I dui + ?i,2 I dwoH 1- "p I ''"j,); then along a cross-cut A^ we have P+ = F_ ; while along i?^ P+=P^- exp 23ri (niTi^ + ?i2T-2, + h iipT,,). Hence Q/P is a function of the place x on T, which is not affected with a discontinuity by a passage over any of the lines A^, B^, and which possesses the same zeros and infinities as Q. This proves that Q/P is one-valued throughout T and continuous at the cross- cuts, therefore Q/P is a function R{w, z). Thus we have proved the converse of Abel's theorem by constructing synthetically the algebraic function R{ic, z), which is 0' at the upper limits and oo' at the lower limits. It may appear, at first sight, as if too much had been proved ; for the theorem appears to imply that the number and position of the infinities y being assigned arbitrarily, there exists an algebraic function R{w, z) which is infinite at these places and nowhere else. The proof, however, ceases to be valid if some of the theta-functions in Q vanish identically. This is a case which always arises when /x^jJ. [Weber, loc. cit., p. 317.] The identical vanishing of a theta-function. We have seen that vanishes at the places Xi, Xo, •■■,Xj„ when the constants k are prop- erly chosen. Make the place x coincide with Xj, ; then, no.ticing that the theta-function is even, must vanish identically whatever be the positions of the places .T,, osj, •••, «,,_,. We shall return later on to this question of the identical vanishing of a theta-function ; but for full information we must refer the reader to the memoirs of Riemann, and the treatise of Clebsch and Gordan. ABELIAN INTEGRALS. 467 § 298. Weber has shown how to deal with the case iJi-'-. shall not vanish for all values of x, t,, fj, ■■•, t,; for the p arguments of this theta-function can be made congruent to an arbitrarily assigned system of periods by a proper choice of x and the t's. In proof of this statement, observe that the p normal integrals of the first kind are linearly independent, and that therefore the p argu- ments can be made to receive arbitrarily assigned increments when thep places x, t.^, t^, ■■■,tp are subjected to independent translations on T. In the fraction ~' y^- Multiplying Q, as before, by an exponential factor P, and assuming the existence of the p congruences, we have an algebraic function Q/P with the upper and lower limits of the /i integrals as zeros and infinities. § 299. When we replace the lower limits c,, c,, •••, c^, which have not been specialized as yet, by the places a^, a^, •■-, a,,, which made their appearance in Clebsch and Gordan's theory as the points of contact of a contact-curve, we can determine the values of the added constants k^ very conveniently. When we pass to the non-homo- geneous form, let the integral of the second kind become /; \^dz {aw + bz + c)Sf/&w Here ip/(aiv + bz + c) is oo^ at one place « of the surface, at whinh nin-i-hs-L n ^ 0^ nnrl 0^ at the p places «.. Let any integral 468 ABELIAN INTEGRALS. of the first kind be | J^ , where <^ = is what the equation of an adjoint curve is 0' at 2j> - 2 places. Hence E, = ("'" + ^^ + f)-^ ^ has 22) zeros, •A among which a is counted twice, and 2j5 poles, namely, the places «« each counted twice. Given that 6) fC-i ("'"(111^ + k^ (A) vanishes at Xj, x.2, •••, x^, we have, making x coincide with ^i, for all positions of .Tj, Xj, •••, x^. Xow let <^ = 0^ at x.2, x^, ■•■, x^ ; and let the remaining zeros be x\_, x\,---.x'j,. Abel's theorem gives, when applied to the equivalent systems a, a, aio, •••, a;,, x\,---,x\ and (Xi, «i, ttj, •••, «^, «2, •■•, «^, %J Oj f =- •-'flit "=2 *yo-K Thus r°dM^+ 2 r'''cZu^ = --J f^cZM^+S P'fZiv|. Hence, on inserting this value in the expression (A), we find that e(^+p+■■■+pdu,+k^. . . . (B) vanishes identically. Since the p — 1 points »,, Xj, ■•■,Xj,_-^ were chosen perfectly arbitrarily, the p — 1 residuals x\, x\, ■■■, x-'^_i must be perfectly arbitrary. We may therefore suj^press the accents in (B). When this has been done, we have two theta-functions which vanish at the same places x^, x^, •••, x^. But this means that the arguments are congruent. Hence the differences of the argu- ments must be congruent to 0, or - fci, w frj, • • •, 2 A;^ = 0. This gives k^ = hJ2 + g,T,,/2 -f g,T^,/2 + • ■ • + g^Tj2, k, = h,/2 + f7ir,,/2 + ^,r„/2 + • • ■ + g^r,J2, K = K/^ + £7in,/2 + g.2r,J2 +■■■+ g^Tj2, ABELIAX INTEGRALS. 469 where g, and ft, are integers. That is, the constants k^ are a system of halt-periods of the theta-function. Allowing x^, x,, ■■•, x^, to coincide with «i, a.,, •••, a we find that 6 S^k-k) vanishes in general at the 2^ places ccj, a,, ••■, a^,, and only at these places. If the function vanish when x is at a, then and the mark is odd. But this occurs only when one of the places «!, a.2, ■■■, «p coincides with «. Now recurring to the theory of § 290, a contact curve (//..^o passes through the n — 2 residuals of the point a, and if, in addition, it touch the basis-curve at «, it has more than n — 2 intersections with the tangent at u. Therefore it breaks up into this tangent, and an adjoint curve 0„. 3 which touches the curve at p — 1 points. Thus, of the 2°'' non-congruent marks, we are led to associate the 2''- '(2^—1) odd marks (§ 229) with improper contact curves, formed of the tangent at a and an adjoint curve 4>n-i'i and the 2^"'(2'' + 1) even marks viii\i proper contact curves ^n--2- AVhen p = 3, the normal curve is a non-singular quartic, and the improper contact curves are an arbitrary tangent and the 28 double tangents. [See Clebsch, Geometrie, t. iii.. Chap- ter I., § 9.] The method followed is that of Weber in his memoir on the Inversion Problem to which reference has already been made. In a different form, it is to be found in Clebsch and Gordan. To these authors, and to Fuchs, Crelle, t. Ixxiii., the reader is referred. In Weber's memoir will be found the proof that to each system of half- periods there corresponds a system of lower limits «i, a.,, ••■, u^ (points of contact of a contact curve), when the theta-function has the prescribed zeros a-'i, x.,, ■•■, x^,. We shall suppose, for the rest of the chapter, that such a canonical dissection of the surface has been selected that all the constants k^^ are zero. § 300. Tlie identical vanishing of a theta-function. The theta-function whose arguments are Cdu.-i C'du, (1) 470 ABELIAN INTEGRALS. vanishes at all places x when the function of- Cdu.-l f'du,) (2), whose arguments are obtained by allowing x to coincide with x^, vanishes for an arbitrary choice of the }7 —1 places x^, x,, ■■■, x^_^. Let the p places x^, x.,, •■•,Xp be tied by a function , and let the remaining moveable zeros of <^ be Xj,^^, x^^„, ■■-, x.^p^^. Then Abel's theorem when applied to the function iJ of § 299 gives 2 I du^+S. \ du^ + I da^ + ) du^ = 0. Therefore the arguments (1) are congruent to du^ + 2 I du^ + I du^. Op— 1 K=l */a.tc »Jap As 6 is an even function, the sign of each argument can be changed, and we then have arguments of the form in (2), in which X and Xp_f.i, x^^^j •••, ^-ip-i can be chosen arbitrarily. Hence we have the important theorem : when the p zeros, which 6 must have, are tied by a function (^, d vanishes identically. [Clebsch and Gordan, p. 211 ; Clebsch, Geometrie, t. iii., ch. i., § 8 ; Eiemann, Crelle, t. Ixvi., or Werke, p. 198.] With the aid of § 283, this may be stated as follows : all the zeros of an algebraic function of the surface cannot be included among the zeros of a theta-function unless the theta-function vanishes identically. JacohVs Inversion Problem. § 301. Introductory remarks. The importance of the inversion problem in such simple cases as V = I — =r=: and I' = I - dx Vl-o,-^ ^0 V(l - ar) (1 - fcV) is well known. It permits the passage from many-valued integrals to one-valued functions. In § 193 we indicated that the problem is altered as soon as the quantity under the radical is assumed to be of higher order than 4 ; for when we pass from elliptic to hyper- elliptic integrals x becomes a function of v with more than two inde- pendent periods, and therefore cannot be a one-valued or a finitely many-valued function of v. The essential distinction between the function of one variable with two distinct periods and the function ABELIAN IKTEGRALS. 471 of one variable with more than two distinct periods is to be sought in the theorem that miwi + m.M., cannot, and 9)Ii(Di + mm., + ■•• + m w can, be made less than any assigned quantity. Jacobi pointed out in Ids memoirs (Crelle, tt. ix., xiii.) that one-valuedness could be secured by introducing functions of p variables in the place of func- tions of one variable. A theta-function affords an illustration of a one-valued function of p variables. Expressions involving 6 can be constructed, on a plan analogous to that of § 297, which reproduce themselves exactly when the variables v are altered as in the scheme (15) below; they are then 2j3-tuply periodic in the strict sense, whereas 6 itself is quasi-periodic. [It should be recalled that 2p is tlie maximum number of independent systems of periods of a one-valued function of j3 variables (§ 193). j The expressions in question arise in the solution of Jacobi's inversion problem. This problem is always sol- uble, and in general the solution is unique. It consists in finding the system of p places x^, X2, •■■,x^ from the j) equations J\hl2 + ( \lll2 + • ■ • -f ( ''du-2 = V.,, du„ (A), where the lower limits are any fixed places, and the v's are arbitrary variables. The paths in any one column are supposed to be the same. We may, if we like, suppose initially that all the paths are drawn on T'; then, removing the restriction that the paths are not to pass over cross-cuts, we have for the most general values of the right- hand sides in (A), the system v'2 = v-2 + 7i, • -f 7(., • IH \-Jtp-0 + ri,T,„ + ^.,T., + ■■■+ gpT.p, y (B). v'j, = Vp + h-0 + Jh-0 + ■■■ +Jip-l-\- giTip + g-jTnp -\ h gpTpp, The difficulty which arises in the case of functions of one variable with more than two periods does not present itself when we consider a function of p variables, v„ v,, -, v, with 2p independent periods. It can be proved by 472 ABELIAN INTEGRALS. ali;obr^ic work that, when the p differences v'^ — v^ are resolved into their real and imaginary parts, 2p — 1 of the 2p resulting expressions can be made arbitrarily small by a proper choice of the integers g, h ; but when this has been done, the remaining expression is not infinitely small. The algebraic lemma upon which this theorem is based, will be found in Clebsch and Gordan (ijp. 130 etseq.). In passing from one-valued functions of one variable to one-valued functions of many variables, a new phenomenon presents itself. The one-valued function ^' ~ y is completely indeterminate when x = 0, ?/ = 0, its value depending on x^ + y"' the mode of approach of x, y to 0, 0. More generally the one-valuedness of a function of ji variables which belong to ap-dimensional space is not necessarily affected by the circumstance that it may take infinitely many values in a space of p — 2"dimensions (Clebsch and Gordan, p. 18G). Definition of Ahelian functions. Let R be any algebraic function of T. Construct the symmetric products of R{x^, -^(Xj), ■■■,R{x^) r at a time, and let it equal { — lyM^. Then the functions M-^, 3Ii,---,Mp, when considered in virtue of the equations (A) as functions of the v's, are said to be Ahelian functions of v^ %, •■-, ^V This definition of Abelian functions is based on that in Clebsch and Gordan (p. 139). Eiemann's definition is altogether different. See Riemann, Werke, 1st ed., p. 457 ; Clebsch, Geometrie, t. iii., p. 222. Inasmuch as for the same system of upper limits, the right-hand sides of the equations (A) are capable of the values given by the scheme (B), the Abelian functions are unaltered when v„ v^, ■■■,v^ are changed, as in (B), by simultaneous periods compounded from the scheme 1 0--0 1 O-.-O Tn Tij Ti3 ^12 "^22 "^23 ' (C), ... 1 Tjj, T2j, Tjp ■•• T^, , and are therefore 2p-tuply periodic. Among the many memoirs dealing with Jacobi's inversion problem we shall only mention those of Weierstrass (Crelle, t. xlvii.), H. Stahl (Crelle, t. Ixxxix.), and Nother (Math. Ann., t. xxviii.).* For the inversion problem when the number of integrals in each of the * The hiptory of the problem is given in the introductory part of Casorati'B valuable work, Teorica delle Funzioni di Variabili Complesse, Pavia, 1868. ABELIAN INTEGRALS. 473 equations (A) is other thanp, as well as for Jacobi's problem, see Klein-Fricke, t. ii. § 302. Clebsch and Gordan solve Jacobi's Inversion Problem with the help of integrals of the third kind. We shall state their solution in terms of the Eiemann surface as is done by AVeber (Orelle, t. Ixx.) and Xeumann. Let the equations be U^'i'i + u/2'2 -\ [-u^^r'p = v, (1), K = l,2,...,p. The quantities v^ are assigned arbitrarily, and therefore the lower limits Ci, Co, ■■•, c^ can be taken and will be taken at the places «!, «2, •••, Uj, of § 290. It is assumed that the places x^, oJj, •••, a;^ are not tied by a function ; and the problem is to express these places, or symmetric combinations of them, as functions of v^, v.,, ••■, v^. Take any algebraic function of the surface, say R. Let it be /ti-placed, and let ^, aud 7;,, where i = 1, 2, •••, /n, be equivalent systems of places. Let them be associated in pairs so that when -q^ moves into the position ^1, ijo moves to ^2, and so on ; and further let none of the /A equivalent paths, which begin at -q^ and end at ^^, pass over a cut B,^. By Abel's theorem for integrals of the third kind we have For simplicity let the places rj^ be poles of R ; then the right- hand expression is log \R{x,) - R(i) I -log \R{a^} - R{i)\- Using the law of interchange of parameters and arguments, we have iogii?(ccj-ij(os-iogi^K)-^(^)s=i,xr'^"^'^' ■ ^^^' The paths of integration in this equation are assigned to suit Abel's theorem (§291). The path of the integral J^^^^^^a ^^ic^ occurs in (1) has not been assigned, and will be taken to be the same as in (2). This may require the addition of periods to the arbitrary quantities v, in (1), but it will be proved later that the periods so added will disappear from the final formula. 474 ABELIAN INTEGRALS. In equation (2) we let A = 1, 2, ••-, p ; there results, if we add the p equations and take the exponentials of both sides, The problem is reduced to the finding of an expression for the right-hand side, in terms of Vi, •Uj, ••■,1'^,. Let E{0=~^'' - ef ( du, ir(0=exp2 f^'dn^ Eemembering (§ 285) that | ■^drr^ , or | dn^ , regarded as a function of $, is infinite at x^ like + log (^ — a;^) and at a^ like — log (f — a^) , it follows that the exponential of this integral is 0' at a;^ and is co^ at a^. Therefore K{$) is 0^ at Xi, x.,,---,x^ and oo' at «i, «2, •■•, «„. The function JI{i) has the same zeros and poles. The functions have no discontinuities at a cut A^, while at opposite places of a cut -B^ each of the ratios H{i+)/H{&_) and K{i^.)/K(^_) is equal to exp 27rt2 I 'du^. (See § 297.) Hence the ratio H{i)/K(i) is one-valued and continuous every- where on T; that is, it is a constant. Thus, H{i,)/HM = K(i)/K{r,J = expk f'dn^^ (4). 6( I du^ — V From(l), H{i)= ^^'^ 'du, Jo. ^'*' therefore, exp 2 {'''dn. =-^ ^^^ , and from (3), (5). ABELIAN INTEGRALS. 475 When the product of the numerators on the left is multiplied out, the coefficients of the powers of R{$) are Abelian functions of Xi, X.2, •••jXj,; and by giving j:) different values to i?(t) we get sufficient equations to determine these Abelian functions in terms of the p/x, places i^, and the quantities i\. The algebraic function E is at our disposal. Xeumann points out that the simplest supposition, namely E = z, is sufficient and con- venient. The equivalent system i, is then of course a system of places in the same vertical, and fx. is the number of sheets. It remains to show that the expression on the right hand of (5) is not altered by the addition of periods to i\. Let the periods added be given by (B), § 301. Then the expression in question acquires the factor '"m n exp 2-iri %hi^ I \lu 1=1 A=l Jri^ or n exp 2 wihi^ 2 I du^^. Inasmuch as no path of integration has been allowed to cross a cut B^, Abel's theorem shows that the factor is 1. § 303. The case ^) = 2. It will be well to indicate how some of the preceding general theory can be brought to bear in the case p = 2, which is next in order to the elliptic case already considered. Following Clebsch and Gordan's method, we take as normal curve a nodal quartic /. An adjoint curve 4>„_. is a line through the node ; so that to places x, x in tlie same vertical on the surface T (§ 281) there correspond points x, x collinear with the node on the curve / From the node 6 tangents can be drawn ; their points of contact cor- respond to the coincident tied places; that is, to the branch-points. We call these points cti, a.., ■■■, f'c- An adjoint curve i//,,.. is one of a system of conies through the node and the points where the tangent at a meets / again. The remaining 4 points in which a conic of the system meets / are coresidual with «, a, x, x ; and the points «i, a., of § 290 are the points of contact of a conic, of the system, which touches / twice. Clebsch and Gordan's theory shows that there are 2* such bitangent conies, corresponding to the 16 non-congruent pairs of half-periods. Of these 6 are improper, and consist of the tangent at « and one of the 6 tangents from the node ; these can be shown to correspond to the 6 odd pairs of half-periods. 476 ABELIAN INTEGRALS. For the sake of simplicity, let a coincide with a branch-point Oj. Then the bitangent conic passes twice through the node, and becomes a pair of tangents from the node. Hence «i, «2 are also branch- points, say a, and a,. The arguments of ^oo are now and the zeros are Xi and Xj. Let s = Xj ; then 1 = */a, tJa i:>»- for all values of x.,, and in particular, letting Xj coincide in turn with a^ and a,. We can determine t and k as follows : In the canonically dissected surface (Fig. 68), let aj be that branch-point which lies within A^ and B^, and let a.,, a^, a^, a^. cin, aj lie in negative order, so that for example a^ and Oo lie within A2, Oi and aj within A^.* The periods across the cross-cuts are given by the scheme \ A\ A,\ B, i B, T12 I 1 I Ti2 I T22, and the integrals from each branch-point to the next are found, as in § 182, to be u,\-r,,r2\ 1/2 |-T^/2| i(r„ + T,2)/2 1-1/2 t*.|-W21-l/2i-T^/2i 1/2 \{r,, + r^)/2\ 0. On the other hand, the pairs of half-periods which make 600 vanish are given in § 236, and, on comparison with the preceding scheme, prove to be congruent with r\ r, r, r, r, r %Jay «-/03 ^^r, *-'<*2 •^°4 *J °-t " Any other arrangement of names for the branch-points would Buit our immediate purpose, but the selected arrangement is in harmony with the notation of Chapter VIII. ABELIAN INTEGRALS. 477 Among these the arguments j ' and j "" are to appear. There- fore t, (c = 3, 5 ; and finally the arguments of ^oo can be written i/"! «/<'3 i/Oj We see that, corresponding to the selected canonical system of cuts, ^00 is associated with the separation of the six branch-points into the two triads (a^, a^, a^) and (aj, a^, a^). The arguments of the other theta-functions are now readily de- termined by the addition of half-periods (§ 232). For example, the half-periods j °cZit^ are (tu + ti,)/2, (1 + tio + t22)/2; from § 232 the mark of the new function is L ., , or (35); hence the arguments to the arguments of Ooo, and are "a x>x:-x: In this way the duad notation, which in Chapter VIII. appeared merely as an algebraic artifice, begins to acquire a vital significance in its connexion with the surface. Each of the ten even theta-functions is associated with one of the ten pairs of triads into which the branch-points can be separated ; each of the six odd functions is associated with a branch-point. § 304. AVe shall proceed to show in the case p = 2 that a theta- function, with arguments j du^, where x and y are arbitrary, can be expressed in terms of algebraic functions of x and y, and of a single integral of the third kind. if-f-r) Let E(.S)= ^ ■ , J.p, • ' ■■ ( f- r- n -"""J» ■"'• *'*• Let I ri' {z + dz-z'){z' + dz'-z) J; (^ '(X'Xi" 480 ABELIAN INTEGRALS. whence ^(' P)= (z-3')^exp n"^" • lim _y£^J_V£l^. dz • dz' (A) Let the value of — ^-^ — —, when ■Vj z= v^ = 0, be denoted by Then e( C' dw.V - 2c'^'^d«, and therefore the required limit is Now the normal integrals Mi, u, are linear combinations of the two integrals | zdz/w, | d«/io (§ 281), and therefore the required limit can be written cz + c' cz' + c' W{Z) 10 {z') where c and c' are constants. Lastly, as we know that ^[ ( ) lias its two zeros at the point a, we obtain ca + c' = 0. Therefore X a factor 2 being introduced to make the denominator on the right agree with that in (6). The connexion of the thetas with the surface is not completed until the constants which appear on the left of equations (6) and (7) are expressed in terms of the sextic. This has been done by Thomse (Schlomilch's Zeitschrift, t. xi., p. 427) for the present case.* But, while it is beyond our purpose to go further into the matter, we can recognise the de term inability of these constants ; and it is • The general problem, as stated by Riemann ("Werke, p. ^31), is to determine log 9{0, 0, •••,0) in terras of tbe 3p— 3 class-moduli (§ 284). The problem is reduced to one of integration by Thomae in Crelle, t. Ixvi., and tbe integration is effected, in the general hyper- elliptic case, by the same writer in Crelle, t. Ixxi. Fuchs (Crelle, t. Ixxiii.) obtains the same results as Tliomae more simply by the use of tbe system of lower limits introduced by Clebscb and Gordan. Klein (Aberscbe Functionen, Math. Ann,, t. xxxvi., p. 6S) considers tbe case p= 3, using the coefficients of the normal curve (tbe plane non-singular quartic) iis cliiss-nmilnli; wberuaa previous writers ou the problem bave assumed a knowledge of the brancb-phkceH. ABELIAN INTEGRALS. 481 found that when they are incorporated with the thetas, there results the same advantage as was gained when Jacobi's thetas (Chapter VII.) were replaced by Weierstrass's sigma-f unctions. In fact, the left sides of equations (6) and (7) are sigma-functions in the case p = 2; but they are not yet in their final form, for which the reader is referred to Klein's memoirs, already cited. The introduction of the higher sigma-functions was due to Weier- strass. See Schottky's Abriss einer Theorie der Abel'schen Func- tionen dreier Variabeln; Staude, Math. Annalen, t. xxiv. The above definition of a sigma-function agrees with that of Staude. § 306. Existence theorems for the liiemann surface T. It is assumed that the surface is spread in a finite number of sheets (say n) over the «-plane, that the sum of the orders of its branchings is 2p+2 n — 2, and that its order of connexion is 2p+l (see § 275). A potential function of the first kind is everywhere continuous on T and its values at any point differ merely by periods due to the description of periodic paths on the surface T. We have stated in § 277 the properties of Abelian integrals on T ; let each integral be expressed in the form $ + it), where ^, rj are real functions of x, y. The functions ^, -q are called potential functions of the first, second, or third kinds, according as they arise from integrals of the first, second, or third kinds. The potentials |, rj are conjugate; when i is given, rj is determined save as to an additive constant which arises from the description of periodic paths. The following simple example shows how f, tj behave at a discontinuity. Regarding ^ as a velocity- potential for a fluid motion in the plane of x, y, the equations f = a constant, ?; = a constant give lines of level and lines of flow. When the combination f-|-!ij = ^ log (2-c), where ^ is real and z — c=pexpifl, we have ^ = A\og p, n = AB ; the lines of level are circles with centre c. and the lines of flow are straight lines which radiate from c. The motion is due to a source of strength 2 irA. The function ?; is many-valued. When A is purely imaginary and equal to IB, i= -B0, 7, = 51ogp, and the motion is that of a liquid whirling round a point c in concentric circles. For a general discussion of the potentials which arise from logarithmic and algebraic discontinuities, see Klein's Schrift, Section I. A potential of the second kind is continuous on T, except at a finite number r of places z^ at which it is infinite like the real part of a function (2_2j ^Jao-f ffi(2-0j +"-+a,^_,(z-2j''«-M, 482 ABELTAN INTEGRALS. and possesses the usual periodic properties at the cross-cuts. In the case of an elementary potential of the second kind, r and A. equal 1 [Klein-Ericke, p. 506]. A similar definition applies to potentials and to elementary poten- tials of the third kind* The materials for establishing the existence on T of potentials of the first kind, and of elementary potentials of the second and third kinds, lie ready to hand. For instance, we have explained in Chapter IX. Schwarz's solution for a doubly connected region when the discontinuity along a cross-cut was an arbi- trarily assigned real period. By an extension of this method there exists a func- tion on a (2p + l)-ply connected Riemann surface with a puncture at a point, which has arbitrarily assigned periods at the 2p cross-cuts which reduce T to simple connexion, and has an assigned value at the puncture. In this way Schwarz's solution of § 2G0 establishes the existence-theorem for potentials of the first kind. The proofs in the text are those given in Klein-Fricke, t. i., p. 504 et seq. Proof of the existence on T of one-valued elementary potentials of the second kind. Let r be a circular region on any sheet of T, and let it contain no branch-place of T in its interior or on its rim. The investigation of § 271 shows that there exists on r a function which is harmonic in the interior, and which takes an assigned system of values on the rim ; and by § 273 there exists on F a potential function ^j which has all the usual properties of the harmonic function except that at one place of the interior it is infinite like the real part of ao/(2 — c). The exclusion of a branch-place is not necessary ; for a branch-place at which r sheets hang together can be enclosed by a region bounded by an r-fold circle, without affecting the solvability of Dirichlet's problem (§ 260). Recalling the fact that the method of combination leads to the solution of Dirichlet's problem for composite regions, it is easy to prove the existence on T of f^. For the surface T can be covered, without leaving any gaps, by a. finite number of over- lapping regions with boundaries which are 1-fold or r-fold circles. In the process of construction of the composite region there are two stages. During the first stage tlie initial region is continued over T without including c, and the corresponding potential function is continuous throughout the interior of the composite region. The second stage begins with the inclusion of c, and the potential func- tion has now the ac'ded property that it is discontinuous at c in the assigned manner. ♦Attention must of courae be paid to the additional periods due to the logarithmic diecODti. nuities of the integral of the third kind. ABELIAN INTEGRALS. 483 To derive general existence-theorems for potentials of the kind met ^vith in considering the integrals of algebraic functions, we can use Uvo principles : the principle of superposition and the principle of juxtaposition. The latter of these is due to Klein (Math. Ann., t. xxi., pp. 161, 102, and Klein-Fricke, pp. 518-522). The principle of suijerposition can be explained immediately. Let ^ be infinite at c like the real part of l/{z—c); then d^^/dc, d%/dc-, ■■■, are oo-, x", ■■■ at c. The potential function dc dc- dc' where the A's are constants, is a potential function with an algebraic infinity of order r at the place c. We proceed to show how Klein derives potentials of the first and third kinds from f,. Let \ be a one-valued elementary potential of the second kind which is infinite at z^ like the real part of c,,/{z - z,,)- Multiply c„ by an arbitrarily small real number dc, and let the result be the stroke dza- There exists on T a one-valued elementary potential of the second kind 4^ which is infinite at Zo ^--^e dzo/(z-Zo) ; to this Klein applies a process which amounts to integration. Let z^ move from a place a to a place b along a chain of strokes dzo, and to each stroke let there be attached the corresponding potential ^^ ; the potential which arises from the juxtaposition of these potentials is X '""^.„, = ^.. (say). As regards its discontinuities 4,j must behave like the real part of X =a Z Zn Preserving the chain of strokes from a to b, construct for each stroke dz an elementary potential i\^ which is one-valued and contin- uous and which behaves at Zolike the real part of idZo/(z-2„). 'Write C a, J — .) I 6 »o Z IT •^'0="' Eegarding the chain of strokes as a barrier on T, the function |',j is one-valued" and continuous on the surface which results when a cut is made along the barrier. The values on opposite sides of the barrier differ by 1, for f., behaves along the barrier like the real part of 2,rA=« z-z; 2r °z-b 484 ABELIAN IKTEGEALS. When the restriction is removed that the barrier is not to be crossed, ^'„j ranges itself as one branch of the system $'^^ ± m, where m=l, 2, 3, •••. Suppose that the chain of strokes begins and ends at Zq = a, and that the corresponding cut does not sever the surface. The potential ah loses its discontinuities on T and has no periods ; therefore it degenerates into a constant; but i'^ passes into a potential which is one-valued and continuous on the surface T after a retrosection has been drawn along the chain of strokes, and which has a period 1 along this retrosection. Let us now consider the surface T and construct with reference to the cross-cuts A^, A^, •••, A^, B^, B.^, ■■■,Bp, 2p normal potentials of the first kind, Ui, Uo, •••■ fzp which have the property that J7, (U^+p) has the period 1 at A^ (BJ where k=1, 2, ■•■,p, and the period at the remaining 2p —• 1 cross-cuts. These potentials are determined completely save as to additive constants; since the co-existence of two distinct potentials of the first kind [7,, U'^ with the period 1 at the xth cross-cut and the periods at the remaining cross-cuts involves the existence of a potential Uoi the first kind which is one-valued and continuous throughout T, i.e. the difference U^ — U', is a constant. From the 2p potentials U, it is possible to construct the most general real potential of the first kind. For the expression U=i{X^U^ + X^^,U^^,) + \ (i.) K=l in which the X's are supposed real, is one-valued and continuous on T' and has the real periods X^, X^.^.,, at A^, B^ which can be assigned perfectly arbitrarily ; and by the preceding reasoning the periods determine the function, a potential of the first kind, save as to an additive constant. To see that the potentials Ui, K, ■••, f/jpi^iust be themselves linearly independent, assume a relation 2 (a,C7, + «.^, [/■«+,) = (ii.). Since the non-vanishing periods of a^U^, a^+jU^+p are a^, «,+,,, it follows that the expression on the left-hand side of (ii.) has periods a„ «,+,,; therefore the equation (ii.) requires that all the coefficients a^, a^^^ shall vanish, and consequently the potentials Ui, U2, •■■, U^p cannot be connected by a linear relation. By associating with each of the potentials t/], U.,, ■••, U^^ its con- jugate potential, we can construct a system of 2p integrals of the first kind ABELIAN INTEGRALS. 485 and the preceding work shows that every integral of the first kind can be expressed in one way, and in one way only, as a linear combination ivith real coefficients \. [Klein-Fricke, t. i., p. 526.] When combinations with complex coefficients are admitted it is possible to express all integrals W of the first kind as linear com- binations of p linearly independent integrals. That it is impossible to express all integrals W=U+iV, in terms of fewer than p integrals, follows from the consideration that this would involve the theorem, that all potentials U of the first kind can be expressed as linear combinations of fewer than 2p selected potentials of the first kind. It is possible to express all integrals W in terms of more than p integrals of the first kind, but the representation will then be no longer unique ; for assuming such a representation, each integral of the first kind can be resolved into its real and imaginary parts, and as a necessary consequence there results the erroneous theorem that every potential (7 can be expressed uniquely in terms of more than 2p potentials. [Klein-Fricke, t i., p. 527.] Enough has been said to indicate Klein's mode of passage from the theorems of Chapter IX. to the Eiemann surface. § 307. The existence-theorem for the general Abelian integral on the Eiemann surface T can be proved by synthesis from the potentials whose existence on T has been established, use being made of conjugate potentials and of differentiation with regard to parameters. It can also be established on Schwarz's lines (Schwarz, Ges. Werke, t. ii., p. 163) . Suppose that the integral is | -f- i-q and that ^ is to be infinite, like the real part ^ of a sum of algebraic functions 2 K2-cJ"*'(«o+«i(2-0 + -+«A-i(2-0^«"'')+&«log(2-c,)i; «=i suppose further that | has assigned periods at the cross-cuts A^, B^, and at the further cross-cuts made necessary by the excision of the places with logarithmic discontinuities. The function ^ — f is free from discontinuities on T" other than the periods due to the cross-cuts, and is therefore one-valued and continuous on T" (for the meaning of T" see § 285). The unique existence of a function with these properties has been established ; hence also the existence of t By combining ^ with its conjugate -q, we get the Abelian integral | -t- ir,. 486 ABELIAN INTEGKALS. This brief sketch of the existence-theorems of Schwarz and Neumann is, we hope, sufficient to show the bearing of the theorems of Chapter IX. on Riemann's theory of tlie Abelian integrals. It must not be supposed that this is the sole application which can be made of existence-theorems. In verification of this remark the reader may consult a memoir by Ritter on automorphic functions (Math. Ann., t. xli.). The discussion of Abelian integrals on an )j-sheeted (2/)-|-l)-ply connected closed surface spread over a plane forms but a small part of the programme drawn up by Riemann and elaborated by Klein, I'oincarO, and others. Mention may be made of two directions in which extensions have been sought. Without altering the surface T it is possible to establish the existence upon it of functions which have properties different from those of Abelian inte- grals or algebraic functions ; examples of this kind are Appell's Functions a muUiplicateurs, Acta Mathematica, t. xiii.* The other extension consists in removing the restriction that a Riemann surface is to be a surface spread over a plane ; passing to the sphere with p handles, Klein has discussed the properties of the associated potentials and has indicated under what conditions a doubly extended manif oldness in hyper-space can be used as a Riemann surface. Beferences. — The materials for this chapter have been drawn mainly from the memoirs of Riemann, and from the treatises of Clebsch and Gordan, Fricke, and Neumann. The importance of Riemann's work in the field of Abelian functions cannot readily be overestimated, but unfortunately it is expressed in too concise a form to be easily intelligible without the use of commentaries. The works of Neumann and I'rym (Neue Theorie d. ult. Funct. ), mentioned below, will be found useful helps to the acquisition of a working knowledge of Riemann's methods. The standard treatise on Abelian functions is that of Clebsch and Gordan. Books on hijpereUiptic and Abelian functions. — Abel's Works, p. 145, Mfemoire sur une propriet6 genferale d'une classe tres-fetendue de fonctions transcendantes ; Briot, Fonctions ab61iennes ; Clebsch and Gordan, Abel'sche Functionen ; Jacobi's Works ; Klein, Ueber Riemann's Theorie der Algebra- ischen Functionen ; Klein-Fricke, Modulfunctionen ; Konigsberger, Ueber die Theorie der Hyperelliptischen Integrale ; Neumann, Abel'sche Integrale; Prym, Neue Theorie der ultratlliptischen Functionen, 2d ed., and Zur Tlieorie der Functionen in einer zweiblattrigen Flache ; Schottky, Abriss einer Theorie der Abel'schen Functionen vom Geschlecht 3 ; Weber, Theorie der Abel'schen Functionen vom Geschlecht 3. Memoirs. — The memoir-literature on Abelian integrals and functions is very extensive. The following short list is added solely with the view of direct- ing the reader's attention to a few of the numerous developments of the subject : — (1) Memoirs based on Weierstrass's work. Weierstrass, Zur Theorie der Abel'schen Functionen, Crelle, tt, xlvii., lii. Henoch, \)e Abelianarum functionum periodis (Berlin dissertation, 1867). Forsyth, On Abel's Theorem and Abelian Functions. Phil. Trans., 1882. ♦Pr5'm stated in his memoir (Crelle, I. Ixx.) the existence of functions of the claps discussed by Appell ; the values of these functions on opposite bauks of a cross-cut are connected by lineo- ^iuear relations. ABELIAN INTEGRALS. 487 (2) Transformation. Ilermite, Sur la thfeorie de la transformation desf onctions ab61iennes, Comptes Kendus, t. xl. I'aris, 1855. Konigsberger, Ueber die Transformation der Abel'schen Funotionen erster Ordnung, Crelle, t. Ixiv. Tliis memoir is written on Weierstrass's lines. (3) Memoirs based on Klein's work. Klein, Ueber hyperelliptische Sigmafunctionen, JIath. Ann., tt. xxvii., xxxii. Burkliardt, Beitragen zur Tlieorie der hyperelliptischen Sigmafunctionen, Math. Ann., t. xxxii. Klein, Abel'sche Funclionen, Math. Ann., t. xxxvi. Thompson (H. 1).), Hyperelliptische Schnittsysteme undZusammenordnung der algebraischeu und transcendentalen Thetaoharacteristiken. (4) Memoirs on special systems of points on a basis-curve, class-moduli, ex- ceptional cases in the theory of Abeliau Functions, reality considera- tions, etc. Brill and Xother, Math. Ann., t. vii. Schwarz, Ueber diejenigen algebraischen Gleichimgen zwischen zwei verSn- derlichen Grossen, welche eine Schaar rationaler eindeutig umkehrbarer Transformationen in sich selbst zulassen, Crelle, t. Ixxxvii., p. 140. Poincarg, Sur un thgoreme de M. Fuohs, Acta Math., t. vii. Painlev6, Sur les Equations diff ferentielles du premier ordre, Ann. Sc. de Vtc. Xorm. Sup., Ser. 3, t. viii., p. 103. Weber, Ueber gewisse in der Theorie der Abel'schen Functionen auftretende Ausnahmefalle, Math. Ann., t. xiii. Nother, Ueber die invariante Darstellung algebraischer Functionen, Math. Ann., t. xvii. Pick, Zur Theorie der Abel'schen Functionen, Math. Ann., t. xxix. Hurwitz, Ueber Riemann'sche Flachen mit gegebenen Verzweigungspunkten, Math. Ann., t. xxxix. Klein, Ueber Realitatsverhaltnisse bei der einem beliebigen Geschlechte zugehorigen Xormalourve der tf>, Math. Ann., t. xlii. EXAMPLES. (1) Prove that | z -t- V? — (? j + | z — Vz^ — c^l = |z + c| + i2- (2) The condition of similarity of two triangles ahc, xyz is = 0. [§10.] 1 1 1 a b c X y z (3) Given a triangle ahc, points x, y can be found such that the triangles axy, ybx, xyc, ahc are similar. (4) In the preceding question, the triangles ahc, xy have the same Hessian points. [§ 33. J (5) In § 25, show that when a and b are complex the question is reduced to the one considered by proper rotations of the two real axes. (6) If M = 2z + z-, prove that the circle |2|=1 corresponds to a cardioid in the w-plane, and that an equilateral triangle in the cir- cle corresponds to the points of contact of parallel tangents of the cardioid. (7) The three points of contact of parallel tangents of a fixed cardioid have one fixed Hessian point. (8) Prove that the Brocard points of a triangle z^z.^^ are the Hessian points of the triangle h^hjc. [§§ 31, 33.] (9) The middle points of the strokes zJ,(k = 1, 2, 3) of § 31 have the same Hessian points as the triangles z„ z.^, z^ a.nd ji, J2, js- (10) Prove that the polar pair of the point oo -with regard to a triangle are the foci of the maximum inscribed ellipse. [§ 35. See Quarterly Journal, June, 1891.] (11) Show how to invert a triangle and its centroid into a tri- angle and its centroid. [Ib.J 488 EXAMPLES. 489 (12) A convenient way of introducing homogeneity is to name any point x by the ratio Xi/x.j of the strokes to it from two fixed points of the plane. Show that the point {Xi + Xiji) / {x.j + \y.,) divides the stroke from a; to 3/ in the ratio A/1. Here \ may be complex. [§ 35. J (13) The points Zj, Zj, Z3, z^ are concyclic if, u, p, y being real, 111 a 13 y Z.Xs + ZiZi Z^Zi + Z.Zi , z^z., + Z-^Zt = 0. (Beltrami.) (14) Through any three of four points Zj, z^,, z^, Zj a circle is drawn. Prove that an anharmonic ratio of the inverses of any fifth point with regard to the four circles is equal to the corresponding anharmonic ratio of z,, 'i> (15) Let a^, 6, (k = 1 to 4) be quadrangles having a common Jacobian. Prove that, when the points are suitably paired, 21/(a.-6,) = 0. [§39.] fC (16) The four points whose first polars with regard to a quartic U have a double point are the zeros of where H is the Hessian of U. [§ 36. See Clebsch, Geometric, t. L, eh. 3, sec. 5.] (17) In the preceding question, the first polar points which do not coincide are given by Zg,U+g,H=0. The quartic having the same Hessian as U is Zg,U-2g,H. (18) The intersections of three mutually orthogonal circles are the Jacobian points of a system of quartics. [§ 40.] (19) Show how to invert a harmonic quadrangle into a square. [Laisant, Theorie des £quipollences.] (20) If M + io be a one-valued function of z, and if normals be drawn to the z-plane at each point z, of lengths u and v, the two surfaces so formed have the same curvature at points on the same normal. [§ 17. See Briot et Bouquet, Fonctions EUiptiques, § 10.] 490 EXAMPLES. (21) A convex polygon which includes all the zeros of a rational polynomial f{z) must also include the zeros of the derived poly- nomial f'{z). [This generalization of EoUe's Theorem iu the Theory of Equations was given by F. Lucas, Journal de I'Ecole Polytech- nique, 1879.] (22) Discuss the amount of truth in Siebeck's paradox, that when y=f[x), to an arbitrary curve in the a;-plane corresponds a curve in the y-plane with fixed foci, namely the finite branch-points. [Siebeck, Crelle, t. Iv., p. 236.] (23) The series Sa-z" , where a is positive and < 1, is convergent in and on the circle | 2; j = 1 ; and the same is true of the series formed by the 9-th derivatives of the terras. But the function defined by the series cannot be extended beyond the circle. [§ 104. See Freedholm, Comptes Rendus, 1890.] (24) The deficiency of the equation ic^ + z^ — oicz" = is ^j = 2. [Kaffy. In his thesis, Eecherches algebriques sur les integrales abeliennes, ch. 3, M. Raffy shows how to find the number p by means of divisions and eliminations, without the solution of algebra equations.] (25) The terms of a series %f„{z),= F{z), are holomorphic in any region T and are continuous on its boundary C, and the series F{z) converges uniformly on C. Prove that (i) The series F{z) converges in every region r' which lies wholly within T {i.e. which has no point in common with C); (ii) The series formed by the successive derivatives of the terms f„{z) converge uniformly in r' and represent the successive deriva- tives of F(z). [See Painleve, sur les lignes singulieres des fonctions analytiques, p. 11. J (26) The integral -—, C^Ml!?. is equal to g{t) or according as t is within or without the contour C. Hence, write an expression which is equal to g,{t) within C„ to g.,{t) within C., •, and to g^(t) within O,, where the contours C,. Co, -•-, C, lie exterior to one another. [§ 139. Compare § 103. An important theory, based on this method of introducing lines of discontinuity is given in Hermite's Cours.J (27) Let ax' + 2hxy + b>/- + 2gx + 2fy + c = 0. If X describe an ellipse whose foci are the branch-points in the o^plane, the two points y describe ellipses whose foci are the branch- points in the y-plane. EXAMPLES. 491 (28) Let F{x, y) = he an algebraic equation, and let the planes of X and y be covered with the Riemami surfaces which belong to the equation. On either surface a non-critical place, at which cl'y/dx- = 0, is such that to a line through it corresponds on the other surface a curve with an inflexion at the corresponding place. Also a non-critical place at which the Schwarzian Derivative vanishes (that is, y' 'Ay' J where accents denote differentiations for x) is such that to a line or circle through it corresponds a curve which has maximum or mini- mum curvature at the corresponding place. (29) The necessary and sufficient condition that the region bounded by two confocal ellipses can be mapped on a circular ring is that the sums of the axes of the ellipses are proportional to the diameters of the circles. [Schottky, Crelle, t. Ixxxiii.] (30) Let F(x, y) become \4,{x, y)\'' by means of a transforma- tion x = Ri{x,y), y=R,{x,y). When the deficiency of F=the deficiency of ,>l, then k = 1. [§ 183. See Weber, Crelle, t. Ixxvi.] (31) The expression d:(;/Vris transformed by meansof z= —H/U into ^dx/V-lz^' — j/,z — (/a, where H is the Hessian, g., and g^ the invariants, of the quartic U. [§ 203. This theorem, which brings the elliptic integral into Weierstrass's form, is due to Hermite. See Halphen, t. ii., p. 360.] (32) If u-irv + w = 0, then crMo-jVo-jM) + o-sMcrVo-iW) -I- foWCiVtrtO = 0. [§211.] (33) If a + l> + c + d = 0, then (e, - e,)a,aaM.ca,c. 2(«o — w'o) w ./ 2{z — tv) tt'o [Weierstrass-Schwarz, p. 89.] (45) In § 221, show that the lines which bisect opposite edges of the rectangle of periods in the ?/-plane map into circles in the «-plane. [Weierstrass-Schwarz, p. 75.] 8(mi,<"2) 3 Tri ' 8((ui, V2) (Uj) 92 12" 8(^3- log A) 8.9/ 8( „+V.f(^rv4, 3 [Klein, Math. Ann., t. xxvii., p. 455.] (49) When two functions Mj, iu are harmonic in regions which have a common two-dimensional simply connected region r, and coincide throughout a portion of T, however small, they coincide throughout T, and their continuations beyond V coincide. [Schwarz, Werke, p. 202.] (50) In the case of the equation x* -\-y*= 1, we can take as the linearly independent integrals of the first kind Cdx/y; j dy/x^, j{ydx — xdy). These integrals are elliptic. [§ 280. See Amstein, Bulletin de la Societe Vaudoise, 1888, where this case is fully worked out as an example of Weber's memoir on Abelian functions of deficiency 3.] (51) Given the equation F{w, z) = 2 If' - 5 If" - z{az' + 2 62 + c)^ = 0, where c, b- — ac are different from 0, prove that ^ \(az- + 2bz + c) + 2 Bicz + Cw- + 2 Ew ^^ P is an Integra] of the first kind, whatever be the values of X, B, C, E. [Eaffy.] 494 EXAMPLES. (52) The number of linearly independent integrals of the first kind may be > p when the basis-equation is reducible. [Christoffel.] (53) Prove that Qj.^^ + Q;. _ + Qjv^^ is an integral of the first kind. [§ 285. See Clebsch and Gordan, p. 22 et seq.] (54) The number of three-sheeted surfaces with ^ assigned branch-points is 1(3^"- — 1). [§ 284. See Kasten, Zur Theorie der dreiblattrigen Riemaun'schen Flache; Hurwitz, Math. Ann., t. xxxix.J (55) The comparison drawn in § 297 between the function _ ■, If the sextic be of the form z' -)- o^, then T„ = T„ = 2 i/V3, T,„ = - i/V3. [A discussion of the values of t„^, for sextics which can be linearly transformed into themselves, is given in section iii. of Bolza's paper, American Journal, t. x.J (57) Between four odd double si gma-f unctions there is the square relation .. .. .. -i i _ a X «^ «v «S x' a^- tty- O'i' 9 '' 9 9 X V V <^« a„ being the branch-point associated with o-„. [§ 305. Compare § 244.J (58) Prove the four product-relations 1 1 1 1 =0, «! Hj ttj a2 ClCl2 0"3; expressed by means of elliptic configuration ; Riemann surface of deficiency one-mapped Ijy means of, 333; theory of, 205 et seq. au, 9-series for, 325; theory of, 305 et seq. ; (T/^u, g-series for, 325 ; theory of, .308 et seq., iu, elliptic function expressed by means of, 304; theory of, 302 et.^ej., • et neq; on the normal surface, 255 ; round dissected Riemann surface, 430 ; use of cross-cut in, 24fi. Interchange of limits and parameters in Abelian integral of third kind, 445. Invariant of qnartic, 34, 35. Inversion, 18, 29. Inversion of elliptic integral, 203; of hyperelliptic integral, 203. Inversion Problem, Jacobi's, 470 et seq ; solution of, 471 ; statement of, 471. Involution, 19. involutions determined by four points, 30. Isogonality, 14. Jacobi's elliptic functions, 309. Jacobian of binary form, 28; of cubic, 2() ; of quartic, 32, 301 ; zeros of, 302. Kronecker's theory of the discriminant, 202; extended to non-integral algebraic function, 271. Rummer's surface, in connexion with 1 double theta-f unctions, 372 ; the co-ordinates of nodes of, 372. I Lacunary spaces, 119. j Laplace's equation, 13, 375. j Laurent's theorem, 175. 1 Legendre's bilinear relation of periods of I elliptic function, 319. ' Like-branched functions, form a system, ' 245. Lima^oii, 15. Limit of a sequence, 42, 44. Linear mass of points, 50. ; Liouville's theory of double periodicity, 287 ct Keq. i Logarithm, 121; chief branch of, 122 ; defined by integral, 25G. I Logarithmics!ngularity,natureof,185,256. . Loops, deformation of, 154; ' on Riemann surface, 212: theorems of Luriith and Clebsch on, 155 I ct seq. ; , theory of. 151. Lower limit. 4b. Many-valued functions, examples of, 36 ' to 39. Marks, tlieory of, 344. j Means, arithmetic, geometric, 7 ; harmonic, 8. ' Meromiu-phic function, 101; consists of fractional part and holo- morphic jiart, 180; determined by zeros and infinities, 181. Minimal basis, definition of, 265. Mittag-Leffier's theorem, statement of, 187; example of, 192 ; proof of, 188. i ilobius's theorem, 20. : Monogenic function, 13. ! Multiple factors in lowest terms of an algebraic equation, 130. Multiplication of argument of pu, 315 ; au, 315. Xeighbourhood, 50, 85. Non-uniform convergence, 70. ^'ormal curve, 451. Normal Abelian integrals of first kind, linear independence of, 435; of second kinerie de puhnanres, integral series; entiirt', integral series. Simply connected (region), (aire) a simple connexio)), in(iiiad€lphe, eiii- furh zusammeii/iiini/eiid. SimultiiDe Bahneii dcr Xiveaupunkte, equivalent paths. Spire, see branch-point. !ite!le. place. ^irei-ke, stroke. Stroke, ftrccke. t^y nectiq iie , holomorphic. Tied, verkniipfl, p. 441. Transcendental integral function, tran- sccndi'iitc ipinze Function, p. 112. Turn-point, Kremumjspunkt. Ufer, bank (of a cross-cut). Ultraelliptic functions, hyperelliptic func- tions in the case p = 2. This terra is falling into disuse. Umyung, circuit. Umr/ebunf/, domain. Unbedinr/t conrerr/ent, unconditionally convergent. Unconditionally convergent, unhcdimjt conrcnjent. Vnendlirh rcr-iuijerte (Convergenz), infi- nitely slow (couvergeuce). Vniforme, one-valued. Uniform (convergence), i/?6icAmi'ssi "* i/lcirhcm Griide. Unilateral surface, one-sided, Dojjpel- fiiirhc. Unstetiijkcitxpunkt, discontinuity, often intinity. Verkniipft, tied. Vcrzu-i-ii/unysulniitt, branch-out. I Vcrziceiifun'jspunkt, branch-point. Voisinai/c, neighbourhood. I VoUstiindiij be'jriinzl, delimited. Werthirjkeit, number of times that a func- tion takes any assigned-value. [An m-placed function has 'Wertki;/- ki'it' »!.] Wesentlicher sinyulUrer Punt;, essential singularity. M'esentliche sinr/uliire Stelle, essential singularity. Winduni/sfliiclw, winding-surface ; ex- ample on p. 207. Windunijspunkt, branch-point. Zusammenhanfi, coimexion. Zusammenhiinije nd , ciiifach, simply con- nected. mehrfitcli, multiply connected. Zicciy, branch. TABLE OF EEFEREXCES. o>«5. nolzmiiller, 215. Hurwitz, 212, 270, 487, 494. Jacolii, 2S0, 280, .312, 319, 326, 327, 340, 470. 471, 4,SI>, 4S7. Joachiinstal, 287. Jordan, (12, 78, 172, 100, 204, 201, 278, 292, 320, 342. Kasten, 404. Kelvin, Lord, 377. Klein, 20, :i(i, 00, 228, 230, 2.32, 230, 244, 251), 201, 274-270, 310, 331, 3;!5, .338, 340, .372, 375, 407, 400, 420, 442, 470, 480, 481, 4N3, 480, 487, 4il3. Klein-Fricke, 110, 244, 202. 2(i5, 270, 324, 335, 340, 420, 437, 4:>8, 441, 442, 458, 473, 482, 483, 485, 4Sli, 402. Konigsberger, 8, 150, 241, 248, 270, 340, 48(i, 487. Kijpcke, 00. Kossak, 51. Krause, 34(i, 340, 373, 486. Krazer, 350, .373. Krazer and Prym, 373. Kronecker, 150, 2()2, 263, 267, 268, 270, 271, 272, 274, 427, 428. Kumraer, 372. Lagrange, 52. Laguerre, 187. Laisant, 480. Lamb, 241, 377. Laplace, 13. Laurent, H., 155, 204, 397. Laurent, P. A., 175. Lecornu, 112. Legendre, 319, 333, .340. Leibnitz, 51. Liouville, 175, 287, 205. Lucas, 400. Lurotb, 127, 155, 235, 2.39. Luroth and Scbepp, 280. Maxwell, 13. Meray, 120. Mertens, 101. Minchin, 13. Mittag-Leffler, 50, 119, 185, 180-189, 285. Mdbius, 20, 21, 227. Molk, 205. Miiller (see Enneper). Neumann, 18, 230, 232, 233, 253, 270, 377, 308, 410, 422, 424, 420, 427, 4:>4, 455, 4.57, 473, 475, 480. Newton, 147. NiJtber, 133, 13>4, 130, 144, 427, 487. Painlevc, 377, 38(), 413, 487, 400. I Panton {see BurusideJ. ; Paraf , 388, 420. I Pasch, (iO. I'eano, 70. Phragmcn, .307. Picard, 77, 78, 89, 04, 101, 105, 203, 204, .•^2, .■■74, 377, 382, 302, 397, 420. Pick, 487. Pinclierle, 02, 104, 105. Pliieker, V.iH, 2:;3. Poincare, 110, 204, 27(), 280, 335, 375, 377, 420, 452, 480, 487. Pringsheini, 78, k;;, s4, OS, 101, 118. Prym, 244, 270, 280, 350, 373, 480. Puiseux, 134, 147. Raffy, 400, 493. I Reichardt, 372. Riemann, B., 13, 18, 41, 49, 52, Ofi, 104, 205, 228, 2:«, 2:«, 230, 241, 242, 245, 24, 487, 403. (See, also, Weierstrass.) Scott, C. A., 146. Seidel, 72, 118. Siebeck, 400. Simart, 241. Smith, H. J. S., 49, 133, 233, 272. Stahl, H., 487. Staudc, 347, 481. TABLE OF REFERENCES. 507 Stokes, 72. Stolz, S, U, 4S, 5i, G2, 85, 98, 109, 126, 133, 138, li4, 378. T.annery, O'.', 72, 118, 119, 126. Thomas, 48, 58, 62, 126, 276, 331, 340, 480. Thome, 72. Thompson, H. D., 487. Thomson (see Kelvin). Todhunter, 283. Tychomandritsky, 280. Vivanti, 280. Weber, 340, 372, .373, 438, 463, 464, 466, 4<)7, 469, 473, 486, 487, 491. (See, also, Dedekiud.) Wedekind, 36. Weierstrass, 41, 42, 48, 51, 58, 59, 60, 62, 72, 84, 86, 94, 98, 101, 10,i, 109, 112, 118, 119, 126, 128, 141, 159, 174, 179, 185, 186, 187, 189, I'.K), 193, 28«>, 295, 301, :!06, 312, 319, 327, 328, 330, 338, .340, 377, 441, 481, 4S(i, 491. Weierstrass-Schwarz, 295, 301, 308, 312, 333, 340, 492. ■Williamson, 337. MATHEMATICAL TEXT- BOOKS PUBLISHED BY MACMILLAN & CO. npecbanicg- "^ RIGID DYNAMICS. An Introductory Treatise. By \V. Steadman Aldis, M.A. ?i.oo. ELEMENTARY APPLIED MECHANICS. Part II. Transverse Stress. By T. Alexander and A. 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