33) CORNELL UNIVERSITY LIBRARIES Mathematics Library White Hall ,„f"«»ELL UNIVEBSITV LIBRARY 3 1924 059 412 910 DATE DUE SEP^ '"'^fc^ wcr j^ m^ CAYLORD pniNTEOINU.S.A. Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059412910 Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39. 48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation. Digital file copy- right by Cornell University Library 1991. INTRODUCTION TO THE THEORY OF ANALYTIC FUNCTIONS. INTRODUCTION TO THE THEORY OF ANALYTIC FUNCTIONS BY J. HARKNESS, M.A. (Cambridge) PROFESSOR OF MATHEMATICS, BRYN MAWR COLLEGE, PENNSYLVANIA AND F. MORLEY, Sc.D. (Cambridge) PROFESSOR OF PURE MATHEMATICS, HAVERFORD COLLEGE, PENNSYLVANIA. MACMILLAN AND CO., Limited NEW YORK: THE MACMILLAN COMPANY 1898 [All Rights reserved.'^ E M CantbriBgt : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. PREFACE. THE present book is not to be regarded as an abridged and more elementary version of our treatise on the theory of functions, but as an independent work. It has been composed with different ends in mind, deals in many places with distinct orders of ideas, and presents from an independent point of view such portions of the subject-matter as are common to both volumes. In the treatise our desire was to cover as fully as possible within the limits at our disposal a very extensive field of analysis, and the execution of this plan precluded the possibility of allotting much space to preliminary notions. At the same time we recognized that readers approaching the subject for the first time could not fail to be hampered by the non-existence in English of any text-book giving a consecutive and elementary account of the fundamental concepts and processes employed in the theory of functions ; subsequent experience and inquiry has only strengthened our belief that if English and American students are not to be placed at a disadvantage as compared with their foreign brethren they should have ready access to text-books which discuss topics of the kind indicated above. It was to an attempt to meet these requirements to the best of our ability that the present volume owes its genesis, and this is its bond of connexion with the treatise. The theory of functions, by virtue of its immense range and vitality and its innumerable points of contact with other branches VI PREFACE. of mathematics, has taken a central position in modern analysis, and has made its influence felt in all parts of the mathematical domain. It is not surprising accordingly to find that such of the current text-books as have been composed in the modern spirit show numerous traces of the inrush of new ideas due to a wider acquaintance with the theory of functions, and that they are, both as regards structure and aim, poles apart from those of the preceding generation. There is, however, much still to be done in the direction of recasting elementary mathe- matics in the light of recent knowledge ; in particular works in English that treat of the scientific parts of arithmetic show little if any trace of recent discoveries with respect to the number- system. As we felt that it would be unsafe to assume any acquaintance with the various modern views on the nature of ordinal and cardinal numbers, and as it was indispensable for the proper comprehension of the succeeding chapters that the meaning of the term ordinal number should be clearly appre- ciated, we have devoted the first chapter to the discussion of what is meant by an ordinal number. This chapter is not and lays no claim to being a scientifically complete account of the matter ; it will serve its object if it conveys to the reader a distinct image of a number divorced from measurement. In places we have gone afresh over old ground ; this has been done either for the sake of organic unity, or in order to emphasize by means of simple examples ideas which appear later in more difficult and complicated forms. It has in fact been our desire to keep the difficulties of the subject apart from those which are merely difficulties of technique. In carrying out this plan we have consistently chosen the simplest available examples. As regards the theory of functions proper we had to make a choice between the methods of Cauchy and those of Weierstrass. While fully alive to the wonderful beauty and power of Cauchy's theory, we decided eventually in favour of Weierstrass's system. Weierstrass has himself stated with his usual lucidity and force the reasons which have led him to PREFACE. VU prefer his own scheme to that of Cauchy and Riemann for the purposes of a systematic construction of a theory of functions. In a letter to Prof. Schwarz (Weierstrass, Ges. Werke, vol. ii. p. 23s) he says : — " The more I reflect upon the principles of the theory of functions, — and I do so incessantly, — the stronger becomes my conviction that this theory must be built up on the foundation of algebraic truths and that therefore it is not the right way if we proceed conversely and call into play the transcendental (to express myself briefly) in order to establish simple and fundamental algebraic theorems, — however attractive, for example, the considerations may be by which Riemann discovered so many of the most important properties of algebraic functions. It is self-evident that all routes ought to be free to the investigator, while engaged on his researches ; I am thinking only of the systematic establishment of the theory." It has seemed to us that for the purposes of an introductory work it was important to secure the advantages of homogeneity, intrinsic logical consistency, and the gradual passage from the simple to the complex in place of the reverse, which form so marked a feature of Weierstrass's system. With this in mind we have, in the main, followed Weierstrassian lines. It would, however, have been mere pedantry to exclude all geometric considerations from a book intended for the use of beginners. This is the justification, — if any justification should be thought necessary, — of the geometric chapters. The bilinear transformation has been discussed in much detail. This transformation is interesting in itself and can be used effectively to bring out many points of importance for the general theory; furthermore it is playing an increasingly prominent part in recent mathematical work and a complete mastery of its properties is now an indispensable prerequisite for the study of the more advanced portions of our subject. Without entering into minute particulars with respect to the remaining chapters we may indicate in a few words the general principles which have guided us in our choice of materials. We have kept steadily in view the desirability of via PREFACE. making the book elementary, wherever that was possible without any sacrifice of thoroughness ; we have sought to remove erroneous notions which we have found prevalent among beginners ; we have tried to avoid a one-sided develop- ment and to bring home to the mind of the student the vital significance of theorems which may appear wholly abstract by using such concrete illustrations as elliptic functions, definite integrals, the potential, etc.; and finally we have excluded, intentionally, all theorems and results which, however beautiful intrinsically, seem unlikely to be of assistance to the student who proposes to carry his studies further. If we should prove to have met with some measure of success in carrying out this arduous programme, we shall be amply rewarded. We have not given many references, partly because this has been done very fully in our larger book, partly also because we have used our material in a way which differs widely from that employed in other books. Mr Arthur Berry, M.A., Fellow and Assistant Tutor of King's College, Cambridge, has been untiring in his assistance in the revision of the proof-sheets. His great knowledge of the subject and keen critical insight have enabled us to make many improvements both in substance and form, and in numberless ways his advice has been of the greatest value to us. To other friends who have taken an interest in the progress of this work we desire to express our sense of gratitude. Finally we must thank the Macmillan Company and the Ofificers of the Cambridge University Press for the help that they have given to us in the publication of this volume, and for their admirable efficiency in the preparation and printing. J. HARKNESS. F. MORLEY. June II, 1898. CONTENTS. CHAPTER I. THE ORDINAL NUMBER-SYSTEM. ART. I. 2. 3 4. S 6. Ordinal numbers Fractions Irrational numbers . The change of origin The decimal system The infinite decimal Distance, point, and angle PAGE I 2 3 5 6 7 9 CHAPTER n. THE GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS. 8. Introductory remarks ... 9. The axis of real numbers . . ... 10. Imaginary numbers, and the axis of imaginary numbers 11. Strokes 12. Complex numbers and the points of a plane 13. Absolute value and amplitude of jr, = f+?i) . 14. Addition of two complex numbers 15. Ratio and multiplication ... 16. The «th roots of unity ... 17. The «th power and «th root of a stroke . 18. To find the point which divides in a given ratio r from «! to ^2 19. The centroid of a system of points the stroke Examples II 12 •3 IS 16 17 18 20 22 23 24 25 26 X CONTENTS. CHAPTER III. THE BILINEAR TRANSFORMATION. ART. PAGE 20. The one-to-one correspondence 27 21. Inverse points .... 28 22. The bilinear transformation converts circles into circles . . 30 23. Coaxial circles . . . 31 24. Harmonic pairs of points 32 25. The double ratios of four points -34 26. Isogonality 36 27. Theory of absolute inversion 38 28. The bilinear transformation is equivalent to two inversions in space . . ... 42 Examples ... - 44 CHAPTER IV. GEOMETRIC THEORY OF THE LOGARITHM AND THE EXPONENTIAL. 29. Sketch of the theory for positive numbers ... .46 30. The logarithm in general ... .... 48 31. Mapping with the logarithm 50 32. The exponential .... . -52 33. Mercator's projection 53 34. The addition theorem of the exponential . . 54 35. Napierian motion 54 CHAPTER V. THE BILINEAR TRANSFORMATION OF A PLANE INTO ITSELF. 36. The fixed points 37. The motion when the fixed points are distinct 38. Case of coincident fixed points 39. Substitutions of period two .... 40. Reduction of four points to a canonic form 41. Substitutions of period three 57 S8 60 61 64 65 CONTENTS. xi CHAPTER VI. LIMITS AND CONTINUITY. ART. PAGE 42. Concept of a limit 67 43. Distinction between "value when ^=a'' and "limit when $ = a" . 68 44. Upper and lower limits 68 45. Every sequence of constantly increasing real numbers admits a finite or infinite limit ... . . . . 6g 46. Every sequence of real numbers has an upper and a lower limit 70 47. The necessary and sufficient condition that a sequence tends to a finite limit . . . . . . 71 48. Real functions of a real variable . . . 72 49. Continuity of a function of a real variable . . 73 50. A continuous function of a real variable attains its upper and lower limits 74 51. Functions of two independent real variables .77 52. A continuous function /{^, ij) attains its upper and lower limits 79 53. Uniform continuity of a function of one real variable . . 80 54. Uniform continuity of a function of two real variables 81 55. Uniform convergence to a limit ... 82 CHAPTER Vn. THE RATIONAL ALGEBRAIC FUNCTION. 56. Continuity of the rational integral function ... 84 57. Instance and definition of a limit .... 87 58. The derivate of a function ... ... 88 59. The fundamental theorem of algebra . . 90 60. Proof of the fundamental theorem . . 91 61. The rational algebraic function of jr 93 CHAPTER Vni. CONVERGENCE OF INFINITE SERIES. 62. Definition of an infinite series . . 96 63. Convergence . ■ -97 64. Simple tests of convergence for series whose terms are all positive ■ • 99 65. Association of the terms of a series . . . loi 66. Absolutely convergent series .... 104 67. Conditionally convergent series .... .106 68. Conversion of a single series into a double series . 107 69. Conversion of a double series into a single series . .110 Xll CONTENTS. ART. 70. 71- 72. 73- 74- CHAPTER IX. UNIFORM CONVERGENCE OF REAL SERIES The need of a further classification . Uniform convergence Uniform convergence implies continuity Uniform and absolute convergence The real power series . . . . PAGE 113 115 117 118 119 CHAPTER X. POWER SERIES. 75. Notation .... .... 76. The circle of convergence ... 77. Uniform convergence of complex series 78. Cauchy's theorem on the coefficients of a power series 79. P^ does not vanish near x=o . . . . 80. Criteria of identity of power series .... 123 125 128 129 J31 132 CHAPTER XI. OPERATIONS WITH POWER SERIES. 81. Weierstrass's theorem on series of power series 82. Remarks on Weierstrass's theorem .... 83. Applications of Weierstrass's theorem 84. Reversion of a power series 85. Taylor's theorem for power series 86. The derivates of a power series .... 87. Differentiation of a series of power series term by term 134 137 139 142 144 146 147 CHAPTER XII. CONTINUATION OF POWER SERIES. 88. The zeros of Px are isolated points .... 89. Continuation of a function defined by a power series 90. The analytic function .... 91. General remarks on analytic functions 92. Preliminary discussion of singular points 93. Transcendental integral functions 94. Natural boundaries .... 149 151 154 156 157 159 160 CONTENTS. Xlll CHAPTER XIII. ANALYTIC THEORY OF THE EXPONENTIAL AND LOGARITHM. ART. 95. The addition theorem of the exponential . 96. The logarithm Examples .... 97. The meaning of a* . 98. The binomial theorem 99. The circular functions 100. The inverse circular functions loi. Mapping with the circular functions PAGE 162 164 167 168 169 170 172 175 CHAPTER XIV. SINGULAR POINTS OF ANALYTIC FUNCTIONS. 102. Existence of singular points on the circle of convergence . 178 103. Non-essential singular points 181 104. Character of a one-valued function determined by its singu- larities 183 105. Transcendental fractional functions . .... 187 106. Limit-points of zeros 188 107. Deformation of paths ... 189 108. The logarithm of Gx . .... 192 CHAPTER XV. WEIERSTRASS'S FACTOR-THEOREM. 109. Infinite products 194 no. Construction of functions Gx with assigned zeros . .196 111. Weierstrass's primary factors .197 112. The factor-formula for sinTrjr 199 113. Formulae for the other circular functions . ... 204 114. The Gamma function and its reciprocal 206 115. 116. CHAPTER XVI. INTEGRATION. Definitions of an integral Reconciliation of the definitions in the case of the power series . ... 209 211 XIV CONTENTS. ART. 117. 118. 119. 1 20. 121. Case where the end-values belong to different elements Cauchy's theorem Residues ... .... General applications of the theory of residues Special applications to real definite integrals . PAGE 214 218 219 222 223 CHAPTER XVII. LAURENT'S THEOREM AND THE THETA FUNCTIONS. 122. Laurent's theorem 230 123. Isolated singularities of one-valued functions . 232 124. Fourier's series ... . . . . 235 125. The partition-function .... 237 126. The theta functions . 240 CHAPTER XVIII. FUNCTIONS ARISING FROM A NETWORK. 127. The network ... .... 128. A theorem on convergence 129. The functions (ru, f«, j?a 130. Series for ^u, (u, a-u in powers of u 131. Double periodicity .... 132. The zeros of ffyu .... 133. Are fp, f, n- periodic ? 243 245 246 249 250 251 252 CHAPTER XIX. ELLIPTIC FUNCTIONS. 134. Case of Liouville's theorem 135. Integration round a parallelogram 136. Comparison of elliptic functions .... 137. Algebraic equation connecting the functions g>a, ff'u 138. The addition theorem for ^» 139. Expression of an elliptic function by means of ^u 140. The addition theorem for fw 141. Integration of an elliptic function .... 142. Expression of an elliptic function by means of a. Here a, a' mean of course integers, positive or negative. Objects considered as a succession from left to right are in positive order ; when considered from right to left in negative order. 2. Fractions. We now attend only to zero, — the object from which (not at which) we begin, — and to the objects on the right of it. We can re-label the alternate objects 2, 4, 6 marking them i, 2, 3,...; see fig. i, where the old names are below the objects, the new above. We must then invent marks for the objects formerly marked i, 3, 5, .... They can be marked ^, I, |,...,or 1/2, 3/2, 5/2,.... }^ 1 ?^ 2 % 3 X X X X X X X 12 3 4 5 6 Fig. I. Conversely we are at liberty to interpolate alternate objects in a given row o, i, 2, 3, ... , only then we mark them 1/2, 3/2, 5/2, and so on. In the same way we can interpolate two objects between every consecutive two of the given row o, i, 2, 3, ... , marking the new objects in order as 1/3, 2/3 ; 4/3, 5/3 ; and so on. In this way we account for the symbols i//, 2//, . . . , where / is any positive integer ; let us call these symbols positive / fractional numbers. 'By positive rational numbers we understand both positive integral and positive fractional numbers. If we interpolate single objects in the row o, 1/2, I, 3/2,..., 1-3] ORDINAL NUMBER-SYSTEM. 3 we have the same sequence of objects as if we interpolate objects by threes in the row o, I, 2,...; and our objects are therefore marked o, 1/4, 2/4, 3/4, I Hence 1/2 and 2/4 are two marks for the same object. When we find two marks attached to the same object we say that they are equal ; thus we write 1/2 = 2^ Arow marked o, 1/6,^/^ 1/3, 1/2, 2/3^^1,... is to be understood as arising from the interpolation of objects by fives ; that is, by introducing the objects 1/6, 2/6, 3/6, 4/6, 5/6,..., or 1/6, 1/3, 1/2, 2/3, 5/6, As 2/6 comes before 3/6 we say that 1/3 < 1/2. We may interpolate as many objects as we like in the natural row, and by the principle of the least common denomi- nator we can interpolate so as to explain any assigned positive fractional marks/,, /j,...,_/5>. Also, given any positive rational mark r other than zero, we can interpolate rational marks between o and r. When no object can be made to fall between an assigned object and o, that assigned object must itself be o. The same observations apply to the negative numbers. We can think of an infinity of objects as interpolated in the natural row so that each shall bear a distinct rational number and so that we can assert which of any two comes first. It is to be noticed that as we approach any of the natural objects there is no last fractional mark ; that is, whatever object we take there are always others between it and the natural object. 3. Irrational Numbers. Progress in other directions leads us to the notion of square numbers ; in considering these 1 % 2 3 X X XXX XX X X X X 1 Z %Z 4 5 Fig. i. 6 7 8 9 l! from the ordinal point of view we re-name our natural row as in fig. 2, where the old names are below, the new above ; and we 4 ORDINAL NUMBER-SYSTEM. [CH. I have to consider how then to bring the omitted objects into the scheme of ordinal numbers. Every object whose new name is fractional had a fractional name, so that the object whose old name was 2 cannot now have a rational name at all. We give it a name which we call irrational ; we call it the positive or chief square root of 2 and mark it '/2 or 2*. As an ordinal number it is perfectly satisfactory, for we know where it comes, whether left or right of any proposed rational number, simply by means of the old naming. Hence it separates all the rational numbers into two classes, those on its left and those on its right. A rational number separates all other rational numbers into two classes ; we put it in one of these classes, and say that it closes that class. Any process which serves to separate rational numbers into two classes, — those on the left, and those on the right, such that the left-hand class is not closed on the right and the right-hand class is not closed on the left, — leads to the introduction of a new object named by an irrational number. If a class, — for example the left-hand class, — is closed by an object, we require no new object. Two numbers rational or irrational, — to fix ideas we will take them both irrational and equal to s and s' , — are said to be equal if the rational objects to the left of s are the same as those to the left of j', and the rational objects to the right of s are the same as those to the right of /. For example, 4*, 2* effect the same separation of the rational numbers. An equivalent con- dition for the equality of s, s' is that every rational number to the left of s shall be to the left of s', and every rational number to the left of / shall be to the left of s. Between two unequal irrational objects s and / there must lie rational objects ; for since s and / are not equal there must be a rational number which is before one of the two and not before the other. It is very important to notice that we have now a closed number-system. When we seek to separate the irrational objects cis lying left or right of an object, either the object is rational, or if not it separates rational objects and is irrational ; in any case //, 3-4] ORDINAL NUMBER-SYSTEM. 5 then it must have for its mark a rational or irrational number, and there is no loophole left for the introduction of new real numbers which separate existing numbers. This is often expressed briefly by saying that the whole system of positive and negative integral, fractional, and irrational numbers is . continuous, or is a continuum. 4. The change of Origin. Let us call the objects associated with the continuous number-system the complete row of objects. This complete row is built into the natural row in the manner described above, the only postulate with regard to the natural row being that of any pair of objects we can say that one lies to the left of the other. In the complete row there are objects integral as to any object x in precisely the same way as the integral objects with which we began are integral as to one of them which was taken as zero. When X is itself an integer this is clear, and has already been used ; it is only saying that we can begin our naming from any one of the natural row. But it follows that every object x^ between i and 2 stands to some object x^, or say x^, between o and i in this relation of congruence ; that is, if we begin our naming from i as origin, then the object x^ will now have to be called x^, or, with the notation of § i, x^-\=x;. x[ x'^ X XX XX XX «o ' •*! 2 *» 8 Fig. 3- We can utilize this periodic property to name all objects by integers and proper numbers, where hy proper numbers we mean names of objects between o and i, o included and i excluded, the last suppositions being made to avoid periphrasis. This is expressed by writing, in the case above, x^= I +jr/. So if ;r, lie between 2 and 3 (3 excluded) we have / 6 ORDINAL NUMBER-SYSTEM. [CH. I where x^ is a proper number. This we can express by and so on. In this way the idea of ordinal addition is established in general without reference to quantity. 5. The Decimal System. We can name our complete system methodically as follows. To avoid too many marks we select the 9 digits, i, 2,..., 9, as marks for objects after zero; and name the following by two marks 10, 11, 12, and so on, up to 19; those following by 20, 21, and so on. In this way all integers are named. We first interpolate 9 objects between every two consecutive integers; those between o and i we name 'i, '2,..., g; the objects between i and 2 that are congruent to these proper 'fractions are named i"i, 12,..., 19, and so on. The system so gained, together with the integers, is the first decimal system. Between every two consecutive objects of this system we in- terpolate 9 new objects, and we name those between o and "i "Oi, 02,..., 09; those between 'i and -2 as "ii, "12,..., -19, and so on. We thus obtain a system which with the first decimal system is the second decimal system. We can proceed as far as we please in this way. And we can freely change the origin to any object so gained, by the addition or subtraction of decimals, performed of course in the usual way, but interpreted solely with reference to before and after, not to how much. The/th decimal system is, whatever / may be, only a part of the rational objects, and includes no irrational object But by ghoosing / large enough we can separate any two rational objects by this /th system. The proof is as follows : — (i) Let the two rational objects be o, r where r is positive. We can take p sufficiently large to make the first proper number other than zero in the pth decimal system lie between o and ;-; since if r have the name />'//>" it coincides with or comes after i//", and therefore we have only to interpolate more than />" objects between o and i to get an object nearer to o than either of i//', r is. 4-6] ORDINAL NUMBER-SYSTEM. 7 (2) And secondly, if the two rational objects be r, / and r, r' have a common denominator /, we have only to take more than / objects to separate r and r. Thus our decimal system allows us to separate any two rational objects by taking p large enough. We can, for example, separate o and any positive rational object by the object i/i--., the object defined here is i —2/3 or 1/3. Thus 1/3 can be defined either by a sequence in positive order or by a sequence in negative order. Generally let d be the limit of an infinite decimal 1 We exclude, in order to avoid two decimal names for one object, the case where every digit after an assigned one is 9. I 6-7] ORDINAL NUMBER-SYSTEM. 9 where d^is o. a^ a,. . .a^ ; then d is an object between o and i (o included) ; and «+ i/io), K+ i/io'),..., (4,4- i/icV), ... is a sequence in negative order, which is not in the form of a decimal sequence ; but i-K+i/io), i-(ar,-i-i/io'),..., i-(4,+ i/io^),... is a decimal sequence and leads to an object ; hence it is clear that the sequence 4+1/10, 4+1/10', ..., dp+ i/icV,... does also name an object. To this object we can assign an infinite decimal d'. It is important to prove that d' is none other than d itself. If not, it is clear that d' is to the right of d ; that is, d< d'. We have then for every/ the following ordering d'p-- define one and the same object, named 2*— i. It is clear that the same reasoning applies to other than proper numbers. 7. Distance, Point, and Angle. The distance from one point to another can be measured in terms of a unit length by seeing first how many units there are, then how many tenths in the part left over, and so on. We thus obtain a decimal, terminating lO ORDINAL NUMBER-SYSTEM. [CH. I or not, which defines a number. By the distance we understand this number. Conversely we postulate that every positive decimal d, infinite or not, that is every distance, can be represented by a terminated line od, or simply by the point d itself, d being on the right of o. Let d' be any other point on the right of o ; if d' is on the right of d we say the distance d' is greater than the distance d, and d less than d', and write d'>d, d A similar postulate is required for angles. We have a natural measure of amount of turn, namely, the circumference of a unit circle. This unit can be divided geometrically into 2^ equal parts. Out of sequences formed from these we have to define the number i/27r, and then we postulate the existence of a point whose distance along the arc is this number. With the aid of infinite series the problem can be simplified. What has been said of the theory of ordinal numbers is necessary for the subsequent analytic theory, and is, we hope, sufficient to render the argument intelligible. As our subject is functions of complex numbers, we shall tacitly assume further results concerning real numbers, as, for example, that we can effect the elementary operations with irrational numbers ; and that nothing new results. For a detailed account of real numbers we may refer to Fine's Number-system of Algebra. CHAPTER II. THE GEOMETBIC REPEESENTATION OF COMPLEX NUMBERS. 8. Introductory remarks. We denote by i one of the roots of the equation :ir' + I = o. The expression |^+«i?, where f and rj are any real numbers, is a complex number. When f = o the number is called imaginary. Thus i itself is a number ; we call it the imaginary unit. We assume that the elementary operations of addition, subtraction, multiplication and division have been defined and justified for complex numbers. What is said in elementary algebra on complex numbers covers our assumption, when we admit, — as we have done, — that we can efiiect these same opera- tions on irrational numbers. We must notice that the totality of numbers f + «i/ is a closed system, with respect to the elementary operations ; that is, the result of an operation is again a number, real or complex. The object of this book is to discuss the properties of such expressions as grow naturally from the repetition of elementary operations, when applied to complex numbers. An adequate geometric representation of a complex number is almost indispensable, and we shall consider in this chapter the repre- sentation by means of directed distances, with some immediate consequences. But in using geometric intuitions, and thus beginning what is called the geometric theory of functions of a complex variable, we must emphasize one lesson of experience : that the intuitional 12 REPRESENTATION OF COMPLEX NUMBERS. [CH. II method is not in itself sufficient for the superstructure. It has been found that only by the notion of number, somewhat as sketched in the first chapter, can fundamental problems be solved. If however we are prepared to replace, when occasion arises, these geometric intuitions, — which are point, distance, and angle, — by numbers, as in § 7, then, and only then, is the use of geometry thoroughly available. This implies no reflexion on the customary applications of geometry to algebra, or conversely, which are made in analytic geometry or elementary calculus. The point is that elementary algebra, as commonly presented, is imperfect and is itself to some extent intuitive. It is at this stage that it becomes necessary to insist on some refinements in our notions of algebra, and we may reasonably attempt a parallel precision in our notions of geometry. The distance and the number must always correspond, and in so far as we can know and say what we mean by the one we must be prepared to know and say what we mean by the other. 9. The Axis of Real Numbers. As already said in § 7, we represent real numbers by distances or strokes measured from an origin along a straight line, which we call the axis of real numbers, a stroke to represent i, or unit stroke, being arbitrarily selected. The unit stroke will always be taken to the right, or eastward. Then all positive strokes are eastward, all negative ones westward. The point which ends the stroke bears the same name or real number as the stroke itself This way of naming points, being already used in analytic geometry, — singly for the line, doubly for the plane, and triply for space, — need not be enlarged upon. We need only remark that in the addition of real strokes, the second is to begin where the first ends, and so on. We next consider a possible interpretation of the sign — . The numbers «, —n are represented by strokes of equal lengths, but opposite directions. Now if the stroke n be re- garded as a material arrow with a fixed end at o, the direction of the arrow cannot be reversed without moving it out of the line. 8-I0] REPRESENTATION OF COMPLEX NUMBERS. 13 The simplest method for securing this reversal of direction is to rotate the arrow, in a plane that passes through it, either positively or negatively, through two right angles. This affords the following interpretation for the sign — . The sign — attached to a positive stroke can be treated as an operator which rotates the stroke about its fixed extremity O through two right angles ; that is, the sign -gives a half-turn to tlu stroke. From this point of view the statement (-)x(-) is + means that two half-turns bring the arrow back to its primitive position. Such an equation as (-2)x3=-6 is capable of the following two interpretations : — (a) the product of the two numbers — 2, 3 is the number —6 ; {b) the result of doubling the stroke 3 and reversing its direction is the negative stroke — 6. In (a) - 2 is a number, but in {b) it is an operator which means ' double and reverse.' 10. Imaginary Numbers, and the Axis of Imaginary Numbers. The real number x is equal to jr x i. Hence x can be regarded as an operator. It is then a product of two independent operators, the one changing the size of the unit stroke and equivalent to a stretch, the other changing the sign and equivalent to a turn ; when x is positive only the former comes into play. While the amount of stretching may be made what we please by a suitable choice of x, the amount of turning is restricted to an odd or even number of half-turns. Is it possible to remove this restriction and allow the unit stroke to be turned as freely as it can be stretched, by allowing x to take imaginary or complex values ? We shall show in § 1 5 that this question can be answered unreservedly in the affirmative. Our immediate task is to find an interpretation for purely imaginary numbers as operators. H REPRESENTATION OF COMPLEX NUMBERS. [CH. II Let the plane of the paper be the plane in which the rotations take place, and let the axis of real numbers go from west to east, as in fig. 4- Let the order east, north, west, south denote the positive order of turning. The operator which turns the unit stroke positively through a right angle we call i, without assuming any connexion between ? and V — I. Hence i.i, or briefly i, is a directed line of unit length which begins at o and ends at the point marked i in fig. 4 ; such a line is called a stroke equally with lines along OE or OW. The operator i, when applied to the stroke i, turns that stroke positively through a right angle and produces the stroke from o to — I. Thus i.i. I =(— i) X I, giving i' = —l,i.l= V — i ; so that i='^ —i. This is the geometric representation of i (§ 8) ; the imaginary unit is represented, as we have seen, by a stroke of unit length along an axis perpendicular to the axis of real numbers and called the axis of imaginary numbers. All the imaginary numbers are built up from the imaginary unit i in the same way as the real numbers are built up from i. The product and ratio of two imaginary numbers are real, just as the product and ratio of two negative numbers are positive. The numbers mi, where m is real, can be regarded as naming either the points at the ends, other than o, of the corresponding strokes ; or the strokes themselves. Further, such strokes as 31, lO-Il] REPRESENTATION OF COMPLEX NUMBERS. 1$ — 3?, can be derived from the unit stroke by a stretch 3 followed by positive turns through one, three right angles respectively. 11. Strokes, Let us now understand generally by a stroke a straight line of definite length and definite direction which lies in an assigned plane. This definition includes as particular cases the directed length that has been used in the preceding paragraphs. Fig- 5- Two strokes ab, cd (fig. 5) are said to be equal when they are of equal lengths and are drawn along parallel lines in the same sense. Observe that ab is not equal to dc, for dc = — cd. Fig. 6. Addition. The sum of two strokes ab, a'b' is defined in the following manner. Draw the stroke be equal to a!b' ; then the sum of a^, «^' is the same as the sum of ab, be, and this latter sum is defined to be ac. Thus the equation ab + bc= ac holds even when ab, be are not in one and the same straight line. l6 REPRESENTATION OF COMPLEX NUMBERS. [CH. II More generally the equation holds not only when ab, be, etc., are strokes along the axis of real numbers, but also when the directions of the component strokes are any whatsoever. Subtraction. Such a difference as ab — a'b' is defined as the sum of the strokes ab, b'a' and is equal to ac' (fig. 6) where be' = — be. It is convenient in comparing strokes to draw them from a common origin o. Then each stroke op is determined by its terminal point/, and there is a i, i correspondence between the strokes op and the points /. The strokes may therefore be characterized by their terminal points /. We shall show in the next paragraph that we can establish a i, i correspondence between the points / of the plane and the numbers in the complete number-system. The following are convenient constructions for op + oq and op — oq. Complete the parallelogram oprq whose sides are op, oq, and produce rp to r' so that r'p, pr are equal in length. Then op +oq = op+pr= or, op — oq = op + qo = op ■'r pr = or = qp. Note. The rule for the addition of strokes applies to velocities, momenta, accelerations, etc. 12. Complex Numbers and the Points of a Plane. Let ^, 7) be real numbers ; the complex number x = ]q ■\- ii) \.s the II-I3] REPRESENTATION OF COMPLEX NUMBERS. i; sum of f and tij, which can be represented by strokes at right angles of length f, 17 as in fig. 8. The sum of these two strokes n-nini ^-ajn's Fig. 8. is ox. Hence the number x can be associated with the stroke ox, or the point x. That is, the point (or the stroke from the origin to the point) represents the number, and the number names the point (or stroke). Here then we have a starting- point for the geometric treatment of complex numbers. The points on the straight lines which are labelled f-axis and 17-axis (fig. 8) represent graphically all real and all imaginary numbers ; and complex numbers f + it) are visualized by points x with rectangular coordinates (^, 17). This method of representing complex numbers was explained in a work by Argand (in 1806). Gauss had discovered it probably as early as 1799, but published nothing on the subject till 1831. A Dane, Wessel, appears to have first published the method in 1797. 13. Absolute Value and Amplitude of x= ^ + it}. Let the polar coordinates of (f, ri) he p, 6; the positive length p is called the absolute value of x and is denoted by \x\. The angle 0, and equally so the angles 6 + 2mr where n is any positive or negative integer, determine the direction of the stroke x. Any one of these angles may be called the amplitude of the stroke. Frequently it is convenient to select a definite one of the set, 6^ ; we shall do so by supposing — IT < dn & IT ; we shall call this the chief amplitude, and denote it when convenient by Kxa.x, while denoting any amplitude by amx M. H. 2 1 8 REPRESENTATION OF COMPLEX NUMBERS. [CH. II The many-valuedness of an amplitude deserves careful attention in this subject, as the explanation of many-valuedness in general depends on it. The following relations connect f , rj, p, 6 : — ^ = pcosd, 7} = p sin 6, p = + 'J^'+n', tan 6 = 7?/f ; hence x= ^ + ii} = p (cos 6 + t sin 6), or, as we may write it, x= ^ + iT] = pcis 6. Later on, when we have defined the symbol e', where x is complex, we shall find that I + z"/ = pe'^- Interpretation of the symbol p cis d as an operator. Each factor of the expression p cis 6 conveys important information. The stroke x can be derived from the stroke i by a stretch which changes the length i to the length p and by a turn of the resulting stroke through an angle 6; hence the following possible interpretation for the equation p cis ^ . I = p cis ^ : — TAe expression p cis 6 can be regarded as an operator which stretcltes the stroke i into a stroke p and then turns this stroke p throicgh an angle 6. This interpretation includes the earlier interpretation of imaginary and negative numbers, as is seen at once by putting 6 = 7r/2, tt, 3Tr/2. 14. Addition of Two Complex Numbers. In algebra the sum of the two complex numbers ^i + ziji, fa + 2% is ?i + f 2 + « (% + %)• Thus addition connects the points (fi, tji), (?2, %) with the point (fi + f^, 171 + ^X But there is no difficulty in proving geometrically that this third point is a corner of the £,+ la.'7,+^2 Fig. 9. 13-14] REPRESENTATION OF COMPLEX NUMBERS. 19 parallelogram constructed on the lines from o to (fi, %) and (f j, rj^); hence the result obtained by algebraic addition of two numbers agrees with the result obtained by the addition of the corresponding strokes according to the parallelogram law. It is evident from fig. 9 that the absolute value of the sum of two complex numbers Xi, x^ is, in general, less than the sum of the absolute values of these numbers. For \x-^\, \x^ are the lengths of the sides of the parallelogram and the sum of these two sides is greater than the diagonal whose length is \x^-^x^\. In the special case where x^, x^ have the same amplitude we have \Xy^r X^\ = \x-\ + \x^\. More generally we may say that the absolute value of the sum of several complex numbers cannot be greater than the sum of the absolute values of the component numbers. This is merely the analytic statement of the geometric theorem that the length of one side of a closed polygon is less than the sum of the lengths of the other sides, the words less than being replaced by equal to when all the other sides have the same amplitude. Subtraction. Since the subtraction of strokes is merely the addition of other strokes and the subtraction of complex numbers is the addition of other complex numbers, the sub- traction of two numbers is interpreted geometrically by the rule given for the subtraction of two strokes. It is well to observe that the stroke from the head of x^ to Ji Fig. 10. the head of x^ is the stroke x^ — Xy, (fig. 10). It is often spoken of as the change or increm.ent of a variable stroke x as it passes from the value x-^ to the value x^. The notation x^—x^ replaces the temporary notation XtPc^. 20 REPRESENTATION OF COMPLEX NUMBERS. [CH. II 15. Ratio and Multiplication. Let a, b, be two com- plex numbers ; then bja in the equation {bja) y. a = b can be interpreted either as a number, or as an operator. From the latter point of view it changes the stroke a into the stroke b (i) by stretching the stroke of length |«| into one of length \b\, (2) by turning the resulting stroke until its amplrtude becomes equal to that of b. Hence the operator bla stretches by an amount M and turns by an amount am^ — am«. Just as p cis Q has a dual signification ( i ) as an operator, (2) as a stroke, so bla can be interpreted not merely as an operator but also as a stroke of length pJ and direction \a\ am3 — am«. This stroke b\a is generated from the stroke i by the operator b\a. -. We have then a geometric interpretation for bja as a stroke ; and we see that the absolute value of bja is the ratio of the absolute values of b and a, and the amplitude ofija is the change of amplitude in passing from a to b. ^, . . \b _ . ~W But we cannot always say, for chief amplitudes, that 3^ f Am ( - ) = Am^ — Ama ; for example, if Am^ = tt, Ama = — tt/s, then Am {^\ = - Tr/a. ^yit^A '"^ ^^-'^-^ "/- ^ 1- * It is easy to construct geometrically the quotient bla. In fig. II, the two triangles o, a, b, and o, i, bla, are directly similar; for the angles at o are equal to am^ — ama and the sides about these angles are proportional. :■ 'y " ■' . ' V Ex. I. Calculate the absolute value and amplitude of \-——. i 1'^^ • 1 = f-rl and am f-J =am^ — ama. Ex. 2. Prove geotnetricaHy that - — F / \P+ta = 1. ~ ■■ '. ' - ' ■ 'U is] representation of complex numbers. 21 We observe that if bja = c, then b = ac; hence fig. 1 1 provides us with a geometric construction for the product of a and c. h.=ac The figure shows that \ac\ = \a\ \c\ and am (ac) = ama + amr. Thus the symbol a in ac can be interpreted as an operator that stretches the length \c\ of c in the ratio |a| : i, and increases the amplitude of c by the angle am a. In words : — The absolute valtie of a product ac is the product of the absolute values of a and c, while the amplitude of ac is tlie sum. of the amplitudes of a and c. The same rules follow also from the direct division and multiplication of pi cis 6^ and pi cis O^. For example, p, cis 6^ X pj cis 6^ = p,p, (cos ^^ + i sin 6^ (cos 6^ + i sin 6^ = p^p^ {(cos 6^ cos 6.^ — sin 0, sin 6^ + i (sin Q^ cos ^^ + cos B^ sin ^j)} = p,p, {cos (^. + ej + i sin (^. + e^] = /3,p,cis(0. + ^,). We have assumed the addition-theorems of the sine and cosine; but it is worthy of note that the matter can be so presented as to prove these fundamental theorems. For since cis ^ is a turn through 6, the meaning of cis ^, cis 6^ is a turn first through B^ and then through B^. It is therefore a turn through B^ + B^, or is cis (6^ + B^. Hence by equating real and imaginary parts we have the addition-theorems. The theorem cis B^ cis ^,... cis &„ = cis (^, + ^, + ... + B„), which is known in connexion with Demoivre's theorem, states 22 REPRESENTATION OF COMPLEX NUMBERS. [CH. II that « separate turns 6„, 6„^i,..., 6^, 6^ amount on the whole to a single turn 6^ + 6^+ ...+6„. Ex. If 2co%d = a+ija, prove that 2 cos «5 = a" +!/«", where n is any integer. 16. The «th Roots of Unity. An important special case of the theorem cis e^ c\s6^ cis ^„ = cis {6^ + 6^+ ... + d„) is that which arises when d^, 6^,..., 6„ are all put equal to 2miTln, where m is an integer. The theorem becomes (cis 2W7r/«)" = cis 2tmr= I. Hence cis 2mirln is an «th root of unity. By making m = o, I, 2,..., «— I, we obtain n distinct roots, and no additional distinct roots are obtainable by giving m other integral values. Geometrically cis zmv/ft is a turn through m/n of four right angles ; and the continuous repetition of the single turn 27r/« will give all the «th roots of unity. The n points which represent the roots form the vertices of a regular polygon. The figure shows this for n = 6. Every equation ^^"—1=0, where m = 3, 4, S,..., has certain roots, called primitive «th roots, which satisfy no equation of similar form and lower degree. For example, .*:"= i is satisfied by two primitive roots of this kind, namely, cis 27r/6 and cis — 27r/6 ; the other roots satisfy one or more of the equations ;jr— 1=0, J^— 1=0, or x'—\=o, and are not primitive. The Fig. II. 15-17] REPRESENTATION OF COMPLEX NUMBERS. 23 primitive nth roots of unity are, then, those turns which, with their repetitions, give all the nth roots of unity. The case « = 3 deserves notice. The cube roots of i are I , cos 27r/3 + i sin 27r/3 ; that is, - I + i^3 - I - ^"'^3 ' 2 • 2 ■ For the second root we shall use on occasion the Greek letter v ; thus V denotes a turn through 27r/3, or the complex number 2 Ex. I. Verify, by a diagram, that the cube roots of unity are i, v, v\ and that their sum is zero. Ex. 2. Draw the strokes which represent the square roots of i and write down these roots in the form | + iij. Ex. 3. Find the fifth roots of unity. Prove that the sum of the «th powers of these fifth roots is 5 or o, according as the positive integer n is or is not a multiple of 5. Ex. 4. Given that c=cis 27r/s, prove that (e^-t^) (f'-e)=5'/=. 17. The nth Power and nth Root of a Stroke. The stroke a" is constructed by the rule for a product. We make the triangles o, i, a; o, a, a' ; etc. all similar ; then the last point of the «th triangle belongs to the stroke a" If we continue the series of triangles in the reverse direction, we get the negative powers ija or a~^, i/a' or ar^, etc. 24 REPRESENTATION OF COMPLEX NUMBERS. [CH. II If \a\= I, the points all lie on a circle of radius i. The curve on which the points lie when \a\ is not i is an equiangular spiral. See § 29. Let d be an nth root of a, so that d" = a. Then the length of 6 is the positive nth. root of the length of a, and the amplitude of l> is an «th part of an amplitude of a. If 6 be one amplitude of a, all the amplitudes of a are given by 6 + 2mtr. These amplitudes will have n different «th parts, namely, B\n, Bjn + l-n-jn, 6/n + 4Tr/«, ..., 9/n + 2{n—i) ir/n ; all the rest are congruent to these n with respect to 27r. Thus we have as in the last paragraph n distinct directions for l>, and therefore « «th roots of a. Clearly the n corresponding points are the comers of a regular polygon ; also it is evident that the remaining « — i «th roots can be derived from an assigned «th root by multiplying it by the «th roots of unity. We shall define the cAie/ «th rooi of a as the one whose amplitude is 0/n, where — tt < S tt. This root will be denoted by a''", any root by Va. 18. To find the point ^hich diTldes in a g:iven ratio r the stroke ttom a^ to a^. Let X be the point. What is meant is this : when a point moves from a, to a^ by way of x the stroke a^ — a^ is resolved into two strokes x-a^, a^—x. And we may say that x divides the stroke from a, to a, (or the points a,, a, in the order named),- in the ratio (x — a^)j{a,—x). We have then x-a^ = r(a^-x), a. + ra, or x= J ^ . I +r To understand clearly what is implied by this formula when r is not real, consider this question : opposite corners of a square are ^j and a^, what names are to be assigned to the other two comers .' Here we are to express x and y in terms of a, and a, (fig. 14). 17-19] REPRESENTATION OF COMPLEX NUMBERS. 25 Now a^—xis obtained by turning x—a^ through a positive right angle. Hence Fig. 14. Similarly a, -y — -i(^y- ^ J.- Here r is i for the point x and — i for j/. Ex. I. Given two opposite points of a regular hexad (namely, the comers «!, flj of a regular hexagon) express the other four points in terms of a-^ and a^. Ex. 2. Mark the points which divide o, i in the ratios ±.{i+i). 19. The Centroid of a System of Points. If we let r= I, then we have for the middle point of the join of a,, a,, or briefly the middle point of a^, a^, the expression — -. In general we define the centroid of m points a^,a^,. ., a„ as the point g, + g^+ ... +a,„ m ' or 2 flx/»«; so that in particular the centroid of «,, a^ is the X=l middle point of the stroke from a^ to a^. Let a, b be the centroids of the sets a,, a^,..., a„ and b^, b^,...,b„; the centroid of the m + n points is S^x + 2^A _ »2i2 + nb m-\-n ' m + n ' and divides therefore the centroids of the two systems of m and « points in the ratio n/m. 26 REPRESENTATION OF COMPLEX NUMBERS. [CH. II EXAMPLES. 1. The two triads a,b,c\ x,y, z ; form similar triangles if I I I =o. a i c X y z 2. If the points x, y, z divide the strokes c — b, a — c, 6-a in the same ratio r, and the triangles x, y, z and a, 6, c are similar, either r = i or both triangles are equilateral. 3. Let a, b, c, d form a parallelogram of which a, d are opposite points. Let a,biX; c, d, y ; a, d, z be similar. Prove that x, y, z is similar to each of them. 4. If a, x,y ; y, b, x; x,y, c are similar, each is similar to a, b, c. 5. If i^+^2= I, prove that x, y are ends of conjugate radii of an ellipse whose foci are ± i. 6. Equilateral triangles are described on the sides of a given triangle, all outwards or all inwards. Prove that their centroids form an equilateral triangle. 7. Prove that X+^u + io^, where X, ^, n are integers whose sum is o .r ± I, represents the points of a quilt formed by regular hexagons. CHAPTER III. :. Jr , . .. THE BILmEAB TBANSFOBMATION. 20. The One-to-one Correspondence. An equation in X and y will be used to establish a correspondence between these variables. It is often convenient to represent the points x and y in different f lanes ; in each of these planes the origin and the point I are selected arbitrarily, except that the unit of length is supposed usually to be the same for both planes. The equation between x and y establishes, then, a correspondence between the points of the two planes, and either plane is said to be mapped on the other. One important way of forming a mental image of the mapping is to draw a series of paths in the one plane (for example* straight lines parallePto an axis, or con- centric circles) and to determine the corresponding paths in the other plane; a jc-path which arises in this way^from an jr-path , is said to be tke map of tke x-path. When the equation is y = {cix-^b)\{cx->rd) (i), the correspondence of the two planes is i, i ; that is, one and only one J/ corresponds to every x, and conversely. There is one possible exception in the finite part of the ;r-plane, namely, the point x= — djc; to remove this exception we treat j/ =00 as a single point and say that there is in the _;'-plane one and only one point at 00 . With this convention ;r = 00 represents a point ; to x=ao corresponds one and only one point in the _y-plane, namely, ajc. And now the correspondence is i, i whatever value, 28 BILINEAR TRANSFORMATION. [CH. Ill finite or infinite, be given to x (or y). The equation (i) is the most general algebraic equation giving a i, i correspondence between the planes of x and y ; hence the importance of the transformation (i), or the bilinear transformation as it is called. It must be noticed that oo is a point only by virtue of a very convenient agreement. It is an artificial point. So we can regard the artificial value or number co as attached to the point 00 . 21. Inverse Points. The bilinear transformation depends only on the ratios of the given constants a, b, c, d; and is therefore given when three pairs of values of x and y are given. Let x^, J/, and x^, y^ be corresponding values ; we have ax+b ax^ + b _(ad—6c){x — x^) ^~-^^~ cx + d ~ cx^ + d ~ {cx + d){cx^ + d) ' (ad — be) (x — x^) similarly y — jc, = (cx+d)(cx^ + d)' Therefore ^HZ- =-:?.±^.^i:£. = ^£z£. (2), y-y^ cx^ + d x-x^ x-x^ where k depends on x^, x^, but not on x. Suppose that x varies subject to one or other of the conditions x—x^ '■ — xj = constant, am ( ^ ) = constant, and that x^, x^ are arbitrary but fixed points of the jr-plane ; what curves are traced out by x} This question must be answered before the full significance of equation (2) can be appreciated. I. Let the amplitude of (x — x^l(x—x^ be constant and Fig- 15- 20-2I] BILINEAR TRANSFORMATION. 29 equal to a. Then the angle x^xx^ is a, so that x can take all positions on a certain circular arc whose ends are x^,x^. Observe that for the complementary arc which makes up the complete circle, the amplitude of our ratio, though still constant, is a — Tr. Hence we say : — When the amplitude of (x — x^\{x — x^ is given, x moves on an arc of a circle. 11. Next let the absolute value of the ratio be constant and equal to p, say. It can be proved that x moves on a circle, and that the circle is in this case complete. This proposition was not given by Euclid, and its fundamental character is insufficiently recognized by many of the modern text-books on elementary geometry. One proof is as follows : — Choose the point a (fig. 16) on the join of x^ and x^ so as to Fig. 16. make the angle x^a equal to the angle ax^x. Then the triangles xjca and xx^a are equiangular ; and therefore the sides of the one are proportional to the sides of the other, giving x-x^ x—x^ - a—x = a-x^ a—x^ a—x The first of these numbers is p ; therefore a — X X \a-x^ a -^. \a — x a- ■^. = P a- ■^, = p that is, Hence a, — which was determined uniquely by the construction 30 BILINEAR TRANSFORMATION. [CH. Ill when X was given, — is fixed for all positions of x for which p is constant. Also \x — a\ is constant, since \x - af =\x^- a\\x^- a\; \ therefore x is any point of a circle whose centre is a and radius |V(«-^,)(«-^j|. Thus the locus of x when it moves subject to the condition ! ^ ^ ! = a constant, ,x-x^ is a complete circle. The converse theorem is true. For if we take two points x^, x^, which lie on a ray from the centre « of a circle of radius p and which satisfy the equation \a-x^\\a-x^\=p\ X X then if x be any point on this circle, = a constant. When the constant is i the circle of centre a desfenerates &"■ into a straight line and the points x^,x^ are reflexions of each other in this line. In the general case the points x^, x^ are said to be inverse as to the circle of centre a and radius p ; and the circle will be said to be drawn about any such pair of points. Any circle through two points is orthogonal to (i.e. cuts at right angles) any circle about those points. For let x^, x^ be the two points and let us draw (as in fig. i6) the circle x^^x, then the join of a and jt is a tangent to this circle at x by virtue of the equation \x-a\'=\x^-a\\x^-a\. It follows that the intersections of two circles which are orthogonal to a given circle are inverse points of that circle. This provides us with another definition for inverse points: — Two points x^, x^ are said to be inverse to a circle, when every circle through x^, x, cuts the given circle orthogonally. 22. The Bilinear Transformation couTerts Circlea into Circles. Returning to equation (2), namely y-y^ x-x' 21-23] BILINEAR TRANSFORMATION. 3 1 = \k\ \y — y we see that \- — =^ This equation shows that when x describes a circle about x^,x^, or, what comes to the same thing, when x moves so that ^ ^ ^ = a constant p, x-x^ ^ then y moves so that I y — y I - — — = /& p = a constant : \y-y, ' "^ that is, y describes a circle about y^, y^- But x^,x^ are arbitrary; therefore circles in the x-plane map into circles in the y-plane, and inverse points as to a circle map into inverse points as to tJte corresponding circle. But if p = I the ;ir-path is a straight line and not a circle, and if \k\p is I the ^^/-path is a straight line. We cannot say, with Euclid's definition of a circle, that a straight line is a circle, but we can say that it is the limit of a circle. It is however usual to say unreservedly that a circle maps into a circle when the transformation is bilinear ; the limitation that either circle may be a straight line being implied. ' " *' " / •' Ex. I. Prove that the map of a circle through x^, x^ is a circle through yi,yi, by equating amplitudes in equation (2). Ex. 2. Let y = ^ ; and let x take the values o, ±i, ±21. Mark the corresponding points,)' ; and determine the centre and radius of the ^'-circle. 23. Coaxial Circles. All circles through two points jr^, ;rj are said to be coaxial; and also all circles about the two points are said to be coaxial. The two systems have been called hyperbolic and elliptic coaxial systems; just as in projective geometry a conic is a hyperbola when it meets the line infinity in distinct real points, and an ellipse when it does not meet that line in real points. So also a system of circles touching at a point is a parabolic coaxial system. It is to be noticed that in the hyperbolic system we are concerned primarily with arcs whose ends are the points x^,x^^ not with the complete circles. 32 BILINEAR TRANSFORMATION. [CH. HI The figure exhibits an elliptic and a hyperbolic system of coaxial circles. Fig. 17. It is evident from the remarks of the last paragraph that any coaxial system of circles in the x-plane, whether elliptic, hyperbolic or parabolic, maps into another such system in the y -plane, when the transformation is bilinear. Ex. Draw two orthogonal parabolic systems of coaxial circles. 24 Harmonic Pairs of Points. Let .£ be a circle about x^, x^, and H a circle through x^, x^ ; let E, H intersect at x^, x^. We shall consider the relation of jr/, x^ to x^, x^. In studying the relation of a configuration and its map we look out for those properties of the one which reappear un- altered in the other. These properties are said to be invariant. Thus four points on a circle, say a cyclic tetrad, have an invariant property, for they map into four points on a circle. And, again, two points oi^^ ciTCt^^nd two points inverse 23-24] BILINEAR TRANSFORMATION. 33 as to that circle, say anticyclic pairs, have an invariant property. In our case the four points x^x^^x^ are on a circle H, and two Fig. 1 8. are on E, while two are inverse as to £■ ; thus the four points, — or rather the two pairs, — are both cyclic and anticyclic. In fig. 1 8 the two pairs (jr,, x^, {x^, x^ lie on the circle H, and interlace so that the points of one pair are separated by the points of the other pair. Hence X. X. X„ —X. am ' - , — am — 7 = it. x^ — x^ x^— x^ Further, because x^ and x^ are on the circle E, Hence li__:!i. = _li_J!l; that is, the points x^' and x^' will divide the stroke from ;r, to x^ in opposite ratios, — that is, in ratios of the same absolute value but of contrary signs. Cleared of fractions the relation takes the form (^1 +^>) « +0 = 2 (x^x^+x^\') ; that is, the product of the sums is twice the sum of the products. In this form the symmetry of the arrangement is brought out. Not merely the points x^ and x^, or x^' and x^', enter symmetrically, but also the pairs x^ and x^, x^ and x,' enter symmetrically. The two pairs are said to be harmonic. If we take the origin at the ceritrt)id of x^,x^, then x^+x^ = o and the relation takes the simple form 1 a ^~ ^~ \% ^ 1 ^~ ' M. H. 34 BILINEAR TRANSFORMATION. [CH. Ill This relation gives •^ _ £l -^2 _ -^1 ■^1 ^i ^i ^i and shows therefore that the strokes from the centroid olx^,x^ to the points x^ and x^ are inclined equally but oppositely to the stroke from x^ to x^. Since the tangents at x^ and x^ to H meet on the join of x^ and x^, the joins of the pairs are conjugate lines as to the circle ; this fact is fundamental in the transition from the present point of view to that of projective geometry'. ^^j, Ex. I. Prove that when x-^, x^ and x^', x^' are harmonic, , , ^^ _ I.I c. 2j - + -. X^ — X^ X^ — Xi X1 — X2 f\ Ex. 2. Taking three circles any two of which are orthogonal, prove that ' the two pairs of intersections which he on any circle are harmonic. Note. Thus we nave three pairs of points any two of which are harmonic. :*/,(/."'. 1 I -'^•'^ Ex. 3. In a regular hexad point out the harmonic pairs. r' 25. The Double Ratios of Four Points. Returning to the equation (2) (§ 21), namely, let x^,y^ and x^,y^ be corresponding points. Then :^3-.r._^^3-^i y*-yx_^^*-^i Hence by division y^ -yi y4-y2_^»- ^, ^* - ^2 ^-^ ys-y^y^-yi ^>-^2^<-^i '" (I)- We have then on the right side of this equation a quantity depending on four points which is unchanged by the bilinear ' The transition is effected by means of the circular points at infinity. In the present case the condition that two pairs of points are harmonic is the condition that the pairs of straight lines from them to either circular point are harmonic. 24-25] BILINEAR TRANSFORMATION. 35 transformation. The quantity is called the double ratio, or Enharmonic ratio, of the pairs x^,x^ and x^, x^, since -^ ^ is, with sign changed, the ratio in which x^ divides x^, x^ (§ 18). If we denote the double ratio by {x^, x^; x^, x^) we have (•*!> -'^8 > ''^1' ^i) ~ ^/'-^S' ■^4 > •*'1> ^2)' (■^S' ^i ' ■*^2' ^^l/ ~ V\-*3> ''^«' •^1' •*^2/' V-*4' ^a ' ''^2' "^^l) ~ \^a> -^t' ^l' -*^2/ ~ C-*^!' -^2 J -'^s' ''^l)' The double ratio depends, then, on the way in which we pair off the four points, and the way in which we associate one of one pair with one of the other. In the above double ratio we have first selected the pairs x^, x^ and ;r,, x^; and then associated x^ with x^, x^ with x^ to get (x^, x^; x^, x^). We see then that since there are three ways of pairing off, and two ways of associating selected pairs, there are six double ratios of foltr' points. If we write (^2 - -^s) (^1 - ^t) = ^. (■*'8 - ^^) (^2 - ^4) = ^' (^1 - ^2) (^8 - ^4) = «. SO that /+m + n = o, then the six are — m/n, — «//, — l/m ; — njm, —Ijn, j^£«^. Special relations of four points, which are not alterable by a bilinear transformation, will betray themselves by special values of the. double ratios. It is convenient in detecting these to suppose a transformation effected by which one of the points passes to 00 . Suppose 7, is 00 , then the double ratios are y^zia y^-ya ya-y^ yi-y/y^-yi' ys-y^' 1 and their reciprocals. Anyone of these determines the shape of the triangle ^,^,7,. When one is real the triangle is flattened out, and all are real. Hence in general : When one double ratio is real, all are real and the four points lie on a circle. ^A-^ ^--r ^Li"' ' ^ 3—2 36 BILINEAR TRANSFORMATION. [CH. Ill Or again, if a double ratio has the absolute value i, then the triangle is isosceles ; that is, two points are on a circle about the third and oo . Therefore, when a double ratio is a mere turn, the pairs corresponding to that ratio are anticyclic. c, i V, ;y ^ y There are two especially simple arrangements, (i) When the triangle is both flat and isosceles ; the four points then pair off into harmonic pairs (§ 24) and the six double ratios reduce to three, namely, — i, 2, 1/2. (2) When the triangle is regular or equilateral ,' the points then pair off in three ways into anticyclic pairs ; and the six double ratios reduce to two, for the ratio "^-J — -^ is either —v or — u^ y.-y» Observe that in equation (i) by regarding x^ and y^ as variable, we have an equation which maps the jr-plane on the _;'-plane when three corresponding pairs of points are assigned. Ex. I. Find the six double ratios of the points o, i, 00 , jr. Ex. 2. If {x, x\; x^, x^ = -v, find x. Ex. 3. From the equation l+m + n = o deduce that the sum of the rectangles of opposite sides of a convex quadrilateral is never less than the rectangle of the diagonals ; and that it is equal to this rectangle when the points are cyclic. Ex. 4. Determine the equation between x and y which maps o, i, 00 into I, V, v'''. • 26. iBOgonality. The derivate of y 2iS to x is defined (just as for real variables) as the limit, if there be one, of the ratio of the change oiy to the corresponding change of x, when the change of x is made arbitrarily small. In the case of the bilinear transformation, we have _ _{ad—bc){x^—x) I ^ ■ ■^' ~^ ~ {cx+d){cx^ + d) ' ^ therefore lim -^J — - = . -^. . x=x,X^-X {cxJrdf In fact the well-known rules for obtaining derivates apply unaltered to this case, so that we can write at once „ ad— be \\ 25-26] BILINEAR TRANSFORMATION. 37 The derivate depends on x alone, not on the change of x. Let these two points jt,, x^ approach x by different paths, — to fix ideas, let it be along two cjrdes, — and let y^, y^ be the ' , (3 is at oo , hence we are led to regard oo as a point in this kind of geometry. Fig- 23. (i) When the radius of the sphere of inversion is the length <97'(fig. 23), the sphere 2 inverts into itself; since OP.OQ=^OT\ Thus any sphere orthogonal to the sphere of inversion inverts into itself. (ii) Next let OP meet any sphere X at Q and R. Fig. 24. 27] Then BILINEAR TRANSFORMATION. 41 OP.OQ = o^, OQ.OR = f, where / is the length of the tangent from to 2. Therefore OP oc OR, and P lies on a sphere. Hence the inverse of a sphere that passes through neither O nor 00 is a sphere. (iii) But if 2 passes through O, let OA be a diameter of 2 and let OP intersect the tangent plane at A in R. Fig. 15. Then OP.OQ = tL\ OQ.OR = OA?; therefore OP oc OR, and P lies on a plane parallel to the tangent plane at A. Hence the inverse of a sphere through O is a sphere through 00 . (iv) Lastly if the sphere passes through O and 00 , that is if it is a plane through O, the inverse is clearly the same plane. If we restrict our view to this plane we have the ordinary plane inversion. in. We have proved that the inverse of a sphere is a sphere. It is now easy to prove that the inverse of a circle is a circle. The points common to two surfaces invert into the points common to the inverse surfaces. Let the surfaces in question be two spheres that pass through an assigned circle; then the inverse spheres pass through a circle which is inverse to the 42 BILINEAR TRANSFORMATION. [CH. Ill given circle. In particular we see that in plane inversion a circle inverts into a circle. Inversion with respect to a plane. When the sphere of inversion becomes a plane, what is to be understood by inverse points ? In the general case let OP meet the sphere at X, Y. Then Fig. 26. X, Y are harmonic with P, Q. Thus when O and therefore X passes to infinity, Y being fixed, PF= YQ. Hence P and Q are reflexions in the plane (fig. 26). 28. The Bilinear Transformation is equivalent to Two Inversions in Space. The relation y = (ax + b)j{cx + d) involves four constants, but these constants appear only as ratios ; hence a knowledge of three pairs of corresponding values {x, y) will determine completely the character of the trans- formation. As corresponding values let us take x=\ 00 \ — d\c\ — bja y = \alc\ 00 I o. Taking any point X exterior to the ;jr-plane as origin, invert the .ar-plane into a sphere through X. The correspondence between the points of the sphere and the values of x is one-to- one ; in particular the value .r = oo corresponds to the point X itself. Let the points — dfc, — bja of the plane pass into points F, Z of the sphere ; and invert the sphere again, this time from Y, on a plane parallel to the tangent plane at Y and at such a 27-28] BILINEAR TRANSFORMATION. 43 distance that Z, X invert into points whose distance apart is - . Taking this plane as the ^-plane and the inverse of Z as the origin, we can adjust the real axis in this plane so that the inverse of X is ale B=f 'a=-# Fig. 27. Now let any point S of the sphere invert into the points P, Q on the two planes, the centres of inversion being X and Y respectively; and let P be the point x of the jr-plane, Q the point y of the j-plane. AP _SY BY XA ~ SX BQ' or ^/'.^(2 = ^^ • -5 F= a constant, and taking for P the particular position — b\a, the constant is Then - a c a 0-- c , or be — ad -? a _ *^ -ad • Hence We have proved then that the two quantities be — ad hi) (-f). are equal in absolute value. 44 BILINEAR TRANSFORMATION. [CH. Ill The plane through X, Y and the centre of the sphere will meet the line of intersection of the planes perpendicularly, say at C. Since CA, CB are respectively perpendicular to the diameters through X and F, CB=^ CA. And since AP and BQ lie in a plane, the angles CAP, CBQ are equal in magnitude. Let us regard angles in the two planes as positive when they are in the senses shown by the arrows, — a natural convention since to an observer standing on the sphere at X angles in the ;r-plane now increase counter-clockwise, and when he moves to Y they still increase counter-clockwise. Then we have /lCAP^-Z.CBQ = o, that is, am {x-\--\-\- am [y — j = constant ; and taking x = , ^ = o. we get am ('n)(-")=-(-^9(-9 Thus f;t: + -j [y — j and — -^ — , which were proved equal in absolute value, are equal also in amplitude. That is, they are equal ; hence, _ ax-vb ^~ cx + d' It is proved, then, that tke bilinear transformation is equi- valent to two absolute inversions. Since we know that a circle inverts into a circle we are led again to the theorem that a circle maps into a circle when subjected to a bilinear transformation. EXAMPLES. V " -I. Given three points a^, a,, Oj, construct the three b^, b^, *, such that Oi, ^1 and izj, oj are harmonic, a^, b^ and a^, a^ are harmonic, and Oj, *, and fli, a^ are harmonic. Prove that the straight lines joining a^ to b^, Oj to b^, a^ to *3 meet at a point, and that the relation of the two triads is mutual. 28] BILINEAR TRANSFORMATION. 45 2. Three points detennine three circles, each circle passing through one point and about the other two. Prove that these circles meet at angles jr/3 at two points. \ 3. Prove that two points on a circle subtend at any pair of inverse points angles whose sum is constant. 4. A point X is reflected in two plane mirrors. Prove that the second reflexion is the point x cis 20 where a is the angle which the second mirror makes with the first ; the intersection of the mirrors being the origin. J(S. Prove that a circle described positively in the ;ir-plane maps into a circle described positively in thej'-plane when it contains the point —d/c. 6. Taking four points a, 6, c, d, prove that when any three are inverted as to a circle whose centre is the remaining point, the triangles formed by the inverse points are all similar. CHAPTER IV. GEOMETRIC THEORY OF THE LOGARITHM AUD EXPONENTIAL. 29. Sketch of the Theory for Positive Numbers. The theory of the logarithm is so important for the general theory of functions that it seems to us desirable to present it from two points of view ; and in this chapter we shall give the geometric view, reserving for a later chapter the numerical view which is indispensable for calculation. The way in which the exponential and logarithm are introduced in Algebra cannot well be utilized here, for the processes used are precisely those which we have to inquire into later. We make acquaintance with these processes in Algebra and subsequently we should inquire into the logic of them. But in the Calculus we rely on what was said in Algebra ; and as the geometric function-theory requires such facts as D log f = i/f, where f is real, there is a dilemma. In the case of sin f there is no such difficulty ; for we have first a geometric definition, and then a geometric proof that Z> sin ^ = cos ^. For these reasons to put the logarithm on a similar basis, whereby we can make use of it without first discussing the theory of infinite series, or even the irrational exponent, is very desirable for elementary purposes. We shall assume the existence of the equiangular spiral, which may be constructed by placing a right cone with axis vertical and attaching a thread AP X.o s, point A of it. The point P is held in the horizontal plane through the vertex O; 29] GEOMETRIC THEORY OF THE LOGARITHM. 47 the thread is then wound round the cone, without being allowed to slip. As the thread is not to lie in a geodesic on the cone, the latter should not be polished. The curve described by P in its plane is the spiral in question, and it can be proved in an elementary way to have this characteristic property: that it cuts all rays from C> at a constant angle. Choose this angle to be 7r/4 ; and then measuring the angle 6 from the point where OP = p has the value i, define 6 as the logarithm of p: — ^ = Logp (I). Thence by a geometric limit-process, namely that which gives the formula pDf,6 = tan ^, where ^ is an angle made by the tangent with the radius vector, we have pD^e = I, and D Log p= i/p. We observe that for a given p there is but one 6 ; so that 9 and 6 +.2-rr are here quite distinct, defining not a direction but the amount of turning. We define inversely p as the exponential of Q, p = exp Q, whence exp 0=1, and D exp Q = exp 6. Take two points of the spiral, p, 6 and p,, 5,, and let 6^-0 be a constant. We have then De^p^ = DePi, that is, when we keep the angle ^^ - ^ constant, the ratio pjp is also constant. From this we infer that pJp depends only on 6^ — 0, and since when p= 1,0 = 0, and p, = exp 0^, therefore pjp = exp (0^ — 0), and Log (j)Jp)-=0,-0 = Log p,- Log p. We are now at the same point, with regard to the logarithm, as with regard to the sine before the series for sin x is proved ; that is before we study analytic trigonometry. 48 GEOMETRIC THEORY OF THE LOGARITHM. [CH. IV There is one other parallelism to be mentioned. In trigonometry we speak of a constant tt, without at first explaining any satisfactory way of calculating it. So here we can suppose known that value of p which corresponds to 6= i, that is exp i. A rough measurement shows that it is nearly 27 ; we denote it by e, and regard its calculation as belonging to analysis, where it is shown that ^=271828x828.... It is geometrically evident that as p passes through all positive values 6 passes through all real values. Equation (i) shows that jdpjp, taken from one value of the positive number p to another, is the change of Log p. 30. The Iiogarithm in g^eneral. We proceed to the logarithm of any number. Let xhe p cis 6. Then dx = dp c\s 6 + pd cis 6. But ddse = (- sin 6+ i cos 6) dO = / cisO d6 ; hence y dx= (dp + ipdS) cis d, and dxlx = dplp + idO. There is then this peculiarity about dxlx as compared with x'dx in general, — that it separates into a real part involving only p, and an imaginary part involving only 6. Thus before dis- cussing integrals of expressions involving x in general we can discuss this very important case; for we have, if initially .r = i, p= 1,^ = 0, I dx\x = I dplp + ?■ de •' 1 •' 1 Jo = Logp, + z0. = 'Log \xj + i a.m x^. This expression, Log p, + id^, we call the logarithm of x^, or The first term is the natural logarithm of p,, — a real number, defined in the last article. It depends only on the distance p of the final x from the origin. The second term iO^ depends on 29-30] GEOMETRIC THEORY OF THE LOGARITHM. 49 the path by which-;i: has passed from i to x^ — that is on the path of integration. It depends on the amount of turn of the stroke X in passing from i to ;ir,. Thus log;tr is many-valued in just the same way as the amplitude. According to the path selected it is capable of an arithmetic series of values differing by 2in ; and the general value of log x^ is found by adding zniri ' to any particular value. " '/ c^'" Ex. Find the values of log i, log i, log— z, log v. In the integration of dxjx we have had to pay attention to the route from the initial to the terminal point. When the variable is real the path of integration of a definite integral is determinate and unique, for the variable is confined to the real axis and must increase or decrease constantly from the initial to the final value. But when the variable can take complex values there are as many (i.e. infinitely many) ways of passing con- tinuously from the one limit of the integral to the other as there are ways of connecting the corresponding points in the plane. We define that value of log x, for which — tt < ^ ^ tt, as the chief value, or chief branch, of the logarithm ; and denote it by Log X. Thus Log X = Log p + i Am x. We have Log (p,/p„) = Log p. - Log p„ ; so that log {x^lx^ = Log p,/p, 4- i {6^ - 6^) = Log p, + id, - (Log p, + id,) = log;r, — log x^ + 2mri. In the same way it is proved at once that the logarithm of a product is congruent (to the modulus 27rz) with the sum of the logarithms of the factors. Since (§15) Am (xjx^) is not always Am x, - Am x^, it is not always true that ^ ,. ,, Log (xjx,) = Log;r, - Logjr„. In the first chapter we discussed two essentially different ways of comparing two strokes or complex numbers x^ and X,. The one way is to consider the difference, or speaking physically the displacement ; the other is to consider the ratio, or the stretch and turn. And the logarithm may M. H. 4 so GEOMETRIC THEORY OF THE LOGARITHM. [CH. IV be regarded as affording the necessary means of transition from the one to the other ; for when x,^ becomes x^ by a turn 9^ — 6^ and a stretch pjp^, then the logarithm, y suppose, makes in its plane a step "Logpjp^ eastward and a step^, — ^„ northward. Observe how particularly simple is the case when there is no stretch. Then pi = p„, and the logarithm is merely i X turn. Whereas in the case where there is no turn the logarithm, though real, can be assigned to a close degree of approximation only by a complicated calculation. With this calculation the reader is already familiar from algebra, where series, such as ^" ^'^ p+i 3VP + I/ P + I 3 V/> ■ are given. The validity of this series for all positive values of p will appear incidentally later on. 31. Mapping with the Logarithm. Let now y = log x, or I' + it}' = Log p + id. We have then f ' = Log p, i) = 6. Thus to the circles p = const, in the jr-plane correspond the lines f = const. in the j/-plane ; to the circles of radii i, e, e^,..., ^~\ ^~^..., correspond equidistant lines f' = o, I, 2,...,— i, — 2,.... To the rays 6 = constant in the jr-plane correspond the lines •»;' = ^ in the j/-plane ; and if the rays are drawn at equal angles the lines are equidistant. Figure 28 shows the mapping. As 6 increases, the line 1)' = 6 moves upwards ; and when 6 is 27r, the map is not 7;' = o again, but 17' = 27r. Thus to one ray in the jr-map correspond infinitely many equidistant straight lines in the_>'-map. If we restrict by the condition — ir<.d&ir, that is if we consider only the chief logarithm Log;ir, then we .' have the whole of the jr-plane, but the corresponding region / in the _j/-plane is bounded by the lines »;'= — tt and ??' =7r, and includes the latter line but not the former. It will be noticed that orthogonal curves map into orthogonal 3O-31J GEOMETRIC THEORY OF THE LOGARITHM. 51 curves, in accord with the property of isogonality (§ 26). For we have here D^y= \\x. V=7r V=37^ T)'=v/l f=-l ■rf=v/A. . r,'=-^A ■n'=-Tr/i ij'=-3ir/4 y— plane. Fig. 28. We have therefore isogonality except when jr=o or when ^=00. We can use this property to determine the curve in the ;r-plane which will map into a straight line in the j/-plane. For the curve must cut all the rays at the same angle ; it is therefore 4—2 52 GEOMETRIC THEORY OF THE LOGARITHM. [CH. IV an equiangular spiral. To obtain its polar equation, let the line in the ^-plane be r =«'?'+ A where a, /8 are constants. Then Log p = ad + ^, which is the equation required. / The angle of the spiral is the angle which the _j'-line makes / / with the lines t)' = const., and is the same for all parallel _y-lines. The orthogonal systems in the .r-plane are two specially // placed coaxial systems of circles. It may be left to the reader ^ to show that the chess-board arrangement in the _y-plane is two specially placed parabolic coaxial systems of circles (§ 23). ^ 32. The Exponential. When j is the logarithm of Jir, we call X the exponential of y ; and write x = exp j. The value of .*: is determined by the equations p = expf', = 1)'; the exponential of the real number f has been already defined in § 29. Thus for a given _;' there is but one x, so that the exponential of a complex number is one-valued. But we know that when x corresponds to y it also corresponds to _y -I- 2n-rrt; that is expj and exp {y + 2«7r?) are the same whatever integer n may be. A function which repeats its value when the argument increases by a fixed amount is called periodic; thus exp_y is periodic and has the period 2-iri ; just as sin^ is periodic, but with the real period 27r. When y is imaginary, — that is when |' = o, — we have p= \, and X is cis 6, while y is iTj' or i6. Thus our definition of the exponential leads us to say that cis 6 is exp id ; in other words that exp i6 = cos 6 + i sin 6. When we can assert, from the analytic theory (ch. XII.), that expjr= i+x+x^/2l+x''js\+... for any assigned x, then we have cos e + isind=i + i0+ {ieyi2 ! + (iey/i \ + ..., 31-33] GEOMETRIC THEORY OF THE LOGARITHM. 53 and cos^=i-072!+^74!-..., sin (?= 61 -(973! + ^75 !_..._ the analytic definitions of the cosine and the sine. In seeking to connect log;ir or exp;ir with x itself we must have in any case the problem of a limit. The limit employed in this chapter is /^p/p taken from one value of the positive quantity p to another. We shall see, in the analytic theory, that hm «(;!:'"•_ I) is Log;r, where .r"" is the chief «th root of ;r. It is well to verify this limit geometrically when x = cis 9. To find the limit in this case we observe that, ^„ being the chief amplitude, (cis^)""=cis(^„/«), and that cis {6Jn) - i is the stroke from i to a point of the unit c«ff„ cis^Bal") Fig. 29. circle (fig. 29) whose arcual distance from i is i/« of the arcual distance from i to x. Thus n{cis{6Jn) - i), as to absolute value, is the sum of the n chords of n arcs whose sum is 6^; while its amplitude is 7r/2 + 6j2n. Hence the limit has the absolute value 6^ and the amplitude -tt/s ; that is, it is z^„. 33. Mercator's Projection. The application of the pre- ceding; ^tincijjlea to the ^Qhlem. of makjjj;^ majj.s. q£ tbft. Eaxt-i'":, surface is of great interest. The discovery of the compass brought with it the idea of steering a course which should make with all meridians a constant angle. This spiral course was called a loxodrome or rhumb line. When the Earth's surface (regarded as a sphere) is inverted from the north pole into a J I ■ 54 GEOMETRIC THEORY OF THE LOGARITHM. [CH. IV plane, say into the tangent plane at the south pole, the meridians become a system of rays in the plane, and the loxodromes become, by isogonality, curves cutting these rays at a constant angle, or equiangular spirals. Now the loxodromes being the important lines, the map so formed by 'stereographic projection' was not sufficiently simple ; what was wanted was a map in which the loxodromes should appear as straight lines. This is done by mapping the inverse of the sphere by means of j = log.y; this is the principle of Mercator's projection. /,: 34. The Addition Theorem of the Exponential. ~ We know that if x^ = x^^^, then log x^ = log jt, + log ;fj + 2mri. Let ;tr=expj, and let j,, j,, j^ correspond to x^, .r,, x^; we see that if exp ^, = exp j, . exp j,, then jTs =^1 +7j + 2tfiri. Therefore exp (jj +j'^ + 2mri) = exp _y, . exp y^\ , ^ or exp ( J, + j/J = exp jf, . exp j,. So exp(jf,-j/,) = expj,/expj»/,. ^ ^ ^ ', ^ Ex. Write exp ni, exp {ni/2), exp ---. , "" as complex numbers of the form ^ + 217. , ^ 35. Napierian Motion. If we consider a moving point x as depending on the time /, then the derivate of x as to t, which we denote by x, is the velocity of x at the time t ; this velocity being itself a stroke whose absolute value, or magnitude, is the speed, and whose direction is along the tangent of the path of X. li X is p cis 6, then V x = {p + ip6) cis 6, r.-. ' J r so that the velocity is the sum of velocities of mc^^itude p in the direction Q, and magnitude pQ in a perpendicular direction. The simplest supposition we can make as to the velocity of a point is that it is constant ; and the simplest supposition we can make as to the motion of all the points of a plane is that they all move with the same constant velocity, — this is a 33-35] GEOMETRIC THEORY OF THE LOGARITHM. JS translation. We suppose any point of the plane to have the initial position x„ when the time t is o, and to take the position pe\ Fig. 30. Xg at any time t ; then on the above supposition we have x^ = d, a given complex number. Hence X(=x^+ bt. If then we are considering a substitution x^=x„ + b, we can suppose the new arrangement brought about by a translation ; x^ being the position of any moving point at time o, x^ at time i. The motion of translation is of course only one of infinitely many ways in which the substitution can be brought about, but it is the simplest. Suppose now that we impose on the _y-plane a translation such that every point y^ becomes y^ + bt at time t ; this imposes on the .jr-plane a motion such that the corresponding x^ (or exp y^ becomes Xt, where Xt = exp {y^ + bt) — exp y^ exp bt = x^ exp bt. Thus at any given time the ratio Xfjx^ is constant. The paths, or lines of flow, in the jy-plane are parallel straight lines ; hence the corresponding paths, or lines of flow, in the jr-plane are equiangular spirals with a common pole x = o and with the same constant angle. Also those points of the _y-plane which lie on a straight line at any instant will lie on a straight line at any other instant ; taking in particular the lines of level, which are orthogonal to the lines of flow in the ^-plane, these will map into another set of equiangular spirals in the jr-plane orthogonal to the lines of flow ; these also are lines of level. The lines of 56 GEOMETRIC THEORY OF THE LOGARITHM. [CH. IV flow and of level are shown in fig. 31, one system of lines being shown by dots. Either system being taken as lines of flow, the other will be lines of level. / Fig- 31- The velocity along a spiral is determined from the equations p = expf, e==i). These give pip = ^',6 = v'> and |', ij' are respectively the constant eastward and northward velocities in the _j'-plane. Thus in the spiral motion the velocity along the radius p is proportional to that radius and the angular velocity d is constant. In particular corresponding to a northward translation in the ^-plane is a motion in the .ar-plane for which p is 0,6 is constant ; this is a rotation of the jr-plane about the origin. And corre- sponding to an eastward translation in the _j'-plane is a motion in the .r-plane for which ^ is o, p oc p ; this is a special dilatation of the plane, of the kind contemplated by Napier in introducing logarithms. In general the motion of the j:-plane is a super- position of a rotation and a dilatation. For convenience this general motion will be called Napierian. CHAPTER V. THE BILINEAR TSANSFOBMATION OF A PLANE INTO ITSELF. 36. The Fixed Points. The bilinear transformation Xi = {ax^ + b)l{cx^ + d) converts a point x^ of the ;r-plane into the point {ax^ + b)l{cx^ + d) ; there is nothing to indicate that the transference is to take place along any special path. But, as in § 35, there is an advantage in connecting the substitution artificially with a special motion of the plane as a whole. Let us begin with the simple form x^ — ax^ + b. There is one finite point which is unaltered, namely the point for which x^=x„, or ax^-^b = x^; that is, the point x^ = bj(i — a). Taking this point as origin, the equation becomes _^ ^ / x^ = ax„; that is, all strokes are to be altered in the same ratio. This is attained by the Napierian motion. If however a= i the above reduction fails ; but the equation is then x^=x^ + b and only a translation is necessary (§ 35). We consider in general the mapping of a plane on itself, by means of the equation x^ = (ax, + b)l(cx, + d) (I); and regarding x^ as the position of any point at the time ^ = o, jr, as the position of that same point at the time ?= i, we seek an appropriate motion of the plane. 58 THE BILINEAR TRANSFORMATION [CH. V There are two points which do not move, namely the points given by x^=x^, or the points given by the equation cx' + (d- a)x- ^ = o. Let this equation have, in the first place, two distinct roots/,/'; these roots are called the Jixed (or double) points of the" trans- formation. With the help of these fixed points we can write (i) in the form ^, = ^^ (2);/^^' for this equation is bilinear and shows, from its structure, that when x^ is f{orf'), x^ is /(or/'). The value of k can be found by letting x^ become infinite ; the value of x^ is then — djc, and we have "-djc-f" or k = {cf' + d)l{cf+d). (Compare §21.) [l. 37. The Motion when the Fixed Points are distinct. If we write z = {x — f)l(x — f), then (2) becomes Evidently the fixed points in the a'-plane must correspond to those in the .*r-plane since z = z^ when x = x^. The new fixed ^points are o and 00 . In this change of planes we have mapped on the s-plane not merely the old and new arrangements of the points of the ;ir-planes but also the intermediate arrangements through which the totality of ;r-points may be supposed to reach their new positions x'. Thus we consider tlie motion of the one plane as mapped into the motion of the other plane. Now when z^ = kz^, a Napierian motion suffices for the ^r-plane; from this we infer a suitable motion in the jr-plane, by which x^ becomes {ax^ + b)\{ac^ + d\ Let us examine this more closely. (i) When \k\ = i the motion of z is along a circle whose centre is O; hence the motion of;r is along a circle about f and 36-37] OF A PLANE INTO ITSELF. 59 /', and the lines of flow are an elliptic system of coaxial circles r (§ 23). The lines of level are the orthogonal arcs of circles ; that is all points which lie at any instant on one of these arcs lie at any other instant on another. The substitution {\)is called elliptic in this case. (ii) When Am k — o,z moves along a ray from o to 00 ;-< hence x moves along a circular arc between / and /' ; the lines of flow are the system of such arcs ; and the lines of level are the orthogonal system of circles. This form of the substitution (i) is called hyperbolic. (iii) In general when \k\ 4= i and Am k-^o,z moves along an equiangular spiral; hence x moves along the map of such a spiral. What this map is like is easily seen from the property of isogonality, for it must cut all arcs between / and f at the constant angle Am k. Fig. v>- 6o THE BILINEAR TRANSFORMATION [CH. V The curve is a double spiral, winding about the fixed points. The lines of flow are a system of such spirals with the same fixed points and the same constant angle. The lines of level are the orthogonal system of double spirals, with the same fixed points. Figure 32 indicates both systems, one of the systems being shown by dots. Since the equation of the equiangular spiral is (§ 31) Logp = a0 + /3, the equation of the double spiral is Log (pIp') = a {9 -6') + ^, where p, 6 are measured from one of the fixed points, /, and />', 6' from the other fixed point /'. For a system of spirals as in fig. 32, a is a constant but /3 a parameter. The substitution is called, in this general case, loxodromic. If we invert the sphere into a plane by using a point of the sphere as centre of inversion, the loxodrome becomes our double spiral ; hence the reason for calling the substitution loxodromic. When the centre of inversion is at the pole the loxodrome becomes the single spiral. Ex. I. The points of inflexion of all paths in fig. 32 lie on a straight line. Ex. 2. The velocity in loxodromic motion is proportional to Ex. 3. In the elliptic substitution the points which move along a straight line have a constant angular velocity about either fixed point. 38. Case of coincident Fixed Points. So far we have con- sidered the fixed points /, f as distinct ; we proceed to the case when they coincide. Since now/' =/, the value of ^ is i ; and the equation ^1 -/ z. ^0 -/ conveys no information. But let /' =/+ if, and let if become o in a specified manner, say with an amplitude 7. The above equation is or, dividing by 8/ and then letting 8/^ tend to zero, I i_ c 37-39] OF A PLANE INTO ITSELF. 6 1 Map the jr-plane on a second plane by writing I c we get ^. = -o + ./+ar' a translation of the ^r-plane. Hence the lines of flow appropriate to this case of coincident fixed points are a parabolic system of circles which touch at / (§ 23). The lines of level are the orthogonal system, also touching at f. The substitutions ( i ), for which tJie fixed points coincide, are called parabolic. The common tangent to the system of circles is itself a circle of the system, namely a circle through f and 00 . The correspondent x^^ of x^='a lies on this circle. But or x^ = 2/+dlc. Hence the common tangent can be constructed by joiningy to 2f+dlc. Ex. The substitution is defined when we assig^n/and the initial and final values of a point, — say that x^ is Xq when x, is x-^. Prove that if we construct a point g so that _/" and g are harmonic with x,, and x-^, then j^i and x^ are harmonic with f and g. This is then a construction for the point x^^ corresponding to any point x^. 39. Substitutions of Period Two. Let x^ = {ax, + b)/(cx, + d), x^ = {ax, + b)l{cx^ + d), and generally x„+i = {ax^ + b)/{cx„ + d) ; what is the condition that x„ shall coincide with x, ? The question can be answered very readily by taking the equivalent equations 2", = kz^, z^ = kz^ = l^z^, ..., z„+^ = kz„ = ... = k'"'^'z, ; the necessary and sufficient condition is that k" shall be equal to I, i.e. k must be an wth root of unity. Since |<^| = i, the substitution must be elliptic. Let us examine the simplest cases « = 2 and « = 3. When « = 2 we have ^ = i ; and the substitution becomes 62 THE BILINEAR TRANSFORMATION [CH. V Taking the upper sign, i.e. k= i, the substitution becomes j: = .r„ and every point remains stationary. When k = — i we have X,— f X. —f x,-f'^x,-f' °' \ so that (§ 24) x^ and x^ are harmonic with / and /'. Combining all the points of the plane into pairs harmonic with two given points /and/', we see that the substitution interchanges these points. This arrangement of the points of a plane into pairs is said to be an involution. Any symmetric bilinear substitution between x^ and x^, that is any equation of the form cx^x^ = a{x^-\rx^-\-b (i), defines an involution. For by a single substitution x^ becomes x^ while ;r, becomes x^ ; hence by two substitutions x^ becomes x^ ; hence k = — \. The above equation between x^ and x-^ is the condition that x^ and jTj are harmonic with the roots of cx'^-2ax-\-b (2). Two pairs of points, say p^, p^ and qt,, q^, will determine an involution ; for the two equations determine the ratios of a, b, c. The relation (i) is then ^■(^i, J^o + ^i, ' =0 (3), AA>A+A> I which can be written (^0 -A) (A - ?i) (?o - ^1) = (-^0 - ?i) (A - ^1) (?o -i>i) (4), or in three other equivalent forms obtained by interchanging x^ and x^, or ;>(,and/i, or ^o^nd^i. Ex. Four straight lines can be paired off in three ways. Let the intersections of the pairs be x^, x^; A) /ij ?oi ?i- Prove by equating amplitudes and lengths in both sides of (4) that these three pairs of points are in involution. ^ Geometric Construction for the Partner of jr=oo . If we speak of the two points which are paired off as partners, then the equation (3) gives the partner of any point xq in the involution determined by given pairs. In or 39] OF A PLANE INTO ITSELF, 63 particular to find the partner of 00 we first divide the first row of the determinant by jto and then let xq tend to «.' Thus we get Xi, I, o =0. ^i(A+A-?o-?i)=AA-?o?i (S)- Here the origin is unspecified. If we take the origin at ^0 itself the equation is ^i(A+A-<7i)=AA. or, if A +A -?! = «'. ^if=AA (6). Since go=o, this gives the following geometric construction for the partner of 00 : — Complete the parallelogram pfffxP\V "^'"^ make the triangle piffiiXy similar to the triangle ggapi . Then x^ is the point required. This point is called the centre of the involution. The Double Points of the Involution. The points which are their own partners are the fixed points of the substitution, but with reference to the involution they are usually called the double points ; for in the pairing off of the points of the plane each is counted twice. To obtain them when two pairs are given, we have, by writing Xx=Xq,=x in (3) or (4), a quadratic in X. But this does not suggest at once a geometric construction. We can construct them very easily from knowing the centre ; for if we write Xy=o in (5) and thus take the centre as origin, the condition that /o and /i, j^Q and q-^, o and 00 are in involution is, as in (6), P^P\ = M\ ; whence the double points are the square roots of /(,/, , and lie on a straight line through the origin which bisects the angle A^A- •^"^ now if we state this in a covariant manner we can dispense with first finding the centre. For the straight line which bisects ptppi is a circle through o and 00 which makes opposite angles with the arcs 0^0°° ^"d oA* • Thus, generally, if AA ^^^ i'o?! ^""e the given pairs, draw the arcs yoA?i and yoA?i> ^""^ draw a circle through q^ and q^ making opposite angles with these arcs. This circle contains the double points. And interchanging the pairs we obtain another such circle, which cuts the former at the points sought. This construction fails, however, when the given pairs lie on a circle and are not interlaced. For then the two circles of the construction coincide with the circle on which the pairs lie. To meet this case we can apply the fact that if two circles A, B intersect, and A ' be orthogonal to both, A ' intersects either circle at points harmonic with the intersections of A and B. If then A' and B' are circles orthogonal with both A and B, each marks off on A points harmonic with the intersections of A and B. Thus if /o, Pi and ^0, ^1 are on a circle or straight line A, and are not interlaced, we draw through /„, /i and ^0. 9\ circles .4' and B' orthogonal to A, and construct any other 64 THE BILINEAR TRANSFORMATION [CH. V circle B orthogonal to both A ' and B'. The points where B intersects A are the sought double points. Ex. I. When /a, /i and q^, q^ are on a circle, and the straight lines ps,px and ^o?i meet at a point r outside the circle, prove that the points of contact of the tangents from r are the double points of the involution determined by /d, /i and jfj, q-^. Prove that the partner of r in this involution is the centre of the circle. Ex. 2. The above general construction for the double points is by means of angles. Discuss the correlative construction by means of lengths. 40. Reduction of Four Points to a Canonic Form. It is often convenient, in discussing involutions, to suppose the double points to be o and oo. The bilinear relation is then merely s^-\- s^ = o, and two pairs are opposite corners of a parallelogram. In mapping two given pairs of points into this parallelogram form, we may further suppose that one of the points maps into i ; the equations which give the pairs of points are thus brought by a bilinear transformation into the forms 5'= I, ^ = 2„». To determine z^, we use the principle that a double ratio is unchanged by a bilinear transformation. The double ratio (2„, -2„ I, -I) z^- \ - z^+ I IS -5 . 2 — , or , ^0 + I - ■S'o - I V^'o + Hence if/„,/, and g^, q^ be the given pairs we can take z^— P" i)' whence we have for z^ two reciprocal values, either of which can be taken. It will be observed that it is not necessary to calculate the double points. Ex. I. Prove that the equation {x^-\){pfl-^x-\-i)=o can be bilinearly transformed into (z^- i) [^■^ + (2 + ^5)2]=o. Ex. 2. When two pairs form a parallelogram, verify the construction for the double points given in the preceding article. Ex. 3. Four points can be paired off in three ways and therefore determine three involutions. Prove, by considering the parallelogram, that the three pairs of double points are mutually harmonic, and that the 40-41] OF A PLANE INTO ITSELF. 65 partners of any point in two of the involutions are themselves partners in the third. 41. Substitutions of Period Three. The case « = 3 of § 39, that is the case in which the substitution, when applied three times, brings each point back to its initial position, arises when ^ = I, whence ^ + i6+i=o, k = exp (+ zirijl). Taking three arbitrary points x^, ;r,, x^, we can of course determine a substitution which will bring them into the positions x^, x^, x^; we now see that this substitution is of the form K -f)l{x, -/') = exp (± 277^/3) . {x, -f)\{x, -/'), where / and /' are to be determined. This way of mapping a triad of points into itself is very closely connected with the solution of a cubic equation. For the essential point is that by writing {ax + b)l(cx + d) = 2 we can map the triad into any assigned regular triad in the ^r.plane ; say into i, v, v^. Thus the cubic whose roots are jr„, x^, x^, can be written in the form by writing (ax + b)l(cx + d) = z. That is, the cubic itself can be written {ax + by = icx + £lf, or say (x-/)' = X(x-/y. Suppose the cubic to be ;i^ + 30^ +^ = o ; then we have, comparing coefficients, /= \/', /' _ x/- = a (I - X), r - V = - /3 (I - \) ; whence //' = -<^. //'(/+/') = - ^> so that /and/' are determined by solving a quadratic. We have then (x-/)I{x-/') = ^X = ^jW, the three cube roots giving the three roots of the cubic. M. H. 5 66 BILINEAR TRANSFORMATION OF A PLANE. [CH. V Ex. I. The points jr„, x^, x^ being mapped into i, v, v\ the points/,/' are at the same time mapped into o, oo . Hence corresponding double ratios of ;r„, jrj, x^, / a.-nA i, v, \i\ o are equal. Hence prove that _ XjX^ + vx ^o +y^^o^i ^ Xf,+vXi+v^Xi ' Xq + v'Xi + vX2 f=—^^-J Ex. 2. Prove that the angle which the stroke from / to /' subtends at any of the points x^fc^x^ is a third of the angle which it subtends at their centroid. Ex. 3. A bilinear transformation can be found which will map a, b, c, d into b, c, d, a only when a, c and b, d are harmonic pairs 1. ^ The periodic « - cui, or figure formed by a point which returns to its original position after « substitutions, can be regarded as the map of a regular « - ad, — that is of the vertices of a regular polygon. This is not the place to enter into geometric considerations for their own sake, but as the figure in question has received some attention from geometers (see for example Casey's Sequel to Eiulid and Analytic Geometry), it is proper to point out that it can be obtained from the regular n-ad\>y a single inversion with regard to an arbitrary point of space ; and that therefore it is a special projection of the regular n-ad. CHAPTER VI. LIMITS AND CONTmUITT. 42. Concept of a Iiimit. The discussion of irrational numbers brings us face to face with difficulties which cluster round the notions of continuity, limit, and convergence. As these notions are woven into the very texture of our subject, it will not be amiss here to indicate to the reader some of their aspects which may have escaped his attention when studying algebra and the differential calculus. For simplicity we will take the variable real. The sequence 4, 3^, 3 J, 3J,... suggests at once the number 3. This number 3 is not itself a member of the sequence, but the numbers of the sequence tend to it (or converge to it) and can be ■made to differ from it by as little as we please. We are therefore justified in saying that the function f + 3 tends to the limit 3, when f passes through the values i, 1/2, 1/3,.... This state- ment is expressed symbolically in the form lim(| + 3)=3. Observe that f did not take the value o, for o is not one of the set I, 1/2, 1/3 Let us now state the matter generally. The numbers fi) ?j> ^8>'" °^ ^ sequence are said to tend to tite limit a, when to every positive number e there corresponds a positive integer /i* such that for f^ and for all later members f„ of the sequence we have ( f „ — a | < e. * The number y. will be different in general for different c's. The number e is said to be arbitrarily small and given in advance. It may be supposed assigned by an 5—2 68 LIMITS AND CONTINUITY. [CH. VI When, with the same notation, we have ^„> ije (or < — r/e) we say that the numbers ^„ tend to the limit + oo {or — oo ). 43. Distinction between "value w^hen ^ = a" and "limit vrhen ^=o." To return to the special example, we have seen that f did not take the value ^ = o ; suppose now that we assign to f the value o, we have at once the value 3 for the function f + 3. We express this in the following way : value (^+3)= 3. In this example lim(f+3) = value(f + 3)- f=o {=0 It is most important that the reader should appreciate that it is far from being true universally that the limit of an expression when f tends to a is the value of the expression when ^ is o. We will consider a case in which the equality breaks down. V - I Suppose that /f = Vz — ^"*^ that a= i. To fix ideas let f approach i through the sequence of values i + i, i + 1/2, f- I I + 1/3,..., then ^ (or f + i) takes the values 3, 2\, 2\,.... The left-hand side of the equation hm i. -= value \. f=. f-i j=. I- I has therefore the definite value 2 ; but the right-hand side is meaningless and the equation breaks down. Y- 44. Upper and Lower Iiimits. The numbers of the sequence 4, 3^, 3^, 3:^,... lie between o and 10; this interval can be contracted to the narrower interval 2 to 6, and this again can be contracted. Suppose that we take the narrowest interval which contains all the numbers ; this is evidently the interval from 3 to 4. The number 3 is called the lower limit and the number 4 is called the upper limit of the sequence. opponent. The opponent names as many numbers c as he pleases, and one has to assign for each a suitable /i. Of course an algebraic inequality, such as /t>i/e, is the usual means of establishing the criterion. In arguments about limits c is restricted to the above meaning, with the consequent advantage that the words 'arbitrarily small ' and ' given in advance ' can be omitted. 42-45] LIMITS AND CONTINUITY. 69 It is clear that the notions upper and lower limit coincide with the notions of maximum and minimum, whenever the sequence contains a maximum number and a minimum number. For example (i) the sequence 4, 3^, 3^, 3 J, . . . has an upper limit 4 which is also a maximum : there is a lower limit 3, but no minimum ; (ii) the sequence 3, 4, 3J, 3^, 3^,... has an upper limit 4 which is a maximum and a lower limit 3 which is a minimum ; (iii) the sequence formed from the numbers 3 + i/«, 4— i/« by giving to n the values 2, 3, 4,..., has the upper limit 4 and the lower limit 3, but it has neither a maximum nor a minimum. 45. Every Sequence of constantly increasing Real Numbers admits a Finite or Infinite Limit. Let the real variable ^ increase constantly ; either there is no real number o which is greater than all the values assumed by f, or there is such a number. In the former case ^ can become indefinitely large ; in the latter case there is a number 7 to which ^ comes arbitrarily close, or to which f attains, but beyond which f does not pass. In the one case we say that the limit of f is +'oo, while in the other the limit of | is 7. In the latter case the limit is also the upper limit (§ 44) ; in the former case we shall say that + 00 is the upper limit. Similarly when ^ decreases constantly it tends to the limit — 00 or else to a finite limit 7'. In the former case the lower limit is — CO , in the latter 7'. The existence of the numbers 7, 7' is not a new assumption ; it is a consequence of the arithmetic definition of irrational numbers. For all real numbers can be separated into the two classes, (i) the numbers which are greater than all the values that f is allowed to take ; (2) the numbers which are exceeded by some of the values of |. From these two classes there is no difficulty in picking out two sequences which will define the number 7 ; and similar reasoning establishes the existence of 7'. •JO LIMITS AND CONTINUITY. [CH. VI 46. Every Sequence of Real Numbers haA an Upper and a Lower Iiimit. First let there be an integer a which is less than all the numbers of the sequence and construct the sequence a, a+ I, a + 2, Passing from left to right along this sequence let (o,, a, + i) be the first interval which contains a number %. Divide this interval into tenths and consider the new sequence a„a,+ i/io, o, + 2/io,...,a,+9/io, a,+ i. Passing again from left to right let (Oj, Oj+i/io) be the first interval which contains a number f. Divide this interval into tenths and proceed as before. In this way we construct a, + i,aj+i/io, , a„+.i/io^'-^ which are sequences of the kind considered in ch. I. (§ 6). They define therefore a rational or irrational number 7'^ we shall prove that this number is the lower limit, that is that there is no number f which is less than 7', and that there are numbers f which differ from 7' by less than any assigned positive number however small. The latter part of this statement is true by reason of the presence of at least one number ^ in every interval (a„, a„ -^,^/io"~'), no matter how large m may be. To prove the former part of the statement, assume that there is a f, say f,., which is less than 7'. Then because a,, a,, a,, tend to the limit 7', it must be possible to find an a, say a^, which is nearer than f, to 7'; thus we have a f which is less than an a, contrary to supposition. It follows that 7' is the lower limit. What has been proved may be formulated as follows : — Given that all the values of a real variable ^ are greater than a finite integer a, tlure exists one and only one finite number 7' with these properties: — (i) no value 0/^ is less than 7', (2) at least one value of % is less than 7' + e, where e is an arbitrarily small positive number assigned in advance. This number 7' is the lower limit of the values of f. 46-47] LIMITS AND CONTINUITY. 71 Secondly, when there is no integer a which is less than all the numbers of the sequence, we say, as in § 45, that -00 is the lower linnit Thus in all cases there is a lower limit. The corresponding theorem for the upper limit can be proved in the same way. 47. The necessary and sufficient condition that a Sequence tends to a finite Limit. When we have reason to suspect that the numbers ^„ tend to a limit o, we can verify whether this is so or not by using the definition of § 42, namely |f«-a| n and n^fi. Then |^„'-^„|S|a-f„| + |a-^„.j /t (fig. 33). By choosing /i' sufficiently large, we can affirm that every point f^'+u ^»«'+2> ?>''+3>--- ^i^s within a new interval p'q, where f^- — /' = q — f ^' = e/2. This interval p'q' may extend beyond p ox q but we can cast away the projecting part since all the points ^^, f^i+i,... He in pq. Attending then only to the interval common to pq and p'q' we observe that its length is at most half the length of pq, and that it contains all the points f^-, f^'+i, Among the numbers /[4'+ I, /i' + 2, /a'+ 3,... we can find a number /*" sufficiently large to secure that all the points ^^", f^"+i, ^^"+2,... shall lie within an interval whose length is at most half that of p'c[ , this interval lying wholly within each of the preceding intervals. By pro- ceeding in this way we can get a succession of intervals whose lengths tend to zero ; furthermore the left-hand and right-hand extremities of these intervals, when connected with the origin by strokes, give two sequences of strokes in ascending and descending order respectively. Hence the conditions of § 7 are fulfilled and the strokes define a point a, which is the limit whose existence we sought to establish. The non-terminating decimal affords a good illustration. When we take the first y. digits after the decimal point, let the number obtained be f^. Then fn'~f»< i/io'', and this can be made less than any assigned e by taking /t large enough. Therefore there is a limit of the sequence f,, f,, f,, If the decimal terminates, the limit is attained. I A sequence which tends to a finite limit is said to be con- vergent. 48. Real Functions of a Real Variable. The most general definition of a real function of a real variable is as follows : — When one and only one real value 17 is assigned to each value of a real variable f, t) is said to be a one-valued function of^. When two or more real values are assigned to each ^, ri is said to be a many-valued function of f . We restrict ourselves in this chapter to one-valued functions. 47-49] LIMITS AND CONTINUITY. 73 This definition admits functions which cannot be represented graphically. Let, for example, ?; be equal to o for all rational values of the real variable | and to I for all irrational values of that variable ; then the points (^, 17) trace out no plane curve in the sense ordinarily accepted for the term ' curve.' Similarly in the case of two or more independent real variables, what is implied by the statement that a dependent variable is a function of these independent variables is conveyed adequately by the adjective ' dependent'; the former is to be given when the latter are given. We lay emphasis on this ; for we propose to consider these general functions in the case where the variable or variables are real, although the only functions of the complex variable x that will be discussed in this work will be drawn from a very general, though not completely general, class of functions known as ' analytic' To keep the two uses of the word ' function ' as distinct as possible we shall use /f , /(|, ij), when simple dependence on real variables is meant ; whereas yjr, where x=^+iri, will indeed denote dependence cfti x, — therefore also on 1, 7;, — but dependence of that special kind which will be expressed later by analytic function. 49. Continuity of a Function of a Real Variable. Suppose that when the real variable f tends to the limit o by increasing values of ^, the function fl^ tends to a limit ; we can denote this limit bylim/^; similarly we use lim/f to denote the a— o a+o A limit (also assumed to exist) for /f when f tends to a by decreasing values. With this notation we can define continuity as follows : Let /f be a one-valued function of the real variable f , which is defined for all values of an interval a — B to a + B, then /f is said to be continuous at a, if lim/^ = liin/l = value /f a— o a+o {=0 The function is discontinuous when any one of the three numbers in these equations is non-existent, and when the three fc» _ _» numbers exist but are not all equal. For example ^^ is not continuous at f = a, because the expression has no meaning when f is equal to a ; here lim \_ = lim ?— r = 2a, but a-o t ^ a+o b ^ f* — a' value ^ is non-existent. f=. ?-a 74 LIMITS AND CONTINUITY. [CH. VI Suppose that when f < i we have 7? =/?= i, and when f S i Fig. 34- we have tj =/^ = o (fig. 34) ; then Hm/f = i, whereas lim/f = O, I— o 1+0 and value/^ *= o. Here again there is discontinuity. The necessary and sufficient condition for the continuity of /f at f = a, where /f is defined for ^ = 0, may also be stated in the following way : — T/ie function f^ is continuous at f = a, wken there exists an interval of the axis of f, — say the interval froin a — Btoa + S, where S is real, — such that at every point f of this interval, we have where e is an arbitrarily small positive nitmber which is chosen in advance. It is implied here that the function is defined when ^ 50. A ContinuouB Function of a Real Variable attains its Upper and Lo^rer Iiimlts. Let f^ be continuous at all points ^ for which aSf ^^8, a and /3 being finite. This in- terval may be called the closed interval (a, /3) ; closed, because the interval contains both extremities a. and ^. Within the closed interval (a, /3) the values f^ have an upper limit 7 and a lower limit 7'. We wish to prove the important theorem that there exists in the interval a point at which /f is exactly/, and a point at which /f is exactly 7'. In other words we wish to show that the values of the continuous function /f admit a maximum and a minimum value, when f takes all values within the closed 49-50] LIMITS AND CONTINUITY. 7$ interval. The following lemma is required in the proof of this theorem : — Lei one of the closed intervals (f„-8, f„+S), (f„ f„ + S), (?o ~ ^. ?o)> "UJliere 8 is an arbitrarily small positive number, be associated with the point f„ of the closed interval (a, ^3), the first, second, or third being chosen according as f „ is interior to or at the extremities a, ^ of the interval (a, /3) ; also let these intervals lie within (o, yS). Then we can assert t/tat there is at least one point fo of the interval (a, /S) such that the upper limit of f^ for the associated interval is exactly 7, no matter how small B may be taken. To establish the truth of this lemma we must prove these two propositions : — (i) Given that (a, ;S') is contained within (a, yS), then tlie upper limit of fl^ for (o', ;3') is at most 7/ (2) Given that (a, /S) is divided into two equal or unequal parts, tJien the upper limit of f^ for at least one of these parts is equal to 7. As regards (i) observe that if the upper limit of f^ for (a', ^') be 7 + 7i where 71 is a positive number, then there are values of y^ which exceed any assigned number which is less than 7 + 7i, and therefore exceed the number 7. This is contrary to supposition ; for there are by supposition no values of fl^ in (a, j8), much less in (a', /3'), which exceed 7. If the upper limit of /f for (a', yS') be + 00 , the upper limit of /f for (a, /3) must also be +00. As regards (2) we can use a very similar argument. When 7 is finite, the upper limits for the two parts may be equal, in which case both must be equal to 7, or they may be unequal, in which case the greater of the two must be equal to 7. When, on the other hand, 7 is + 00 we must have at least one of the two new upper limits equal to + 00 . We can now proceed to the proof of the lemma. Divide (o, /3) into two equal parts ; for one at least of these, say (a,, ^,), the corresponding upper limit is 7. Divide (o,, ^,) into two equal parts and select a half, say {tt^, ^^, which yields an upper ^6 LIMITS AND CONTINUITY. [CH. VI limit 7. By continuing this subdivision, we are led to two sequences O. O]. *ai ^s ««. /8,A./3,./3 ./3», , which are composed respectively of ascending and descending numbers. These sequences define a rational or irrational number fd, since a„— /8„, = (a — /8)/2'', tends to zero when n tends to infinity ; this number f„ lies within (a„, /8„). Within (a„, /3„) the upper limit of /^ is 7 ; hence the upper limit of /^ in any interval (f„— S, f„ + S) which contains (a„, /3„) is at least 7; since (a„, /8„) shrinks indefinitely as n grows, we can make the interval (^„ — S, ^„ + S) as small as we please. Again since (^„ — fi, f„ + S) is merely a part of (o, /9) we can assert at once that the upper limit of f^ in the former interval is at most equal to the upper limit, 7, of /Iq in the latter interval. Since the upper limit of/| for the interval (?o ~ ^' ?o + ^) is at least equal to and at most equal to 7, it must be exactly equal to 7. This is on the assumption that 7 is finite; when 7 is + 00 the upper limit of /"f for the interval (f„ — 8, |^„ + S) is not less than 7 and is therefore + 00 . This completes the proof of the lemma. It is evident that the lemma is still true when the words 'upper limit' are replaced by 'lower limit,' and 7 by 7'. We have spoken of 7 = + 00 as if it were a possibility, but the lemma shows that 7 cannot be + 00 when /f is continuous. For (a) the supposition 7 = 4- 00 implies, by the lemma, that the values ofyf increase indefinitely as ^ tends to f„; (b) the supposition that /f is continuous at f, implies that the values of /| differ very little from the finite value /f„ when ^ approaches very near f^ ; and the statements (a), (b) are irreconcilable. We can now prove the theorem that a continuous function f^ attains each of its two limits at least once. By reason of the continuity of /f at f„, there exists an interval (f, - 8, ?« + 8) such that for every point f of this interval we have l/f-/^.|<6 (i). SO-5I] LIMITS AND CONTINUITY. JJ where e is an arbitrarily small positive number assigned in advance. And by reason of the lemma we can assert that there may exist within (?«- ^. ?o + ^) a point f such that /f = 7, and that there must exist a point ^ such that /f >7-e. Thus, for the same values of f as in (i), we have l/?-7l /^ 4, c, \ *, \^ c^ d, V 1 fe d 4 y ( \ «4 =>. Fig. 35- rectangle into four equal parts as before, and select a part which yields the upper limit 7 for the associated values of f ; let this part be c^c^c,c^ {c^ = b^, and proceed as before. Then we get a sequence of rectangles '^A^»«4' bx^-PJ>K^ '^I'^i'^lf*^ ^A^i'^i each of which contains all that follow; the areas of these rectangles tend to zero and there is one and only one point (?o' vd which lies within all the rectangles of the sequence. This assumption is of the same nature as the assumption made in § 7. For each rectangle provides two abscissae and two ordinates ; for example c^c^c^c^ provides the abscissae of f, (or c^ and Cj (or r,) and the ordinates of c, (or c^ and c^ (or c,). And hence the rectangles lead to ascending and descending sequences of abscissae which converge to an abscissa f„ and to ascending and descending sequences of ordinates which converge to an ordinate tj^. The combination of the sequences leads to a point It is proper to define here the expression neighbourhood of (^, 17). All that is meant by this is a region that consists of points on and within some circle whose centre is (f, 17), and whose radius is not zero. We can see, precisely as in § 50, that : — The values of the real function i, of the two real variables f , 17 for the region V of the plane admit the same upper limit for every 51-52] LIMITS AND CONTINUITY. 79 neighbourhood, contained within V, of the above point (f,, %) as for the whole region V. It is almost unnecessary to say that when a neighbourhood of (f„, »;„) lies partially outside V, — this will happen for example when (f„, i;^) is on the boundary of F, — the theorem will still apply, provided that account be taken only of such points of the neighbourhood in question as lie within or on the boundary ofr. (Ill) Continuity of/(f, 77). Suppose that /(f, 17) is defined for all points of a region V which covers a continuous portion of the plane ; for instance F may consist of all points on and interior to the ellipse ^+-4=1- We shall say that the function f{^, rj) is continuous at a point (f„, ?;„) in the interior of F, when there can be found a neighbourhood of the point such tftat for every point (^, 17) of this neighbourhood, we have |/(l,'?)-/(fo.'7.)i<^. where e is an arbitrarily small positive number given in advance. 52. A Continuous Function /(^, 17) attains its Upper and Lower LimitB. It is now possible to prove the following important theorem on the upper and lower limits of a continuous function : — There is at least one point of F at which the value of f{%, i)) is exactly 7 and at least one point at which the value is exactly 7'. Let (f„, i7j be found by means of the sequence of rectangles as explained above. Then /(^<„ ■»?<,) must be equal to 7; for if possible let it be equal to 7 - S, where 8 is a positive number. {a) Because /(f, »?) is continuous, it follows that there is a neighbourhood of (^„, 17,) for every point (£ 7?) of which |/(?,'7)-/(^..'7o)ly- S/2, for the values of /(f, 17) must approach arbitrarily close to 7 and some of them must therefore exceed 7 — 8/2. As (a) and (d) contradict each other, it follows that /(fo''?o) = 7- Similarly it can be shown that /(f, rj) attains the value 7'. 53. Uniform Continuity of a Function of one Real Variable. Suppose that a real function /^ of a real variable f is continuous at every point f of a closed interval (a, /3) given by a ^ f S /3. Because of the continuity of _/f there exists for every point f between o and yS an interval (^ — /t, f + ^) such that for every pair of values f ', f " included in (a, /3) and in (? — ^, f + ^), we have i/r-/ri o ; (ii) tion-uniform continuity, when A = o. Fig. 36- Suppose that A > o ; the essential fact to grasp in this con- nexion is that one and the same value of k, namely ^ = A, will serve for the inequality (i), whatever be tlte position of ^ in the interval (a, /3). The following theorem disposes of the possibility of the continuity being non-uniform : — If a real function f^ of a real variable f be cotitifiuous in a closed interval (o, ^), tlte continuity must be uniform. As soon as it is shown that jO is a continuous function of ^, the theorem follows very readily. For in this case p attains its lower limit A at some point c of (a, y3), by § 50 ; and A = o means that there is no interval at c within which the oscillation is less than e, contrary to supposition. Let ^, be a point of the interval (f— p, f-t-p), and let p, denote p (f,). Evidently the interval (f, - p,, |, -I- p,) that belongs to f, must extend at least to the nearer and at most Fig- 37- to the farther of the two points ^-p, ^ + p- In fig. 37 we have drawn f , a little to the right of ^, and in this case In any case, when f, is near f, |p-p.|S||-fJ; and this inequality implies the continuity of p at f, which is what we had to prove. 54. Uniform Ooatinuity of a Function of two Real Variables. Let /I be replaced by/(f, 17), where | and r, are real ; and let the interval of continuity {a, b) be replaced by a region of continuity, r, the boundary of r M. H. 6 82 LIMITS AND CONTINUITY. [CH. VI included. Here, of course, (^, i)) is treated as a point referred to rectangular coordinate axes. Finally let _/(!, ij) be real, one-valued, and continuous at each point (|o, i)o) of r. To simplify matters let (|o, 170) I'e in the interior of r. The continuity of /{^j v) ^^ (^01 Vo) 's expressed in the statement that the inequality l/(f,'!)-/(lo.'!o)l Va)i °f Ae values p can be used in place of p'. To prove that the continuity of /(f, 17) in r is uniform it is necessary to show that the lower limit of the values p (f , rj) in r is greater than zero. This is done as before by showing that p (^, tj) is continuous. Let (|i, i/i) be a point at a distance 8 from (^, tj). We get, by an argument similar to that used above, the inequsdity and infer the continuity of p (^, t)) in T. The novelty consists in the use of circles with centres (^, if), (^,, tjj) instead of intervals with centres ^, fj. The functions /x of a complex variable x( = $ + iri) which we shall consider will all be expressible in the form u (^, ?;) + iv (^, i;), where u (|, 17) and z/ (f, i;) are real functions ; it follows that when /x is one-valued and continuous in a region T situated in the finite part of the jr-plane, the continuity is uniform. 55. TTnifonu Convergence to a Limit. Let /^ be a continuous function of a real variable ^ in a closed interval (a, 6). At each point f between a and i l'm/(l+^)=/l) by reason of the continuity ; so that at $,/{i + A) tends to the limit y| when A tends to zero. Because the continuity is uniform it is possible to assign a positive number 8 which zs itidependent of ^ and such that |/(f-t->^)-lim/(^ + -%)| 1 ' ^i ^) is said to converge uniformly or non-uniformly to its limit J^ii, if). Here again it is essential to notice that when A>o, a definite value 8 = A will serve for the inequality, ivhatever be the position of{^, 17) in T. If in the preceding work we use only one variable ^ instead of two variables |, tj, we must replace the region r by an interval. Cantor has illustrated the possibility of non-uniform convergence to a limit by means of the function j , where | is real and confined to the closed interval (o, i) and « is a positive integer which tends to infinity. Take any definite value ^ in the interval ; then L 2=0 ; but we cannot assert that there exists a positive integer ^ such that when n'^li "Vie have, for all values | in the interval, For this reason the convergence to the limit is non-uniform. The full significance of this example can be appreciated best by tracing some curves of the family "1(1- ^) 1 „2^2 + (,_|)2 for values o< f < i. As the integer tt grows the curves tend on the whole towards the axis of f ; but since 17 has a maximum value 1/2 when f=i/(«+i) each curve is a wave of height 1/2. Let ^ take an assigned value ^Q which may be as small as we please ; when n tends to infinity the parts above the interval (!„, i) tend to coincidence with that interval, and ultimately the crest of the wave lies at a finite distance 1/2 above some point of the interval (o, ^0). Fig. 38- 6—2 CHAPTER VII. THE BATIONAL ALQEBEAIC FONCTION. 56. Continuity of the Rational Integral Function. The rational algebraic function includes the rational integral function of the complex variable x, defined by _;j' = a„+a,jr + a,jr'+ +«„jr» (l), and the rational fractional function, or rational fraction in x, which is a quotient of two rational integral functions of x. It is at our option whether we represent y on the jr-plane as in ch. V. or use a separate jz-plane as in ch. III. An essential distinction is to be drawn between the «th power and an «th root of x. When y = IJx there are, in general, n values o{ y for each x, i.e. y is a« n-valued function of x; whereas the function x'^ is one-valued. One of the first questions to be asked about any function is whether it is one-valued or not. In the equation {\) y is evidently one-valued (see § 20) as regards *• ; we shall show presently that x is n-valued as regards y. Some preliminary theorems and explanations are necessary before this can be proved. The term function is to be applied to the various expressions in X as they are introduced ; the above functions will be found to be included in the general class of functions to be discussed later on under the name of analytic functions. 56] THE RATIONAL ALGEBRAIC FUNCTION. 85 In this and succeeding paragraphs we shall be concerned with rational algebraic functions, including as special cases rational integral functions and mere powers ; but the definitions that we shall give for continuity, limit, derivate, etc., will apply to the functions which are to be introduced successively in later chapters. The definition of continuity (§ 49) has to be modified. The interval (a — 6, a + 6) on the ^-axis is to be replaced by the assemblage of all points x which satisfy the inequality I j; — <2 I S S, where S is real and positive; that is, the interval, composed of all points on the real axis at a distance from a not greater than 2, is to be replaced in the plane by the circular region which contains all points;!: whose distances from a do not exceed S. The definition of continuity runs now as follows : — A function fx of a complex variable x is said to be continuous at the point x = a, when there exists a circular region (|.r— «| S 8) such tJiat at every point x of this region tJie differettce between fx and fa is less than an arbitrarily stnall positive number e chosen in advafice. It is implied that the function is defined when x = a. The following theorem serves as a basis for the proof of the continuity of the rational integral function oi x: — The rational integral fiinction y = a^ + a^x+ ajc" + . . . + «„;tr", which takes tlie value fl„ when x=o, ca7i be made to assume a value as near as we please to a^ by taking x small enough. It is necessary to prove that \y~a^\ can be made less than e. We have \y-a^\ = \a^x->ra^-^ ...■YanX''\ Let ^ be the greatest of the numbers | aj, | a, | | «„ I, then whence, if |;r|< i, we have \y-aA<- \ ' |. 86 THE RATIONAL ALGEBRAIC FUNCTION. [CH. VII If then we wish |^ — a„| to be less than e we have only to choose X so that \-\x\ or \x\< e/{^ + 6). As such a choice o{x is always possible, the theorem is proved. Closely related with this theorem is the following : — It is always possible to find for \x\ a value X such titat for this and all greater values of\x\ we shall have I «»-^ + «n-i^"-' + ... + fl^ + fl„ 1 > 7, where 7 is an arbitrarily great positive number which is chosen in advance. For the rational integral function under consideration has for its absolute value I I I j and because the absolute value of the sum of two quantities I I I is not less than the difference of the absolute values of these quantities, it follows that the second factor in the expression for \y I is not less than I I I I But, by the preceding theorem, can be made less than e by making I 6 , — , < where /*' denotes the greatest of the numbers \a^\, |«,|, \a^\, ..., ]a»_i| Hence, when l^i.>(M' + e)/6, we have bl>k"l-{k«l-e}. 56-57] THE RATIONAL ALGEBRAIC FUNCTION. 87 To fix ideas let e = ^| a„ |. Then b|>i|^„||;ir|»; and we see at once that \y\>'y for a// values oi x which make I «n r We may take therefore for X the real positive value of /,-?5L vu„r Note. Let x and the a's be real. Then the theorems that we have proved show respectively that the term a^ governs the sign of the whole expression when the values of x are sufficiently small, and that the term Ojtar" governs the sign when the values of x are sufficiently large. Ex. I. Prove that for sufficiently large values of \x\, the absolute value of the last term in where r is an integer which is less than n and greater than o, is greater than the sum of the absolute values of the remaining terms. Ex. 2. Prove that the sum and product of two functions, both continuous at a, are continuous at a. Prove the same for their ratio, when the denomi- nator is not zero. . | 57. Instance and Definition of a Limit. Let us use the example of § 43 in a slightly modified form, and let x and a be complex. The function fx = {x^ — a^)j{x — a) is one-valued and is equal to x + a for all values of x other than a. For x = a the function is not defined at all ; this does not affect the possibility of the existence of a limit when x approaches a. When \x — a\ S, fx — b\<€, then b is said to be the limit of fx when x tends to infinity. If when \x — a\ i /e, it is said that the limit offx is infinity w/ien x tends to a. If when \x\>h, \fx\ > ije, it is said that t/te limit of fx is infinity when x tends to infinity. Hence we see that a limit b can exist even when fa is x"" — a^ undefined as in the case ; further we see that in order x — a that fx may be continuous Sitx = a, there must be one and only- one limit when x tends to a, there must be a definite value fa at x=a, and this value must be equal to the limit. 58. The Derivate of a Function. The limit of which a particular instance has just been given is one of capital import- ance. A slight generalization is effected by taking ;tr" instead of x^, n being a positive integer. Let then y=x'\ and when x = a let y = b,%o that b = a": Then y — b=x" — a"', and O - b)/{x -a) = (;tr" - «")/(;r - a). Write h for x — a; the expression on the right-hand side becomes [(a + hY — d^]lh ; or, by the binomial theorem, «rt"-i + « (7? - I ) a"-2 /«/2 ! + + nali^- + /?"-i. By § S6 the difference between this expression and its constant term can be made as small as we please by confining h (4= o) to a suitably chosen small circle about h = o. Therefore by the definition of a limit lim {y — b)/(x — a)= na"~^. The limit thus found is called the derivate of x^ for the value x = a. Since a can take any position in the ;tr-plane, we may replace a by x, and say that tlie derivate of x^ at t/ie point x is Let us generalize and take y =fx, where fx is restricted 57-58] THE RATIONAL ALGEBRAIC FUNCTION. 89 provisionally to the meaning 'a polynomial or fraction in x.' Let fx have a definite value fa at a ; the limit, if existent, of ifx —fa)/{x — a) for x = a is called tlie derivate of fx when x = a, or the derivate of fx at a. Writing x for a we have the law of the derivate {or general values of x, excluding certain special positions. The derivate is a new function of x, — where the word function, like the symbol fx, is used only temporarily in the restricted sense of polynomial or fraction, — which is closely related to the given function ; it is denoted sometimes by an accent fx, sometimes by the capital initial letter D of ' derivate,' Dfx. A third notation is dfx/dx ; the origin of this notation, and of its accompanying name the differential quotient, may be worth explaining. The definition of the derivate is in words " the limit of the . change of function , ,, , r ^t. • ui i. j ratio -; ^ r ^-TT-, when the change of the variable tends change of variable to zero." Call this limit, — supposed to exist,— ^Z?;^. We define the differential of a function for an assigned x by the equation dfx = Dfx X (an arbitrary change of x). If the function be merely x itself. Dfx = i and the differential of X, namely dx, is simply the arbitrary change of x. The equation may be rewritten therefore as follows : — dfx = Dfx X dx. Thus the ratio or quotient of the differentials dfx,dx is the same as the derivate Dfx. The formal rules for constructing the derivate of fx are the same whether we use complex or real values. Thus, when the definition of a function applies to complex values of x as much as to real, the derivate will have the same form in both cases. The existence of a derivate which is not zero ensures isogonality (§ 26). Taylor's theorem for the rational integral function, namely that f:,=fa+f'aix-a)^ra^^+...+f-a^^^. is proved precisely as for real variables. The theorem enables 90 THE RATIONAL ALGEBRAIC FUNCTION. [CH. VII US to arrange a series of powers of ;tr in a series of powers of x — a, in other words to change the origin. 59. Fundamental Theorem of Algebra. The funda- mental theorem to which we refer is that every equation fx=a^+ a^x + a^x^ + . . . + <2„_i ;ir"~^ + fl„;ir" = O, in which the ds are complex {real values of course included) and an 4= O, ]ias a root. With this proposition stands or falls the associated theorem that every algebraic equation lias n roots. It is obvious that if the first theorem be not true, the second is not true, let us assume then that the first theorem is true and prove that the second follows from it. Let _/iir = o be satisfied \iy x = x^. Then fx = a^-\- a^x + a^x"^ ■\- ... +a„x'*, = a„ + a^x^ +a^x^+... +a„x/'; hence /x = a^(x-xj + a^{:(' -x^'') + + «„ (jr» - jr,»). As every term on the right-hand side is divisible by ;ir — jr,, Jx must be divisible by ;ir — ;r,. Let us write then /x = (x-x;)(d,+ i>^x + d^x'+...+ ^^1 Jr"-0 = (^ - ^,) /.^. where the second factor is necessarily of degree n— i. By hypothesis,yjjr=o is satisfied by some value ;rj ; a repetition of the preceding argument shows that /^={x-x^)f^, where /^ is a rational integral function of degree n — 2. By continuing this process we arrive finally at the formula /X = ar,{x-X^) (X-X^) ... (X-Xn). This formula shows at once that ^ vanishes for x = x^,x^, ...,Xn and for no other values of x. In the next article we shall give a rigorous analytic proof that every algebraic equation has at least one root ; in the present one we adduce some geometric considerations which throw light on the analysis. When X traverses the whole of the ;r-plane, y must traverse S8-6o] THE RATIONAL ALGEBRAIC FUNCTION. QI the whole or part of the j/-plane ; in the former case one of the values of y is o, in the latter there are certain regions of the jz-plane which are never reached and one of these regions may contain y = o. When X is at any point a, such that /'a ^ o, let j be at ^. If ^ is zero, the equation has a root ; suppose then that b^o. Then y can be brought nearer to the origin than b is. For by the property of isogonality angles at b are equal to angles at a; hence by a proper choice of the direction at a we can make the direction at b what we please. The condition f'a^o ensures isogonality. The points a excluded by this condition are at most n — \; these points, being finite in number, can be avoided. Thus we can bring y nearer and nearer to the origin ; but this does not show that y reaches the origin. 60. Proof of the fundamental Theorem. We have proved that there exists a circle, with origin as centre, which is such that for all points on and exterior to this circle, 7 being an arbitrary positive number assigned in advance. All the roots of the algebraic equation, in case they exist, must lie within this circle. Let a be a finite value of x and let b =fa. . Suppose that b^o. I. We propose to prove in the first place that there is a value a^■h\x\ the jr-plane which gives a point y whose distance from the origin is less than that of b ; that is, which makes |/(. + .)|<|/«|,or|/(^+^^ < I. To take the most general case assume that in Taylor's expansion the coefficients /'a, /"«,..., /'-'a vanish, while /V + o. Then f{a + h)_ hr f^a h^+' r+'a , ^ /^ 'fa ~ r! fa^r+\\ fa ^'"^n\ fa' I f*a Let h = p{cos6^-isme), j^y- =p,(coses + tsin0,), 92 THE RATIONAL ALGEBRAIC FUNCTION. [CH. VII where s = r, r+ 1, ...,7t; then /{a+ /^ ^ J ^ prp^ |-j,og (^^ ^ 0^y ^ ,• sin (^^ ^ ^^)J fa + p''+Vr+i [cos (r+ I + 0r+i) + i sin (r+1 + ^^+0] + • ■ • + p"Pn [cos {nO + On) + 2 sin {fid + On)}. Now let 6 be so chosen as to make rd-v6r = ir; then fa I — p''pr + terms m /3''+\ p ,r+l _r+2 whose absolute values are p^^^pr+i, p^'^^pr+2, ■■■> P^Pn ', and let p be sufficiently small to secure the inequality p^pr< i- The absolute value of the expression on the right-hand side cannot exceed I - p'pr + p''+-'pr+i + ... + p^'pn-, that is, f{a + h) fa S I ■P Pr Pr Pr — pn-r Pn PrJ By the theorem of § 56, the expression Pr pr Pr can be made to differ from unity by as little as we please, provided we choose p sufficiently small. It is therefore possible to find a value of /i such that P Pr i-pei±^-... Pr is positive and less than unity. For this value we have \f{a + /i)\<\fa\. The proposition that we have established can be stated as follows : — Given t/iat | fx \ does not vanish for an assigned value x=a, it is always possible to alter x so as to diminish \fx\. II. Secondly let us consider the lower limit 7' of the values of |y^|. Since |^| is a continuous function of the real variables f, ij, this lower limit is attained and is therefore a minimum 6o-6l] THE RATIONAL ALGEBRAIC FUNCTION. 93 value of Ij'^I (§ 52). If possible let this minimum value 7' be different from zero. Since there is a value x for which |^| is 7' exactly, it follows from what we have shown above that it must be possible to find a value of |_/i:| which is less than 7', contrary to the supposition that 7' is the lower limit of the values \fx\. Hence 7' must be o, and the value o is attained. This proves that there exists a value of x which makes fx vanish. Note. The theorem that the values of \fx \ can be made smaller and smaller by successive alterations of jr is not, by itself, sufficient to establish the existence of an x which makes fx = o. The initial value \/a | might for example be 3 and the diminishing values produced by the successive alterations might be 2J, 2J, 2J, 2^,..., a sequence whose lower limit is 2. As soon, however, as we are certain, from our knowledge of the properties of continuous functions of two real variables, that the lower limit of this sequence is attained by \/x\ for some value of x, we see that still smaller values can be attained, and that 2 is not the lower limit of the complete system of values of \/x\. Thus the lower limit of this complete system is o, and drawing again on the above-mentioned property of the continuous function [/x\, we know that the value o is attained. Here is a concrete example of the necessity of examining whether a variable quantity does or does not attain its lower limit. Since every algebraic equation of degree « in ;ir is satisfied by n values of x it follows immediately that the equation ^Str— j/ = o gives n values of x for each value of j/ ; this justifies the statement of § 56. 61. The Rational Algebraic Function of x. Let jy =/x = (flo + a^x + a^x^ +...+ a„;tr")/(3„ + b^x + b^x'' ■¥...+ b^x'^) ; it is evident that y is in general a one-valued function of x. To make this statement universally true we suppose the fraction in its lowest terms ; further we must assign one value to j/ when x= , and regard j/ = 00 as a value as in § 20. " The value of jc when ;ir= 00 " is in itself meaningless, because 00 is not a definite number, but lim^ often exists and then we define value^ to mean Hm^. Thus in the present case y has one value when ;r = 00 ; 94 THE RATIONAL ALGEBRAIC FUNCTION. [CH. VII for when \x is sufficiently large anx", b^x"^ become the all- important terms of the numerator and denominator and their ratio tends to oo , a„/^m, or o, according as «>,=,< m. When n>m we can divide the denominator into the numerator, getting a quotient Co + c^x+C2X^+ ... -\-Cn-mX^~™ and a remainder d^ + d^x + d^x^ + . . . + ^m_,;ir"^^ Hence ^0 + d^x + d^x'' + . . . +^™_,jr™-> J/ = ^0 + ri;r + T^;*:^ + ... + c„_^;tr"-^ + b^ + b^x+ b^x"" + . . . + b^x'^ Partial Fractions. If x-^, x^, ..., x^ be the roots of the equation formed by equating the denominator to zero, these roots occurring to the orders of multiplicity m^, m^ m^, so that »2i + »22 + . . . + vt}. = nt ; then, as we know, the fraction can be resolved into partial fractions and y takes the form Co + c^x + c^x'^ + . . . + t„_m^" + 2 r=\ A„ A„ ^"», "I X — Xj. (x—XrY '" (^ When n = m, the part external to the sign of summation dwindles to a^lbm, and when nm\ and the part C„ -H C^X -I- f^jr^ -1- . . . + Cn-mX'^~^ 6l] THE RATIONAL ALGEBRAIC FUNCTION. 95 exists and its most important term is Cn-mX"-^ when |^| is large. We say that n — m is the order of infinity of y when x=co. Let us frame the definition in such a way that it may be applicable also to other one-valued functions than the rational. If we divide by x'^-^ the function ceases to be infinite when ;»r= 00 ; it takes in fact the value c„_m. But the function, when divided by any lower power of x than the {n — m)th, is infinite when x= aa . We say that the function has an infinity, at x=«i , of order s, where s is a positive integer, when tlie function fxjx' is finite for x=n the function has a zero at x=ao ; this zero is said to be of the order m — n, just as when m • • • ) and the act (whether of thought, writing, or speech) by which we pass from the sequence of terms to the sequence of sums, — this is the infinite series. 63. Convergence. The sequence s^, s^, ..., Sn, ... may tend to a finite limit s. When and only when this is the case M. H. 7 98 CONVERGENCE OF INFINITE SERIES. [CH. VIII we say that tlie series is convergent. In this case the series leads to a number, namely s. When there is not a finite limit the series is divergent. Mathematical notation is concerned es- pecially with what results from operations. When we write a-\- b, the operation is the replacing of two numbers by a single one ; the notation, so soon as we are used to it, suggests that single number. A notation introduced to signify an operation ends usually by signifying the outcome. So in this case we first agree to denote by «! + ^2 + . . . + a„ + . . . , by S«„, or by San, the series or operation itself, and then when n=l the limit s exists we agree to use the same notation for s itself Lastly, as an almost unavoidable result of using the same notation for both the limit and the operation, the limit of a convergent series is frequently spoken of as the series itself. But it must not be overlooked that when there is no limit we must recur to the definition of the series. Note. In most English text-books a series is said to be divergent only when it tends to c» . Thus i — i-f2 — 2-I-3 — 3-I-... is (in this view) neither convergent nor divergent. The definition adopted here has the support of Cayley {Encyc. Brit., Art. Series) and Stokes {Camb. Phil. Trans, vol. viii. p. S3S), among English authorities, and is prevalent elsewhere. We use L to signify the limit when « tends to 00 , if this limit is known to exist. We mean by a real series one with real terms alone ; by a complex series one with complex terms. A necessary and sufficient condition for the convergence of S(2„ is that corresponding to every positive number e given in advance there shall exist a positive integer /a such that the absolute value of «7H-i + '^n+s + .. ■ + «n+p shall be less than e for every integer n equal to or greater than fi, and for evety positive integer p. That is, I SnJrp — ■?« | < f when n^ ft,. See § 47. The one integer /u. is to serve for every selection of «(5/x.) and of/. Up were assigned, instead of being free, the condition 63-64] CONVERGENCE OF INFINITE SERIES. 99 of convergence, though necessary, would cease to be sufficient. For instance in the series i + 1/2 + 1/3 + 1/4+ ..., the sum of an assigned number of terms following the «th can be made as small as we please by increasing n ; whereas the sum of n terms following the «th is at once seen to be greater than 1/2. When in calculating the limit of a convergent real series we decide on a degree of approximation, — say we want to be within i/io' of the limit, — the least corresponding number /i. measures the rapidity of the convergence. The expression j — j„ is called the remainder or residue of a convergent series. It is- of course itself a limit; it will be denoted by r„. Since y. s = s,, + rn, n'-> and Lj,i = s, it follows that Lr„ = o. 64 Simple tests of Convergence for Series with terms all positive. Whether a series is convergent or not can be decided in some simple but important cases by a com- parison term by term with a standard series. An especially convenient standard series is the geometric series Sjt" This is convergent when x is any number, real or complex, such that \x\< I. For since we have 1 s,, — I -X': \ I -X\ 1 ^]n+i or if I .r I = . where a > o, ' ' I +a- I I and ^ < e when « > 0(1+ a)" u{i+fta) I — ae 0(1 +«a) ae Here then /. is any integer >^-^ < and there is the limit I —X lOO CONVERGENCE OF INFINITE SERIES. [CH. VIII (i) Let 2rtn nnd 2a'„ be series with terms all positive and let 2a:„ be convergent with the limit s. If a'n = an, from an assigned valne of n onwards, then also S^'n is convergent. For if s'n denote the sum of n terms of the accented series we have .a„, 'Sa'n is divergent. (2) If^a^ is convergent and if when n = m ^ n+l ^ ^n-^-i a n a^i then 1a',i is convergent. For if ^ ?"^^ ^ '^^"+^ and so on, then by multiplication, -;- ^ — , when 72 ^ ;;/. CO 00 00 Since S (2,1 is convergent, so is S ^^ and so is S ajam- n=l 71=971 n=m Hence, by the preceding theorem, 2 ci'^j^'m is convergent ; and n=7n ao oo therefore so are S «'« and S d^. n=tn n=\ Ex. If 5«„ is divergent and, from an assigned value of k onwards, iL!i±i > ?2±i then 2fl'„ is divergent. (3) fF/?^« L rt„+i/a„ f;t:/j^j awi/ w /,?jj ^/«(2« i, the series is convergent. 64-65] CONVERGENCE OF INFINITE SERIES. lOI Let the limit be i — o where a. lies between o and i ; then to every e there is a /i such that when « S fi, I -a-e< - -< I -a + e. «n Choose a definite e, < a, so that i -a + €< i. Let the corre- sponding fi he m; then comparing a„ with a geometric series whose constant ratio is i — a + e we see that the conditions of the preceding theorem are satisfied. Ex. When L -^i* exists and is greater than i, the series is divergent. If the limit is i or if there is no limit, other tests must be used. .See § 76. (4) W/ien L a,i"" exists and is less than 1 , the series is convergent. For proceeding as in (3) we have «„'" < I — a + e < I, n = ni. Since «„ < ( i — « + e)" and since the geometric series whose «th term is (i — a + e)" is convergent, the conditions of (i) are satisfied. 65. Association of the Terms of a Series. By the sum of two series Sa„ and 2a'„, real or complex, we understand S (a^ + T'm)- It is clear that, if ] i-„+p — j„ 1 < 6 and j /,,+p — /„ j < e, then |j„+p + /„+p-J„-/„|< 2e (§ 14). Hence the limit of 2 («„ + «'„) is the sum of the limits of 2a„, 2a'„. The theorem that 1 2(i:„| < 2 |rt„| is of great use in handhng series. 1 1 Observe that if 2 | a„ | has a limit 5 all that the theorem tells us is that if also 1 2a„ I has a limit U | , then \s\: 3. Take the terms by threes, that is, consider the series (I + 1/3- i/2) + (i/s + 1/7-1/4)+ (3)- We prove first that this third series has a limit. Let a„ denote i + 1/2 + 1/3 + ... + i/«- Then a„/2 is 1/2 + 1/4+ ••• + i/2«, ando2„-or„/2 is i + 1/3 + i/S + ... + i/(2«-i); so that if Jjn is, as above, I - 1/2+ 1/3- ...- l/2«, The sum of 3« terms of (2), that is of n terms of (3), is ■ /an = I + 1/3 + •■• + i/(4« - I) - (1/2 + :/4 + • •• + i/2«) = Clin — 0(2n/2 — ^n/S = ^471 + ^ml^- I04 CONVERGENCE OF INFINITE SERIES. [CH. VII [ Hence L/3,1 = L jj„ + Ls.2,iJ2 = Log 2 + J Log 2 = fLog2. Thus (3) is convergent, but has not the same sum as (i). We have now to see whether we may remove the parentheses in (3) so as to get (2); that is whether (2) has the same sum as (3). The sum of 3« +1 or of 3« — i terms of (2) differs from the sum of 3« terms by i/(4w-f i) or by i/2«; that is by a number which has itself the Hmit zero. Thus in the series (2) the limit is the same when we take three terms at a time, whether we begin our sets of 3 with the first, second, or third term; but this amounts to taking the terms one by one. Ex. I. Prove, by means of e and fi, that when La„=o and the series is convergent, so is the series Sa„. Ex. 2. Prove that if we take first p positive terms of the series for Log 2, in their natural order, and then $■ negative terms and repeat this process indefinitely, the sum is Log 2 + J Log ^/f. The easiest way to prove this is to assume Euler's theorem that i + 1/2 + 1/3 + ...+ i/« — Log « has a finite limit when w is « . See § 109. 66. Absolutely Convergent Series. Let !«„ be any convergent series ; and let the terms be rearranged. When the first series is finite, the two series have the same sum ; when the two series are infinite (if one is, so is the other) it is by no means necessary that they have the same limit; and it is of cardinal importance to know when this is the case. First let 2fl„ be a convergent series with positive terms. Let the new series be 2(z'n ; that is, the terms ^IJ ^2) ^'sj •'•J ^?ll ••• are rearranged in the order ^ It ^21 ^3j •••» *^7H ••• J the term in every assigned place in the old series has a definite place in the new ; and conversely. There is to be no omission and no repetition ; but for the same term there is to be a one-to- one correspondence of the old place and the new. 65-66] CONVERGENCE OF INFINITE SERIES. 105 Take now n terms of the old series ; these will be found in the new series, say they are in the first m terms a\, a\, ..., a',n, where m^n. And all these m terms will be found in the old series, say in the first n +p terms, where n +p = m. Hence if j„ and s'n denote the sums of « terms of the two series, s,i = Sjfi^ and j^ = i",j_^p. Since then L j„= 'Lsn+p, =s say, L/^ is neither greater nor less than s, that is, L s'm = s. Next let 2«„ be any convergent series, and let A„ be the absolute value of of the old, where « S ;« S « +/. We can no longer assert that j„ S s'm, but we can assert that the terms which are in s'm, but not in j„, are all among the terms «»+!> «n+2, •••) «n+p- That is s'm — Sn is made up of some or all of the terms in s„+p — j„ ; and therefore \ s'm -Sn\ = the sum of some or all of the numbers A^+i, An+2, ■■•, ^«+j). and a fortiori S the sum of all these numbers. If then An+i + An+2+ ... +^,i+j, can be made as small as we please by suitably selecting n, we can assert that I •^ m -S^n I < ^; and therefore that L /^ = L j„. We are thus led to a classification of series. For any convergent series ! ^n+i + ^ii+2 + • ■ • + ^n+p I < ^ > but if further the absolute values of the terms form a convergent series, then An+r + ^n+2 + ...+An+p ^31, — It is important to notice that the i, i corresponderice provides against omissions or repetitions of terms. Suppose that a (i, i) correspondence has been established between the single numbers i, 2, 3, ... and the pairs (/>, q), and that in this way the terms of an absolutely convergent series Sa„ have been arranged in rectangular array, the following two questions suggest themselves: — (i) Is the series formed by each row absolutely convergent .' 68] CONVERGENCE OF INFINITE SERIES. 109 (2) Granting that such is the case, is the series formed by the sums of the first, second, third, .... rows absolutely con- vergent, and is its sum equal to that of the given series ? These questions are answered by the following theorems : — Theorem I. W/ien infinitely many terms d,ni, ^ma, ^ma, ■■■, are selected from tJu absobitely convergent series S«m, the series 2/5',„,i is absolutely convergent. For the more terms we take the greater is 2 | bmn \ ', but it is less than S | «„ |, which by hypothesis is finite. Hence, by the theorem of § 45, 2 | b^n \ has a limit. Theorem II. If by means of a i, i correspondence betweett tlie numbers n and the pairs {p, q) we arrange tlie terms of tlie absolutely convergent series 2«„ in the rows b^, bi^, b^, ... to infinity, b^, b^, b^, ... to infinity, b^i, b^i, b^^, ... to infinity, and so on, then the series co7nposed of tlie sums of tJiese rows is also absolutely convergent and has the same stem as the original series. By Theorem I. the series bm\-Vbmi-\-bim+ ... is convergent; let it have the sum bm- The series that we have now to consider 00 is 2i^m- Consider the/ rows following the w«th one. 7n = l ■Let 1^m,v ^ ^m+l "I" ''m+2 "r ^)n+3 + • . • + ^m+p ! and let the a with minimum suffix used in the p rows be a„+i. Then y^.p consists of terms selected from r„, where r„ = fl„+i + a:„+2 + a„+3 + ... to infinity, and we infer that I ^m, p I < 1 ^71+1 I + I ''^n+a 1 + i '^n-^-i \ + • ■ ■• As the expression on the right-hand side tends to zero when n tends to infinity, and as n tends to infinity with m, it follows that 'Lrm,p = o, independently of the value assigned to p. This establishes the convergence of the series tbm- To complete the proof of the theorem let Sn = ai + an + ...+an, tm = bi + b-2 + •■■ + ^m, and let a„+i be the first of the a's which does not occur in t^. no CONVERGENCE OF INFINITE SERIES. [CH. VIII The difference A„ — j,j consists of terms a selected from those that occur after a,,, and therefore tm~ ^n\<- I '^n+l | + | <*n+2 ] + | (^n+s I + Hence, when m and « tend to infinity, \tm — ^n\ tends to zero, and we have This proves that the sums of the series S«,j, S^^ are equal. 69. Conversion of a Double Series into a Single Series. In the preceding article we have converted an absolutely convergent single series into a double series ; let us now examine the reverse process. Given an array of elements \b^q\, or Bpq, let tlie rows yield convergent series with sufns Bi, B^, B3, ... atid let tlie series SB^ converge also. Then the series formed by adding the sums b^, b^, b^, ... of the rows in the array of elements bpq will converge absolutely and so also ' will the single series b\i + bji + b^ + ^13 + ^22 + ^31 + • • • ; and these two series will have tJie same suin. By the conditions of the theorem each row in the array of ^'s yields a convergent series ; hence the rows have sums bi, b^, b^, .... Since ! "ml + ^?n2 + i>m3 + . • • | = ^m\ + ^ma + J^rra + ■ ■ •, we have \bm'\ = B^, and therefore 1,bm is an absolutely con- vergent series. Reference to the second array of § 68, shows that bx\ + ^12 + ^21 + ^13 + ^22 + i^si + • • • + b-ji + b^_^ + . . . -(- (Siftj is the sum of the terms in the first h diagonals. We have to show that the series continued to infinity is absolutely con- vergent. Consider the rectangle of terms B^^ where p ^ terms of (i), n+i-ni) + • ■ • is absolutely convergent, and its sum is the product of the sums of the given series. Deduce the binomial theorem for a negative integral exponent. CHAPTER IX. UNIFORM CONVEEOENCE OF REAL SERIES. 70. The need of a fUrther classiflcation. Hitherto the general terms of our series have not been considered as dependent on a variable x\ now we shall make the terms dependent on x. There is clearly a difference between the problem of convergence of say Xar" for an assigned x and that of the same series when x is treated as a variable ; for in the latter case we have to consider, not a single series, but an infinity of series arising from the various values of x in some assigned interval or region. We consider in this chapter the series /i^+/2^+/3f+ •••+/»?+•■•, where the terms are functions of a real variable as defined in ch. VI. § 48 ; these functions are supposed to be one-valued and continuous for the interval considered. The remainder after n terms will be denoted by r„(f). The geometric series 2 p converges only when | f | < i ; this n=0 shows that the convergence of the series is to be considered for intervals, — the open interval (—1, i) in the present case, and CO for the general case 2yj,f intervals such as a g f s /3, a < f S y9,\ n=0 , ogfLog,„i/|f|' when f is given. But equally we can always choose ^ so that Logio i/e '^'^Log.oi/I^l' when fju is taken arbitrarily and then fixed. The point is that the least /i will in general depend on f and the values of the /t's for all values of ^ in the interval may or may not have a finite upper limit. Only when this finite upper limit exists is the convergence uniform. Non-uniform convergence in an interval is due to what is known as infinitely slow convergence near certain points of that interval. For an assigned f let /tj, or briefly /*, be 8—2 1 16 UNIFORM CONVERGENCE OF REAL SERIES. [CH. IX the least number of terms of a convergent series which will make | r„ (f ) | < e for all values /« S /i ; then /it is a measure of the rapidity of the convergence. If for example we wish to calculate from a series the value of tt with an error i/io' we should say that of two series used for this purpose that one converges the more rapidly which needs the fewest terms to furnish the desired result. Hitherto ^ has been fixed ; now let it traverse an interval in which the series converges but not uniformly. As ^ tends to certain values in the interval we have to take more and more terms without limit to secure the desired approximation. The convergence near these points is said to be infinitely slow. Thus in the example 2p(i— |), the con- vergence is infinitely slow as ^ approaches i. It must not be supposed that because /x tends to oo as ^ tends to i, therefore /x is infinite when f is i. In fact when f is I, each term of the series is o and therefore \rn{^)\^- -- and in order that | r„ (^) | may be < e/2 when « ^ /i, we must have (,.+ i)|r|>l/e + (1/6^ -!)>'=, where we suppose e < i. Here as f tends to o, /i tends to oo . Ex. Prove that if a series is uniformly convergent in an open interval (a, /3) it is convergent at a and /3. 73. Uniform and Absolute Convergence. It must be noticed that uniform convergence does not imply absolute convergence, nor conversely. Thus the non-uniformly convergent real series (i-f)+?(i-?)+r(i-f)+-- is absolutely convergent in the interval O S f S I. On the other hand take any uniformly convergent series. If it is absolutely convergent we can at once turn it into a series conditionally convergent throughout the interval considered, without altering the sum or the uniformity of the convergence, by adding to each term the corresponding term of the series I-I + l/2- 1/2+1/3- 1/3 + ... which converges conditionally to zero*- A sufficient but not a necessary test for the coexistence of these two kinds of convergence is given by the following theorem : — * See a paper by Osgood, Bull. Amer. Math. Soc. 2nd ser. vol. iii. p. 73. This paper will be found valuable in the way of clearing up many of the diffi- culties of the subjects of limits and convergence. 72-74] UNIFORM CONVERGENCE OF REAL SERIES. 1 19 The series S /"«? is absolutely and uniformly convergent in an interval when the absolute values of its terms for that interval are less than the corresponding terms of a given convergent series 2 «« whose terms are real and positive. The series converges absolutely by § 64. But further the two inequalities a„^.i + a»+2 + . .. < 6 (« = /i), and kn(f)|S|/n+i| + |/n+2| + ... show that I r„ (^) I < e, whatever be the position of ^ in the interval. This establishes the fact that the convergence is uniform in the interval in question. Ex. I. Prove that every series obtained by multiplying the terms of an absolutely convergent series 2a„ by functions/i,^_/3,... of the variable ^ which have finite values within an assigned interval, converges absolutely and uniformly throughout that interval. Prove in particular that cos X(,| — i cos Xi^ + J cos Xjf - J^cos X3I+ ... is absolutely and uniformly convergent in every closed interval of the axis of real numbers. Ex. 2. Prove that 2 ( — i)"->f ~V«^ 's absolutely and uniformly convergent in the closed interval (-1, i). Ex. 3. Prove that the series \-irx 7.x iifi 2x'"-' :+Zs — : + z2 — : +•••+- i-x X^-l X*—l i^"— I is uniformly convergent along a circle whose centre is o and radius < i. 74. The real Power Series. We shall discuss the case of a power series, that is, a series of ascending positive integral powers, a„ + aif+a,p+.... Suppose that it converges absolutely for a given value fo- Then by the theorem of § 73 it converges absolutely and uniformly for every | such that | f | < | fo i- So far as absolute convergence of the above series is con- cerned, the following theorem gives important additional infor- mation. I20 UNIFORM CONVERGENCE OF REAL SERIES. [CH. IX If for a given value ^o of ^ we have, for every n, |a»?o"|^7. where r-^ is a given positive number, then tlie series is absolutely convergent for all values of ^ such titat | ^ i < ||o I- For |ann<7l?/for, whence the series of absolute values is less than the geometric series 7(i+ir/ioi+ir/roi^+-x whose sum is 7/(1 ~ I ?/?o |)- The additional information is this: that if the series converges, absolutely or not, for %=%a, it converges absolutely for | f | < ! f„ I. For we can extract from the known fact of convergence at |o an inequality |a,.fo"|<7- If this be not obvious, observe that the inequality I Onfo" I < fo « S /i., where eo is any assign^ positive number, combined with the existence of a maximum value among the /i terms lool, |ai?o!, •••, k^-ifo"-'!, implies such an inequality provided that 7 > 60 and 7 > the maximum value just mentioned. It follows that if the series diverges wlien ^ = fi, it diverges wlun 1^1 > I fi|. For if it converges for a value of f such that I ^ I is greater than | f 1 1 it must converge for f, itself. There must then be a positive number p such that Somf" converges absolutely when \^\< p and diverges when \^\> p. We know then the interval of absolute convergence of the series (i), namely it is the interval (— p, p); but the interval will be closed or open, that is, it will or will not contain the frontier points p and — p, according to circumstances. We know, so far, that (- p + y8, p — /3) will serve as an interval of uniform convergence ; here /3 is any assigned positive number less than p. The notion of uniform convergence once firmly grasped it will be evident that it does not matter whether we make (— p+ ^, p — ^) open or closed or partially open ; for 74] UNIFORM CONVERGENCE OF REAL SERIES. 121 uniform convergence in any one case is accompanied by uniform convergence in the others, since there is convergence at the ends. We have now to examine the more extended interval Now if the series is divergent at p and at — p we have to leave the interval as it stands; it cannot come up to the point p, that is, nearer than any assigned distance /3, for (§ 72) uniform convergence up to p would imply continuity of the sum at p. But if the series be convergent at p the interval does come 7ip to p. Take first a special case, the logarithmic series ?-r/2+f/3- (0- That p=i appears from the fact that when f = i we have the conditionally convergent series Log2 = i -1/2 +1/3 -1/4+.... Let j„ denote the sum of the first n terms of this series. Then the coefficients of (i) are j,, s.^-s^, s^-s., ..., and the sum of the terms after the «th is rn (?) = (Jn+i - Sn) f"*' + (^»+= " -fn+i) P^' + • • • = - J„ r+' + r+'( I - ?) (-f-i+i + -J n+2 ? + -fn+s f + • • • ), or, for every ? such that | ? | < i, ^« (?) = r+' ( I - ?) [^«+I - J" + (•fn+. -Sn)^ + {Sn+. - Sn) ? + • • • ] (2). Now we know that \sn+p-Sn\<€ when nS.fi; hence the absolute value of the expression (2), when o < ? < i, is less than that is, <£?»+» I jr, |. Hence there is a frontier value R such that when | ;i: | < .^ there is absolute convergence but when |;ir|>.^ there is diver- gence. That is, within the circle (7?) the series is absolutely convergent and without the circle it is divergent. The circle {R) is called the circle of convergence. The open 126 POWER SERIES. [CH. X region (J?) is called the domain of the series. This domain supplemented by those points of the circle at which the series is convergent gives the region of convergence. With regard to R we must mention the two extreme cases R =o, R = . Series for which R = o, — i.e. which converge only when x = o, — are to be cast aside as useless ; series for which R = 00 are of great importance ; they will be considered in detail later on in this book. The following theorem gives useful information relative to the convergence of Px. If for some positive number g and for a certain value x^ of x we luxve AnX^^ zle. n Hence when |j?|=i and x^i, \sn+p — Sn\ satisfies the original criterion for convergence. But since tijn is divergent, the convergence is only conditional. Hence tke logarithmic series Ix'^jn is conditionally convergent at all points of its circle of convergence except at x= i , wlure it is divergent. Ex. The series ;r+rV2+;r'/3+r4/4+ar«/s+j:l/^. 77. Uniform Convergence of Complex Series. In speak- ing of a power series Px we understand, unless the contrary is specified, that the ;r is a point within the circle of convergence. Also Px will be often used to denote the sum of the series, where that number exists ; to repeat an earlier remark, the same symbol denotes very conveniently, both a definitive process and the number that results from that process. het /yf, /^x, ...,/nX, ... be functions (defined or to be defined) which are one-valued and continuous in a region F of the ;r-plane ; and let the series /iX+/2X+ ... +fnX+ ... converge at all points of F. Then if j„ (x) be the sum of the first n terms we can, for an assigned x, find a positive integer /j. such that \sn+p(x)-s„(x)\rX^^ , where the exponents are positive or negative integers, tfien \b^\=g, where g is the maximum value of I fx \ on the circle (p). Let be chosen as in the lemma ; construct the function _ fix) + /(fa) +f(d-x) +... +f(d-^x) ^ ' ' n ' where n is a positive integer. This function is equal to -{nba + b^x'^^{i +^'"■ + 0™" + ... to n terms) + ... + drX""-{i +6"^ + 6'"^+... to n terms)], that is, to ^ijr™' ^""'' - I ^jjr^^"^-! b^x"^0"'^-i "'^ n ^"'"^i "^ n e-^-\'^"'^ n ^"-- 1 ' The absolute values of 6™^\ 0^'^', etc., are equal to i ; consequently the second factors in all the terms after the first are finite, for their denominators are finite and different from zero and their numerators do not exceed 2 in absolute value. Hence when n tends to infinity, <^(.r, n) tends to the limit b„. Since each of the terms f{x), f{6x), ... is less than g in absolute value when x lies on (p), it follows that | <^ | &g; there- fore \b^\&g. II. Let us consider now the power series Px = a^ + a^x + atX^+ ...+a„x'^ + .... We find, after division by ;r", that x-^Px = {a,x-^ + ajjr-»+i +...+«„ + an+^x + . . . + a^+^xs^^'') 78-79] POWER SERIES. I3I The infinite series in the lower line can be made as small as we please in absolute value by taking h sufficiently great, whatever be the position of x on the circle (p). Thus I x~^^Px I £ I the expression in the upper line | — e ; that is, 6 + I x~^Px I ^ I the expression in the upper line | £ ^„, by the first part of the proof But the maximum value of \x~^Px\, for the points of (p), is p~^g. Hence An&glp". .which is the theorem that we set out to prove. Notice that g is the maximum value of | Px \ taken over a line (a circle), not over a region. When the meaning of ^ is changed to 'a number greater than the maximum value of \Px\ on (p),' we have AnKg/p". The above theorem has been stated only for the ordinary series ; but it can be extended with ease so as to cover the case so of a Laurent series S a^x'^. The domain of convergence is no longer circular but annular, consisting namely of the points exterior to (.^1) the circle of — oo convergence of 2 a„x^ and interior to (R^) the circle of 00 convergence of 2 anX", Ri being supposed less than R^. The «=o number p is chosen so that Rio. Then Cauchy's theorem states that An=glp^- Hence for points x in the open region (p) we have ^gX/(p-X). Hence if gXI(p-X) O : W/ien Uo vanisJies and a^, a.^, ... do not all vanish, there is an open region {X^ within which Px vanishes at no point other than o. For let Px = x!^ (a,i + a,j+jj: + a^j^^^ + •••)> where n > o. The series in parentheses has the same circle of convergence as Px ; furthermore it has a constant term which does not vanish. Hence by the last theorem there is an open region (X^) within which the series does not vanish. Evidently, then, this region satisfies the requirements of the theorem enunciated above. 80. Criteria of Identity of Power Series. 1/ Px vanish within every circle (p), where p is arbitrarily small, at points distinct from O, tlte coefficients of Px all vanish. This is a direct consequence of the theorems of the preceding article ; for they tell us that when the coefficients are not all zero there can be found a circle (p) at no point of which, excepting perhaps o, does Px vanish. We define a zero of Px as a point at which Px vanishes. This definition accords with that of § 61. The following distributions of zeros require that all the coefficients of Px shall vanish : — 79- 8o] POWER SERIES. 1 33 (1) They fill a region which contains o ; (2) They fill a curve which passes through or terminates at o; (3) They form a system x^, x^, x^, ..., x^, ... to infinity, such that there are numbers of the system X^, X„, X^, ..., Xn, ... to infinity which are greater than o and less than e. An example is afforded by the system jr„ = (— i Y~^jn. In (i) and (2) a^ = o because Px = o when x = o. Let Px = xQx; we must not assume that x = o satisfies Qx = oand prove that aj = o by putting x = o. The proper proof is the same as that by which we show in (3) that ^0 = 0; namely Px cannot be continuous at ;r = o, if Po^o, while Px= o at points arbitrarily close to o. To meet all cases, we need a new word. When the points of an infinite system are distributed, as in (i), (2), (3), so that one or more points, distinct from jr„, lie within {x„, e) however small we choose to make e, then the point x„ is said to be a liitiit-point of the system. The above theorem can now be stated thus : — If the zeros of Px have o as a limit-point, the coefficients of Px all vanish. The case where the zeros are infinite in number, but do not have o as a limit-point, will be considered in § 88. The following theorem is an immediate consequence : — 00 00 Wlien the equality 2 a^"" = 2 b,^'- is satisfied by values of »=0 n=0 X which have o for a limit-point, we have (2„ = bn for all values of n. For the zeros of 2 («„ - <^„) ;«■" have o for a limit-point, and therefore an = bn for all values of «. CHAPTER XI. OPERATIONS WITH POWEB SEKIES. 81. Weierstrass'B Theorem on Series of Power Series. Let Uo + Ui + U2+ ... + Ug+ ... be a series of power series with the general term «? = ag, + agiX + ag^x^ + ... +aq„x'^+ ..., and let (/?) be a circle within which the separate terms Ug and the collective series S«g converge. Arranging the series 2«, in the form of a rectangular array in which «, occupies the (q+ i)th row, it is desirable to know whether the sum by- columns is equal to the sum by rows; for the sum by columns is formally a power series in x, and should it be equal to the sum by rows the series of power series can be condensed into a single power series. The following theorem, due to Weierstrass, supplies a sufficient condition, different from that in § 69. Given that tJie separate series Ug atid the collective series 'S.Ug converge within the circle {R) and that the latter series laig C07iverges uniformly along every circle (Ri), where R^ ^mo + 'mi-*^ + 'ma-*" + ••■ + 'mn.*' + (2). The former of these two series converges in the open region {R), for each of its m component series ?^„, «i Mm_i converges in that region. Let us see whether the same is the case for the latter of the two series. 1 36 OPERATIONS WITH POWER SERIES. [CH. XI By Cauchy's inequality, we have Hence, using capital letters to denote absolute values, r,„„ + 7\^,x + T^^x-' + . . . + r„„,x» + . . . g 6 i x-^ir^\ n = But when X < R^ the series IX^/Ri^ is convergent and has the sum RJ(Rj — X); therefore 2 A„,jJ^" converges absolutely for all values ^<.ffi. This means that it converges absolutely in the open region (R) ; for, if x be any selected point of this region, R^ can be chosen between X and R and the absolute convergence at x follows at once. It results from the addition of (i) and (2) that the series QO S «nA-" converges within the circle (R). III. Finally we have to prove that 'S.Ug = 2a„;ir" within the circle (R). Select any value x within (R) and then take R^ intermediate between X and R. For this value of x and for the same value of ;« as before, we have to prove that m — l on 00 CO 5=0 q = m M=0 n=0 or removing the first terms on the two sides of the equation, since these are known to be equal, we have to prove that QO 00 2 ?<, = 2 ^,„„jr" (3). q^m n=0 This can be done very readily by the use of the inequalities that have been found above. For ( i ?/, - i /™„;r» S e + eR^\{R, - X) ; \q=m n=0 and whenever it can be asserted of any fixed quantity that its absolute value is less than an arbitrarily small positive number, that quantity is necessarily zero ; hence (3) is proved. We proceed to some important applications of the theorem. 8l-82] OPERATIONS WITH POWER SERIES. 1 37 82. Remarks on Weierstrass's Theorem. This theorem of Weierstrass's on series of power series in x can be extended at once to series of power series in i/jt. It must be remembered that now the domain of convergence of a power series is the part of the plane exterior to some circle (o, K). The theorem as extended runs : — If the terms of the series 2«,, where converge outside t/ie circle (R') and the series itself converges uni- formly on the perimeter of every circle {R^) where R^ > R', then for every point x exterior to tlie circle {R') t/ie series Xuq is expres- sible as a single power series in i/x. Combining this result with that given earlier we obtain at once a sufficient condition for the composition of a series of +00 series 2 a^x" into a single series of that form. Instead of a n= — GO region interior to (R) or a region exterior to (R'), we have the annular region (/?', R) where R' is supposed less than R. A convenient modification of Weierstrass's theorem replaces the uniform convergence of the series 2 Uq over every circle g = (^j), where Ri < R, by the uniform convergence of the series for the closed regions (.^1). Evidently the former condition is contained in the latter. We shall, in general, use Weierstrass's criterion for the con- version of the double series Xug into the single series Px, in preference to that of Cauchy (§ 69, corollary). There is nothing in either method to indicate whether the region for which the equation is proved to hold good includes all points that satisfy the equation %Uq = Px; later on we shall see that Taylor's theorem furnishes an example where the information furnished by these criteria, — exact of course as far as it goes, — proves to be incomplete. At first sight Cauchy's criterion seems both simpler and more effective than that of Weierstrass ; that this is not always the case appears from the series X— I I + :a"--(f^)"^ <■>■ 3-;r \3-x/ \3-^' 138 OPERATIONS WITH POWER SERIES. [CH. XI or " / I 2 '' \" The series (i) converges for all values ofx such that X— I S a, where a is a proper fraction. The component series 3-^! i.e. for all values of x for which the real parts are less than 2 ; and it converges uniformly for all values of x such that X— I have 3 for a common radius of convergence ; hence all the con- ditions of Weierstrass's theorem are satisfied by taking R to be 2, and we see without further discussion that (2) can be ex- pressed as a power series whose radius of convergence is at least as great as 2. Cauchy's method uses the new series 122 2^-1 replacing in (i) by -H y] and requires for the con- vergence of this new series the inequality 2 X-i 3 3-^ that is, Jf<3/2. It has given us therefore less information as to the domain of convergence of the final series and has required the summation of a new series. The two series X x^ x' l—X I —X^ 1 —x' X X' X' :+ (3). i-x''^i-x*'^r^''^ ^4). in which X< 1, can be represented as double series by expand- ing the individual terms as power series ; furthermore they arise from one and the same array X X'' x" X* .. X' x^ x^ x^'' .. X' X^o ^16 ;r» .. 82-83] OPERATIONS WITH POWER SERIES. 1 39 by adding by rows and columns respectively. The double series is absolutely convergent, for the sum by rows of the absolute values of the separate terms is the convergent series X X' X' i-X^ i-X^'^ i-X''^"" Hence Cauchy's criterion applies and the sum by columns must be equal to the sum by rows, and therefore the series (3), (4) have the same sum. To see that Weierstrass's criterion applies to this case, observe that for any assigned value of x such that X < i , we can take R between X and i and get I -x"" R«i+i ^ I - E' and therefore ^ a series which is uniformly con- vergent within {R). It follows that the series (i) is uniformly convergent within (R) and therefore comes under the operation of Weierstrass's theorem. Ex. Prove that X ■2.x'' yfi X !?■ 3? T^x'^T^"' i-^^-~{.\-xr^ {i-x^Y^ {^.-x'f •■• when |j:|'i,^2, ..., j/^) can be expressed as a power series which converges within {R). 84. Reversion of a Power Series. Let y = PyC, and let Qy be expressed as a power series in x. Since Pfi is o, the condition in (ii) is merely that Qy shall have a radius of con- vergence greater than o. Let us examine whether Qy can be identified with x itself. We have y = a^x + a^x' + . . . x=bt + biy + biy''+ ... = d„ + b^a^ 4 (^1^2 + b^x) ^ + ( Vs + 2<5sa,flj + ^stZi^) j^ + . . . , so that the conditions to be satisfied are b^ = o, hai. = I, b-^a^ + b^^ = o, i^jfla + zb^-^a^ + (^3(2i' = o, and these equations determine b^, b^, b^, b^, ... uniquely. Hence if the series x^ P-^y whose coefficients are obtained by the above equations is shown to be convergent for values of y other than O, we have one solution of tJie equation for x. This very important process is called the reversion of a power series ; and P-^ is said to have been reverted. Of course under the same conditions the equation J/ = flo + /"^ can be reverted in the form x=Px{y-a^. The same process is not applicable to j/ = P^ when n > i. For 83-84] OPERATIONS WITH POWER SERIES. 143 then Q(F^) contains no power of x^ less than the «th and cannot be identified with x. To this case we shall return later and we shall then be able to establish the convergence of the reverse series. Meanwhile we wish to point out an important peculiarity which arises when we map the domain of PnX on the _y-plane. Taking more generally we have first when n= i, y -fo = «i (^ - ^0) + a^(x-x„y + ..., and lim (jy —y^)l{x — x^) = Oi, showing that there is isogonality at Xo (§26). But if ai = a2= ... = an-i = o, an=^o, the isogonality breaks down ; we have then \im(j/—j/o)/{x — XoY = an, whence an x=x, angle at j/,, is n times the corresponding angle at x^. Let ^ —jft = an(^ — x^oT y then j —y = /"n+j, (x — ;r„), where / is some positive integer. When x describes a small circle (jT,, p),y describes a small circle, n times as fast, about _y„. Also (j>-y)/(J -y,) = Pj,{x- X,) ; hence |jr — jr„|, or p, can be so chosen that \y—y\l\y—yo\ is as small as we please, for all points on the circle {x^, p). It is usual to say that y describes a small circle about y^ n times as fast as X describes its circle about x^ ; it will be understood that this is strictly true only for y. The path of y is, for small enough values of p, as nearly circular as we please. The case « = i is the ordinary case ; the case « > i means that when y =yo there are n values ol x equal to x^. This peculiarity will be discussed in chapters XX. and XXI. Given that y = a^x + a^x^ + a^x^ + ... arises from x = diy + b^y- + b^y^ + . . ., it is desirable sometimes to have a formula which connects bn with y. Differentiate (see § 87) x = b^y-\- b^y^ + ... + <5„j>'" + ... with respect to x and divide by j^". Then 2=yrA + ^^^ + -^„+... + ^» + («+i)^„,.+ .,' The only term on the right-hand side which contains i/x to the 144 OPERATIONS WITH POWER SERIES. [CH. XI first power is the term ndny/j- For example y/y arises, save 00 as to a constant, from i/j''^~\= 2 c^^"", by differentiation ; and after such a series has been differentiated term by term there can be no term in i/x to the first power. By equating the term in ijx in nbny'ly to the corresponding term in i/y, we get 1/0! where [ Ji/jj means the coefficient of ijx in the expression contained within the square brackets, when that expression is given as a series in x. Thus the reverse series of y = a^x + a^x' + a^x^ + . . . x= 1. ■^- J/" Vx 85. Taylor's Theorem for Power Series. Let x and x + /i lie in the domain (R) of Px. The series for P(x+ k) is flo + «i (a- + h) +a^(x + hy + «s (jr + ^)' + . . . , and may be written a^->ra^{x+h)+a._(x''->r2xh + k')-^a^{x? + y(;'h->rixh^->rh^)-ir ... . Can we arrange this series in powers of k ? Treat ;ir as a fixed point within (R), refer /t to;r as origin, and let Ug (§ 8i) be ag{x+/i)'". The series Xug itself is P(x + k), and is uniformly convergent along any circle concentric with and smaller than the circle \x + /i\=R, i.e. for values /i within (x, R-X). Take then for the Fig- 39- 84-85] OPERATIONS WITH POWER SERIES. 1 45 circle of Weierstrass's theorem the circle (r, R - X). And suppose, as above, that x lies within (R) and that ^ is a point interior to the circle (x, R - X) which touches internally the circle of convergence {R). With these limitations on x and h we have P {x + h) = Px+ hP'x + —. P"x + ... +—,P^x + ..., where the series P'x, P"x, ..., P^x, ..., known as the ist, 2nd, ..., «th derived series of Px, are P'x = ai + 2aiX+ia^'+ ...+(«+ i)«„+i;r"+ ..., P'x= i.2aj + 2.3a^ + 3.4a^^+...+(« + 2)(«+ i)a„+^x^+ ..., and so on. The series P'x is derived from Px by a differen- tiation term by term, and a repetition of this process gives successively P"x, P"'x, etc. By this use of Weierstrass's theorem we have proved the following theorem which is the exact analogue of Taylor's theorem for real variables : Let X be any point in the domain (R) of the series Px and let a circle be described with centre x to touch (R) internally ; t/ten for all points X -^ h within this latter circle tJie series P{x-\-h) can be represented by the series k^ h^ Px + hP'x +~P"x-^...+\p^x+..., 2 ! n\ fin in which the coefficient of — ^ is obtained by differentiating each term of Px n times and adding the results. Ex. Prove the same by means of § 68. Replacing x by x^ and ;r + // by x, the equation runs This form of Taylor's series is especially useful in the discussion of power series, for it establishes a connexion between the original series in x and a new series in jr — jr,, where x^ lies in the domain of Px. We now know that the radius of convergence of the series in x—x^ is at least as great as R — X^; we shall see presently that M. H. 10 Px=Px,^-Fx,{x-X,)■^■'^{x-X,y■>r...->r'^{x-X,Y^■. 146 OPERATIONS WITH POWER SERIES. [CH. XI there are many cases in which it can be greater than this quantity. Maclaurin's theorem Px = Po + P'o .X + P"o .x''/2 \ + ... +/^o..r»/K!+ ... is of course merely a special case of Taylor's theorem. If we start with a series P{x — Xo) in place of Px, and select a point Xj within the domain of P{x — x^ instead of the point x,, in the domain of Px, Taylor's theorem will run P(x-x,) = P{x,-x„) + P'{x,-x,).(x-x,) + P"{x,-x,).{x-x,yi2\+ ...+P^(x,-x,).{x-x,YJn\-^ .... Tfie domain of this series in x — x^ is at least as extensive as the open region bounded by that circle of centre jr, which touches internally ttie circle of convergence of P (x — x„). 86. The Derivates of a Power Series. The radius of convergence, R', of P'x will be proved equal to Ji by establishing that (i)R' ^R, and (2) R g R'. Since each column in Weierstrass's method for combining a series of power series into a single power series gives a convergent series whatever be the position of x within (R), the series P'x that arises from the first column must give R' £ R. To see that R = R' notice that the domain of convergence of Px is the same as that of 2 a„x"-^, and compare this latter »=i series with Px, or 2 «fl„jr'^'. As the general term of the former series is evidently less than that of the latter series in absolute value, the domain of Px is at least as extensive as that of P'x. This completes the proof It is an immediate deduction that P"x, P"'x,.,., P^x,... have a common radius of convergence, namely R. Hence all the derivates of Px have ttu same domain as Px. Ex. Prove that 2a,^+V(«+i), 2 fl„^" **/(«+ 1 )(« + 2),..., —series ob- ?l=0 rt=0 tained by successive uses of the mechanical process of integration term by term, — have a common domain of convergence, namely (R). 85-87] OPERATIONS WITH POWER SERIES. 147 Let us now find the value of DPx, the derivate of the sum of the series. At every point of the domain {R) the value of Pix ■'r fi\ — Px DPx is the limit of — ^^ j^ when x-vh tends within the h domain {R) to x. There is no difficulty in finding this value ; for the fraction is equal to a power series in h whose first term is Px, and therefore tends to P'x as a limit. Hence DPx=P'x. This theorem required proof The existence of Z>f/f is not a necessary consequence of the continuity of f^ when f is real ; hence the statement that DPx exists cannot be regarded as a truism. The theorem shows that we get the same result whether we first sum the power series and then differentiate, or first differentiate the separate terms and then sum the resulting series. In the present case then we have a double limit in which the order of operations does not matter. 87. Differentiation of a Series of Power Series term by term. Let u„, «i, u^, ..., Uq, ..., R have the same meanings as in § 81 and let tug be uniformly convergent within every circle (^j) where R^ is less than R by an arbitrarily small amount. Take any point x^ within the circle {R) and expand the 00 series zi^, u^, u^, ..., Uq, ... and a, = 2«g, in ascending powers 9=0 oix—x^. Then i «<'»'(^o)-(^-^o)™/'«! = i fv»'(;t:„).(;r-jr„)™/w !...(!). Since the individual series u^, «,, Wj, ..., «,, ..., and the collective series ?/ = ao + M, + M2+ ... +«9+ (2), converge uniformly in the closed region {x„, R' - X^, we can apply Weierstrass's theorem to the double series in (i) and reduce it to the single series 2 {z/„"«i(;.r„) + «,""'(;ir„) + ?/2<™»(jr„)+ ... +a,'"«(J^o)+ •••}(^-^o)'"/»«!. m=0 148 OPERATIONS WITH POWER SERIES. [CH. XI Comparing the terms in x^^jm ! in this series and in the series on the left-hand side of (i), we deduce that at every point of the open region {R). In words this theorem runs as follows : — Under the specified conditions tlie m"' derivate of tfie sum of a series of power series is equal to the sum of the /«'* derivates of the individual terms. CHAPTER XII. CONTINUATION OF POWEE SEBEES. 88. The Zeros of Px are Isolated Points. With the help of Taylor's theorem it is easy to generalize the theorem of § 80. In its generalized form the theorem runs : — When Px has infinitely Tnany zeros within a circle (R') con- centric with and interior to t/ie circle of convergence (R), the coefficients of Px are all zero. Let the circle be enclosed within the square whose sides .R' IE 0-™ « -»R Fig. 40. are the tangents at ± R', + iR'. Divide this square into four smaller squares as in the figure; in at least one of these new squares there must lie infinitely many zeros of Px. Select a square of this kind and subdivide it into four squares as in the figure. Select from these new squares one in which the number I50 CONTINUATION OF POWER SERIES. [CH. XII of zeros of Px is infinite and subdivide this square in turn into four new squares. Proceeding in this way we get a sequence of squares, each one of which contains infinitely many zeros of Px. The reasoning of § 51 shows that these squares determine a single point a which is interior to them all. This point is a limit-point of the zeros (§ 80), for within every circle of centre a, no matter how small may be the radius, there are zeros which are distinct from a. Hence the series %P"a.(x— aY'/ttl has zero coefficients, and therefore Px vanishes at all points interior to {a,R — \a\). Within the circle just mentioned take a point i> ; the point l> being a limit-point of the zeros of Px we can use the same reasoning as before and show that Px is zero at every point interior to {b, R — \b\). Thus step by step we can prove that Px vanishes over the whole of its domain (R) ; and then we see that Px must have zero coefficients (§ 80). The theorem here established does not exclude the possibility of a series having infinitely many zeros in its domain of convergence. When the zeros are infinite in number the series has zero coefficients, or else the zeros of the series have no limit-point in the domain {/?) and at least one limit-point on the boundary of the domain, that is, on the circle of convergence itself. Corollary I. W/ien two power series in x converge witkin a circle (R) and are equal for infinitely many values within a circle {R'\ where R' < R, then the two series are identical ; that is, tliey liave the same coefficients. This corollary is an immediate consequence of the theorem, and needs no proof. Corollary II. The points within t/ie dotnain of Px at which the series takes a determinate value b^ have no limit-points interior to the circle of convergence of Px, provided that Px does not reduce to ttie coristant b„. This corollary can be proved by applying the theorem to the series ao-ba-\-aiX+a^x^+ ...^anX'* -^^ ... , instead of to 2 «„jr». »=o In the theorem and in the two corollaries the points which are considered are members of infinite systems confined within 88-89] CONTINUATION OF POWER SERIES,. 151 finite regions. When the points of such a system have no limit-points in the interior of the associated region, they are said to be isolated within the region ; the points of a finite system are isolated, whether the region be finite or infinite. 89. Continuation of a Function defined by a Power Series. Let us now return to Taylor's theorem. And first let us show by a simple instance that when from a series Px we deduce a series P{x — x^ by means of the theorem, the new circle of convergence may cut the old one. Take the geometric series I +x + x"-+ ..., whose sum is when \x\< i. I —X ' ' Here then Px = , and by differentiation I —X ^ whence Taylor's theorem takes the form r i-x„ (i-x,y {i-x,y < I, that is, if X is nearer to x^ than i is. The series in parentheses is again a geometric series, and is convergent if | I Xq Thus the new series defines the function in the circle through i whose centre is jr„, while the original series defined the function in the circle through i whose centre is o ; and manifestly the new circle will in general cut the old one. Thus ground is gained ; we have a new series equal to the old in the region common to their domains, but serving to define the function for an exterior region. In this very simple case we do not need the aid of power series ; we know all about the function to begin with. ' I —X ^ 152 CONTINUATION OF POWER SERIES. [CH. XII The use of the power series will, however, be appreciated as we proceed ; the present point is that the domain of the con- tinuation of a series may very well extend beyond the original domain. Ex. Prove the binomial theorem for (i-jr)", where |:ir|< i and « is a negative integer, from the series used in this article. When we have a power series P{x — Xa) with a radius of convergence Ro, and a point Xi in its domain, we have by Taylor's theorem a new power series in x — x^, say P(x — x), which is known to be equal to the old series for all points in the circle (x^, i?o— l-^i — ^oD- Let the radius of convergence of the new series be .^i ; and suppose that, as in the instance just given, the circles of convergence cut each other. We must prove that the two series are equal at all points which are in both circles of convergence; these points constitute an open region which can be called the common domain of the two series. It is sometimes convenient to denote the new series by P{x — Xo\ Xi) ; the notation indicates that the series is deduced Fig. 41. (by Taylor's theorem) from the given series P{x — Xo). It is also convenient to denote by d the circle of convergence of a 89] CONTINUATION OF POWER SERIES. 153 series P(x — x„), and by Cnm the circle which is concentric with Cn and touches Cm internally (fig. 41). With this notation, we know that our two series are equal within the circle Cm and we have to prove that they are equal in the region shaded in the figure ; to this shaded region the boundary points on Q and d do not belong. Take x^ within Cio and with it as centre draw a circle Cm to touch Co internally and a circle C^ to touch Ci internally. We shall need only the smaller of these ; let it be C^. We can deduce a series in x — x^ directly from P{x — Xo); let this be P{x — x„\Xi). Then we know that P(x-Xo) = P(x — Xo\x„) within Cjo. Again we can deduce a series in x — x^ from P(x — Xo\x^\ let this be P (x — x^\x^\ x^). Then we know that P {X — Xo\Xi) = P (X - Xo\Xi\ X,) within Csi- Hence within the smaller circle Ca both equalities hold. But also we have within C^o P{x-x,) = P{x-x,\x,). Therefore in the region common to Cw and C.^, we have P {x - Xo\Xi) = P {x - Xo\xi\ Xi) ; whence, as the series are both series in x — x^, and x^ is in the region common to do and di. they are one and the same series (§ 80). Hence we have within di P{x-Xo) = P{x-Xo\ x^), and P{x-Xa\Xi) = P{x-Xn\x^; whence P{x-x^ and P{x — Xo\xy) are equal within C^. The series have been proved equal in that part of the shaded region lying within Ca- We now add this part on to do; and by selecting suitably a new point x^ in the region so extended, drawing the circles di and do and taking the smaller, we again 154 CONTINUATION OF POWER SERIES. [CH. XII increase the region of equality of the series ; and the process can clearly be continued until the extended region includes all points of the shaded region, that is until the equality at all points of the common domain is established. The question of the equality of the series on the boundary of the domain is left an open one. "'>[''>]'' '^i^^'t^ ' There is one possible power series^which requires special mention. In the case of i/(i— ;r), there is an expansion in oo powers of ;r- 00 or ijx, namely S (- i)"/;p"; this series con- n=l verges at all points exterior to (o, i ) ; take any element P{x- x^) of ij{i—x) whose domain lies partly outside (o, i). At all points outside (o, i) and in the domain of P{x — x^) we have 00 2 (— \Ylx'^ = P{x-Xo); and therefore the series in i/:»rcan be «=i called a continuation of P{x — x^). More generally when we have an expansion Piijx) con- vergent at all points exterior to {R), and equal to P {x — x^) at all points in their common domain, then P(i/x) is called a continuation of P(x — x„). 90. The Analytic Function. An analytic ftmction is defined by an aggregate of series composed of a primary series and its continuations; the separate series are called elements of the analytic function, and the primary series is called the primary element. Among these elements is to be included P{i/x) in case P{i/x) is a continuation of an element P(x—x„). Let us see how the value of the analytic function can be found at a point d which lies outside the domain of the primary element. It is necessary to form a chain of series P{x-a),P{x-x,),P{x-x,), ..., P'(x-x„), in which the first link is the primary element while the last link is a series whose domain includes d. The method of construction of this chain will be understood clearly by examining the way in which the second link is attached to the first. 89-90] CONTINUATION OF POWER SERIES. 155 Let jTi be situated in the domain (a, R) of P{x — a). Then P{x — x^ is that series P{x — a\x^ which is deduced from P{x — a) by writing x— a = x^ — a ■'r {x — x^ and using Taylor's theorem. The radius of convergence of P(x — a\x^ is at least as great as the shortest distance from Xy, to the circle {a, R). Hence if we take x^ within the shaded region (a, R/2) we can be certain that a will lie within the domain of the new series in ;r-:r, ; so thatP{x- a) can be deduced from P{x-a\xt) by a direct application of Taylor's theorem. Let P (x — a\xy\x^ he & series in jr — ;r2 which is deduced from Fig- 43- P{x—a\x^ by the use of Taylor's theorem, and let Xj be chosen in the region {x^, Iii/2), where {x^, Rj) is the domain of P(x — a\ x^. Continuing in this way it may be possible to reach an element P(x — a\xi\x^\ ... |jr„) whose domain contains b, and from this final link we can deduce, by immediate continuation, an element P{x—b) whose constant term denotes the value of the analytic function when x=b. In the case of the analytic function 1/(1 - x) the value at b does not depend on the special chain that happens to be selected; for we can deduce from the series 'WiX — x^Si series in • b and this series can only be 2 (;r • n=0 ^)"/(l - jr)"+', into which 156 CONTINUATION OF POWER SERIES. [CH. XII x^,x.2, ■■.,x„ do not enter. But with a different analytic function the value at d of the function might very well depend on the manner in which Xi, x^, ..., x„ are interpolated. In this way it is possible to allow for several or even infinitely many values at a point. The chain described above permitted the passage from the series P (x — a) to the series in ;r — (5 by a succession of im- mediate continuations. The choice of the points Xi, x^, -...Xn was made with a special view to the reversal of the process. The series in :tr— ^ can be chosen as the primary element ; for the series in x — Xn is an immediate continuation of it, and the series in x — Xn_i is an immediate continuation of the series in x — Xn' etc., and finally P (x—a) is an immediate continuation of P{x — a\ Xi), so that it is possible to pass from the series in ;r — ^ to the series in x—a and thence to every element of the analytic function by a chain of immediate continuations. A chain with this property of being available each way may be called a standard chain. We have proved then that atiy one of the elements of an analytic function, in particular of the analytic fufiction I /( i — x\ can be treated as the primary element, all the other elements being deducible from this by the process of continuation. This theorem implies that each element contains in germ the whole of the analytic function. 91. General Remarks on Analytic Functions, (i) In the case of the elementary analytic functions the arithmetic expression comes first, the power series second. It is otherwise with a majority of functions. Take for example the case of a differential equation ; when it is found impossible to write down in terms of a finite number of known analytic functions a solution of such a differential equation as ^^= ^^y> what is done is this. We select an ordinary point x=a and find ^ as a series P{x-a), and then deduce the analytic function from P{x—a) by continuations. (2) The question which is suggested by the consideration of such an equation as dx ^ ' 90-92] CONTINUATION OF POWER SERIES. 157 where f'x is a one-valued analytic function, is this : given that Fx is one element oi f'x, is it safe to infer from the solution y = Px that the equation is satisfied by the analytic function of which Px is merely one element? Or we may ask the closely related question: are the first, second, ..., derivates of the continuations of P{x-a) the same as the continuations of the first, second, ..., derivates of P(x-a)? The answer is in each case in the affirmative. Without entering into any details, — for these would carry us far into the subject of differential equations, — we may say that the general principle involved is that a property possessed by one element of an analytic function is possessed also by the others. (3) It would be a serious misconception of Weierstrass's views to suppose that he wished to employ no other means of investigating the properties of any selected class of functions than those employed in the continued use of Taylor's theorem. In all cases he had recourse to some functional property : — an algebraic equation /(.r,j') = o, an addition-theorem, a differential equation, and so forth, and in this way gained a control over the subject-matter, that could not have been obtained from the series alone. (4) The passage from a to ^ in § 90 was by a succession of steps a to jTj, x^ to jTj, and so on. The number of interpolated points may be decreased ; it is certain that it can be increased as much as we please. There is often an advantage in thinking of a continuous route or paiA from a to 6 ; the meaning then is that the various sets of interpolated points all lie on this line and furnish standard chains. (5) Starting from an initial point a and an initial series /"(jr— a) and confining the paths to an assigned region it may happen that the value at every point b of the region is unique, whereas this ceases to hold when the region is replaced by a larger one. In this case we have for the first region a one-valued portion of a many-valued analytic function. 92. Preliminary discussion of Singular Points. Analytic functions exhibit every variety of behaviour with regard to con- tinuation beyond the domain of the primary element. The nature of the function depends, in fact, on the nature of the obstacles, called singular points, which limit the domains of the series. What these obstacles are will appear further on in more detail ; but even at this stage we can give a plausible description of the ordinary obstacles. In the case of the series i -f jr-t-jr^'-f ..., the obstacle is the point x=^i, at which ^— - becomes 00. Naturally, since a power series is defined for each point of its domain, no circle of convergence can enclose this point; but all circles may be expected to pass through it. 158 CONTINUATION OF POWER SERIES. [CH. XII In the case of the series i +x^ +x* + ..., = ~ — - , there are I — x^ two obstacles, ;r = + i. These and these only are what stop the circles of convergence. A circle of convergence with centre Xn will pass through tke nearer of these points; that is, if Xo = ^o + ivo, through i when fo is > o, through - i when fo is < o, through both when fo is o. In the case of the series for (i —xy^, namely I I ^ _ I ■ 3 ^ the obstacle is ;tr= i. If the circle of convergence of the series can enclose this point, then when x describes a small circle about i (fig. 44) the series is unaltered and the sum is the same, namely (i —xy^. But \l{i—x) is not the same after describing the circle, for the ampli- tude of I —X has increased con- tinuously through 27r ; whence the amplitude of a selected square root has increased continuously through '^' **' IT, and the values (i —x)^'^, — (i -x^ interchange. Hence the point is not in the circle of convergence ; it is however on it. In the first case there is no ordinary power series about the point I, but there is a negative power, namely —(x— i)~'. In the second case there is equally no ordinary power series about the point i, but there is a power series with a negative power oi X— i ; since ii x= i +Xi, l-X- 2Xi+xi' 2Xi\ 2/ ' or if |;fj|<2, -i L+i_^ + .... I -x<' 2jri 4 8 In the third case there is a fractional power series which has only one term. Thus when about a point jr„ there is no element of an analytic function there may be a series P ij{x — x^, with a 92-93] CONTINUATION OF POWER SERIES. 1 59 finite or infinite number of terms, or there may be a Laurent series P il(x — x„) + P{x — Xt), or again there may be a fractional series such as P\x — Xo, or the fractional series may be a Laurent one. This is merely an indication of possibilities, not a classification of them. In the above cases we know the sum of the given series, that is, we have direct expressions for our functions, and it is to these that we look for information as to the obstacles. This will be observed also in other cases. In general we rely on the theory of this chapter for the regular behaviour of a function, to infer for example its continuity ; but for its peculiarities, just those things which mark it off from other functions, we may do well to look in a different direction. 93. Transcendental Integral Functions. When Px has an infinite radius of convergence the series defines a transcen- dental integral function. Its properties resemble in several respects those of the rational integral function, but these two classes of function behave very differently at infinity. There is no need to discuss continuations, for now the single series Px defines by itself the complete analytic function. Among the simplest transcendental integral functions are » ir those defined by the series 2 x"/w!, 2 (- i)":j^+7(2« + i)!, n=0 m=0 2 (— i)"j^/2«!; these functions are discussed in the next »=o chapter. A special notation Gx is used for integral functions, whether rational or transcendental. When Gx is not reducible to its first term, there must be points in the finite part of the plane at which \ Gx \ is greater than any assigned positive number, however large. Assume, if possible, the existence of a finite upper limit 7 for I Gx\. Then A^&'ilp^, however large p may be (§ 78). This requires that «,, a^, a^, ... shall all vanish, contrary to supposition. Hence the theorem is true. This important theorem is due to Liouville. l6o CONTINUATION OF POWER SERIES. [CH. XII For a rational integral function Gx there is a radius R such that for all points exterior to {R) we have | Gx \ > y, where y is an arbitrarily assigned positive number, however great (§ 56). 94. Natural Boundaries. Another case must be men- tioned, where there is a finite radius of convergence, but on every arc of the circle of convergence there is an obstacle. There is then no possible continuation beyond this circle. In this case the circle constitutes what is known as a natural boundary for the function. As such functions belong to the higher theory, — where however they present themselves in an unsought way, — we merely give an instance. Lerch (Acta Mathematica, vol. x. p. 87) has shown that, if q and r are integers, the point e^sTi/r on the circle of convergence (o, i) of S ;r"', = fx, is an obstacle. For let x = pe't^'^ (p< i ), and let p n-l increase ; then as p approaches i the part of the series from the rth term onwards, namely 2 p"\ tends to infinity. This would n=r be impossible were the point e^"'" situated in the interior of any immediate continuation of the power series. It is clear then that there are on the circle (o, i) infinitely many obstacles, and further that on any arc there are infinitely many obstacles. Several distinct Analytic Functions defined by one and the same Arithmetic Expression. Tannery has given a very simple example of this peculiarity. Since lim 1/(1 -;ir")= I, o, according as |4r| <, > i, it follows that the series whose arithmetic expression is I - or ^= 1- 2 X' I —X »=ij:' - I has I or o for its sum under the same circumstances. 93-94] CONTINUATION OF POWER SERIES. l6l Now this series is uniformly convergent in the closed region {R) where R is any positive number < i ; for the remainder r„;tr= /-/ ' y^ I —x^ is less than R^^/i i — R'^") in absolute value. The series therefore fulfils the requirements of Weierstrass's theorem of § 8i and can be transformed into a series Px when \x\< i ; similarly when |jr| > I it can be transformed into P(i/x). The arithmetic expression, then, gives parts of two distinct analytic functions within and without the circle. These functions happen to reduce to the constants i , o. Let /iX, f^x be any two one-valued analytic functions of x with a finite number of singular points, then the expression 2 2 ^^ defines, within and without the circle (o, i), parts of two distinct analytic functions f^x, f^ ; the circle itself is not a natural boundary for either of these functions. M. H. " CHAPTER XIII. ANALYTIC THEOEY OF THE EXPONENTIAL AND LOQABITHM. 95. The Addition Theorem of the Exponential. The simplest instance of a transcendental integral function is the exponential, defined by y = exp jr = 2 ;r"/« !. w = The test l.Ar^ilAn=ilR (§ 76) shows that here R = oo. One marked peculiarity of this function is that »=o it was on this fact that the geometric theory of ch. iv. was based. The function was deduced indeed from this fact, with the con- dition that,;/ = I when x=o. By direct multiplication of power series (§ 83) we have the ' addition theorem ' : exp jr, . exp jTj = exp (x^ +X2) ; for the product of the series for expjr, and expjTj is = .+(.,+.,)+(£L±_f^V(£i±£^V In particular '\i x^= —x^, we have expjri.exp(— Xi) = expo= i, and exp(— jr,)= i/expjr,. 95] ANALYTIC THEORY OF THE EXPONENTIAL. 163 If the argument be an imaginary irj, and if, separating the series for exp it} into a real and an imaginary series, we write exp?J7 = ^'+iV. then exp (— tT)) = f — ir)', and by multiplication i = ^'2 + ^'2. Hence exp ?'»; cannot vanish for a finite r). That exp f, where f is real, does not vanish for a finite ^ is clear when f > o from the fact that the series is formed from positive terms beginning with i ; and when ^ < o, expf = i/exp(— f), where the denominator is finite. Since, ii x=^ + it], exp jr = exp ^ exp ii] and neither factor is zero, we infer that the exponential has no zeros. When the real number f tends to + 00 , exp f tends to + 00 ; hence exp — ^ tends at the same time to zero. It may seem there- fore that we ought to say that expjr has a zero at infinity, on the same grounds that we say that ijx has a zero at infinity. There is this important difference between the two cases: — As in § 61, we mean by "value of ijx when jr=oo'' merely " the limit of ijx when x = oo " this limit being independent of the manner of approach to the artificial point jt = 00 ; whereas we can assign no such meaning to " value of exp x when ;r = 00 ," for now there is no unique limit (§ 57), that is, no limit independent of the manner of approach. For this reason the value of exp x dX x= is left entirely undetermined. This is a very good illustration of one kind of singular point (§ 103). We are justified in speaking of 00 as a singular point of e*. For if there were a series P(i/x) good outside any circle (R) we should have /'(i/|) = exp f for all values ^ > R. But if f tend to + 00 , P(il^) tends to its constant term whereas exp f tends to + 00 ; hence the equation cannot be true. The addition theorem admits of an evident extension, namely expxiexpx.i... expx„ = exp(xi+x,+ ... + Xn). In particular if the arguments are all equal to x, (exp xY = exp nx, where « is a positive integer ; and, if ;r = i , exp« =(exp i)", 1 1 — 2 164 ANALYTIC THEORY OF THE [CH. XIII where exp i =s-i +1/2!+ x/3! + ... = 2718281828 ..., the number denoted by e (§ 29). Let the number of terms *•„ be unlimited, but let the series Xr„ be convergent with a sum s; and let j — j„=r„. We shall prove that L exp Sn = exp L s„ = exp s. We have s„ = s — r„, exp j„ = exp s . exp — r„, I exp j„ — exp J I = j exp {—r„)—i\\ exp s | = |-r„ + r„72!-... j I exp J I S(kn|+knV2!! + ...)|expj|; and I r„ I < e when « > /i, therefore, for such values of «, I exp j,j — exp J I < (e 4- e'/z ' + ....) | exp s \ , <(e + 6=+ ...) |expj|, € I — € yl\ -^o^' sin 2 -|-^ where Tan-' a signifies that number in the open interval (— 7r/2, ■77/2) whose tangent is a. These formulae hold for all values of ^ and when oSo-^ i, except for the combination ^ = tt (mod. 27r), o- = i (§ 76). 1 66 ANALYTIC THEORY OF THE [CH. XIII For points on the circle ^ = 26„ (mod. 27r), and the series take the simpler forms cos — ^ cos 2— ...= Log (2 cos o/2), sin — ^ sin 2 — . . . = o is the chief amplitude corresponding to (j). The behaviour of the last series in the neighbourhood of the point TT calls for especial notice. As increases up to tt the sum tends to 7r/2 ; as decreases down to tt the sum tends to — '7r/2 ; when ^ is tt the sum is zero. Thus there are distinct limits for the two modes of approach, and the value at the point of discontinuity is the mean of the two. These real series illustrate another point. Take, for instance, sin^-^sin2<^H-^sin3(^- .... It is convergent and in spite of discontinuities has a definite sum for every 0. But if we differentiate term by term we have the series C0S(^ — COS2<^ + COS30— ..., which is not convergent since L cos n is true when Am ajd = Am a — Am d, that is, when — IT < Am a — Am d Sir; in our case Am(i +z)- Am(i -z), or 6^- 60, is shown by a 96] EXPONENTIAL AND LOGARITHM. 167 glance at fig. 45 to be > - 77/2 and < 7r/2. Hence for all points within and on the circle, the points z= ± 1 excepted. Log ^— ^ = Log ( I + 2) - Log (I - ^) = 2{z+z'l3-+s'/s + ...) (4). If we map the unit circle on to a new plane by writing i+z ^ t- I = t, z= ; l-z /+ I and remark that for points on the circle am ^ = ■jrJ2 so that t is imaginary, and that t=i when z = o, we see that the circle maps into the imaginary axis in the /-plane, and the interior of the circle into the half-plane on the right of this axis. Thus for all points / of this half-plane and the imaginary axis, except the points o and 00 , we have t-i 1 ft-iy I //-i\» ) , . Thus this series for the principal logarithm covers the important case where / is a positive number. . Log t = 2 EXAMPLES. 1. For everj' point considered every term of the series (5) is a power series in t. Hence (§ 87) we can differentiate term by term. Verify that we get in fact Z>Log/=i//. 2. For what values of x is the equation valid.' 3. The equation defines the principal logarithm for all values of x provided we understand by 3^'^ that square root which lies either in the right-hand half-plane or on the upper half of the imaginary axis. 4. Prove from (4) that when 6 is real sinfl+ Jsin3d + ^sin5tf+...= ±»r/4 or o, distinguishing the cases ; and that cosfl + Jcos3fl+Jcos5fl + ... = Log 5. Prove that I + 1/3- l/S - 1/7+ 1/9+ I/I I + ■.. = 7r/2^\ I -1/3-1/S+ l/7-l-i/9-i/ll-... = 2"'Log(2"-+l). 1^1 cot- 2 1 68 ANALYTIC THEORY OF THE [CH. XIII 97. The meaning of cH'. In elementary algebra a meaning is assigned, perhaps prematurely, to expressions such as a^'*, where / and q are integers, by an appeal to the ' Principle of Permanent Forms.' The explanation leaves matters in this state that the student regards a^" and ^a'^ as equivalent symbols, each capable of |^| values. There ensues this difficulty, that «^"' has two values but a' has one. When the exponent of a is irrational or complex, the principle referred to becomes quite inadequate. The fact is that a' when x is any number follows exp;r in the order of difficulty, though not historically. When we accept exp X, we can proceed without ambiguity as follows. We saw (§ 95 ) that when n is a positive integer ^" = exp «. Whatever number x may be, let ^ = exp X. Thus ^"^ is the positive number exp 1/2 ; e" has a single definite value. So we define a^ as exp (x Log a), which again is a definite number. But it must be observed that a different definition is also in use ; namely a^ is exp {x log a) where log a is of course any log- arithm ; here O' would have in general infinitely many values, for it is exp (x Log a + 2xmri) ; and so would ^ have in general infinitely many values, namely q-k^ {x + 2xn7ri). Observe that the definition we adopt agrees with our previous convention as to «''", namely (§ 17) that it is not any «th root of a but the chief «th root. For now /J ^i/n _ gjjp f - Log a = exp - (Log I « I + ? Am a) = exp - Log I a j . exp - Am a = 1 Va I cis - Am a. n 97-98] EXPONENTIAL AND LOGARITHM. 169' For «* we have the series I + JT Log a + (x Log ay/2 ! + ..., a*— I whence lim = Log a. Compare § 32. Ex. I. Prove that, when a and b are neither o nor 00 , the equation a^-=b leads to the equation l>^i'=a when x lies in a certain strip of the :ir-plane which includes the origin. Ex. 2. Prove that lim (i +(5jr)>'' = exp b. 98. The Binomial Theorem. We have ( I — «)-* = exp — X Log ( I — «), or (i -«)-* = n-« + z<72! + «V3 ! + ••■, where « = — jrLog(i —a) = xa+xa'^l2+xa?li-\-..., \a\ being < i. By Weierstrass's theorem (§ 81) this series in « can be regarded as a power series in a, say I -irx-fi+x^^l2 ! +jr3a»/3 !+ ... ; for each term u'^jn ! can be regarded as a series in a, uniformly convergent within the circle (o, i), and the exponential series in u is uniformly convergent in any circle. We have to determine the coefficient, x^, of «"/« !. It will arise from 11, u^, ..., ?(", and will consist solely of terms ranging from (« — i) ! jr to jt". That is Xr,=Ji^+... + {n-i)\x. But when — jr is a positive integer m, where mr^lti\-.... We have Dxy=\ -x^\2\-\-x^lif\- ..., = cos;ir = Vi — sin^.r, the square root being that (perfectly definite) square root which is cos;r and not — cos jr. So long as |sinjr|< i the square root can be expanded by the binomial theorem, and that expansion applies which begins with i, namely cos j.-= 1 — ^sin*;jr-l- ... . Hence we have, when \y\< i. D^y=\/i-y-, 174 ANALYTIC THEORY OF THE [CH. XIII where the square root is no longer definite for an unrestricted y, but is definite so long as \y\< i, being that square root which begins with i when expanded, i.e. (i — j/')'". or dx=dy\i +iy+^-^y+.. , \ y' \ .% y^ and x=y-ir~.— -K ^ - + ..., 2 3 2.4 5 so long as \y\< i. Since Sin~'_y is o with y, this element is Sin~^_y for small enough values oly. We now prove that the element, throughout its domain, is Sin"" j/. When f = ± 7r/2, y = sin (+ 7r/2 + irj) = + cosh t). Thus the edges of the fundamental region map into parts of the real ^-axis, from i to | 00 | and from — i to — | 00 |. Only when y crosses these parts can x pass out of the fundamental region. Thus the element, which is Sin~'^ when j/ = o, is Sin~' j when \y\ whence (§ 96) an element of tan-'^* is where \y\^ i, the points + i excepted. As a fundamental region we take the strip - 7r/2 < f S 77/2. The points of this strip form the chief branch of tan"'^', this branch is denoted by Tan~'^. When f = + tt/z, y = tan (+ 7r/2 + itJ) =±i coth r). Thus each edge of the strip maps into parts of the imaginary j/-axis from i to i\°o\ and from —i to — « | 00 1. Hence when !_;' I < I X cannot pass out of the fundamental region, and the above element is a part of the chief branch. ? = V2 101. Fig. 46- Mapping with the Circular Functions. We have sin (f + iv) = sin f cosh ij + i cos f sinh 17, 176 ANALYTIC THEORY OF THE [CH. XIII whence if y—^' + W = sin jt = sin (f + iij), f ' = sin f cosh tj, t) = cos f sinh 77. If then the jr-path is f= constant, the ^jz-path is half the hyperbola (f/sinf)»-(V/cosf)»=i, since cosh'' t; — sinh^ 1? = i , and cosh 77 Si. It is the right half when sin ^ >o, and the left when sin f < o. If the ;i:-path is 77 = constant, the j/-path is the ellipse (f /cosh rif + (V/sinh 1;)= = i . All these conies have the same foci ; namely the points y= ± \. Therefore they cut at right angles ; but this is merely a consequence of the property of isogonality (§ 26). Since cos ;r = sin (;r + Tr/2}, the very same curves serve as the map of the orthogonal systems of straight lines when _;/ = cos jr. The corresponding maps for_j' = sec x or esc jt are obtained by writings' = ijy, and are the inverses with respect to the centre of the confocal conies. For ;»' = tan x the matter is simpler. We have iy = {^ - l)l{^ + l). Now as X describes a line parallel to an axis, ix describes an orthogonal straight line ; and 2ix describes a line parallel to this ; hence, \{ z = e^, the map of the orthogonal lines on the ^- plane is (§31) a system of rays from the origin and circles round the origin ; and since iy = {z— i)/{z+ i) the map on the ;/-plane is a system of coaxial circles, through and about the points which correspond to .sr = o and 00 , i.e. y=±i. To a vertical jr-path, f = constant, corresponds a horizontal path for 2Mr, a ray from O for z, and an arc between + i for y ; but to a horizontal jr-path from x to jr + tt corresponds a com- plete circle about the points + {. Thus for instance to a real X corresponds a real y. When we make the restriction that j is not to cross a selected arc between + i, we restrict x to lie in a vertical strip of the ;i:-plane of breadth tt ; we may regard the j-plane as actually cut along the arc, and if we include one edge of the cut and J -51] EXPONENTIAL AND LOGARITHM. 177 <-xclude the other, then we include one edge of the strip and not the other. On the cut 7- plane the function tan"'.*- is separated ii>to distinguishable branches. In agreement with the definition of Tan-i jr we make the cut along the imaginary j-axis from i to i I 00 I and from —i 00 | to — « ; and we include the right edge of the cut since as any complex x moves from left to right y describes its circle from the left edge of the cut to the right, and we agreed that the right edge of the strip for Tan-' x was to te counted in. This notion of a cut, in selecting a branch of a many-valued function, was in common use before the introduction of the Riemann surfaces (ch. xx.) ; and when these surfaces are employed it is useful in connexion with functions which have thereon logarithmic singularities. The discussion of sin~'jr from this point of view is facihtated by the consideration of a Riemann surface. For JT = sin - 1 _)' = - log (y/ + V I -y ) ; and here to consider completely the dependence oi x otl y we first render Vi -y^ one-valued for allj^s ; and then render the logarithm one-valued by a cut. Ex. I. Discuss the mapping of parallels to the jr-axes by means of cot x. Ex. 2. Prove that a cut along a complete hyperbola, fig. 46, separates the branches of sin-'j/. Ex. 3. We have, \iy = co%x, 2y = z+lJ2 where z=e^. Hence when x moves horizontally or vertically, determine the map on the 2-plane and thence that on the ^-plane. Ex. 4. From the equation -^^dx, deduce by equating amplitudes that the tangent of an hyperbola bisects the angle between the focal distances ; and by equating absolute values that (ds f where the real integral is taken round an ellipse of which s is the arc and the length of the conjugate radius. XI. H. CHAPTER XIV. SINGULAR POINTS OF ANALYTIC FUNCTIONS. 102. Existence of Singular Points on the Circle of Convergence. The general theorem of which special examples have been given in § 92 is this : — Tlie circle of convergence (R) of Px must pass through at least one point x, which has the property that it is impossible to find a power series Q{x— x^) which shall coincide with Px in the common domain of the two series. The domain of the series cannot contain such a point ; what we have to prove is that the points x^ do not all lie outside the circle of convergence ; i.e. at least one lies on that circle. We shall prove first an auxiliary proposition : — Given that Px converges in tlie open region (R') and that the radii of convergence of all series P (x \ c), wJiere c is any point of this open region, have a lower limit D, tlien tJte radius of conver- gence of Px is R' + D. FT+D-s Fig. 47- I02] SINGULAR POINTS OF ANALYTIC FUNCTIONS. 1 79 The figure shows that every point of a circular region {R' +D — B) can be reached by immediate continuations of Px, B being taken as small as we please. We shall use this in the proof of the theorem. Observe, in the first place, that Px and the aggregate of series F{x\c) give one value at each point x in the open region (i?' +D — B); for any two members of the aggregate coincide in value in their common domain, when the domains happen to overlap. Let us use /x to indicate the common value given by the complete system of series at any point x within {R' + B); it is legitimate to use R'+D instead of R' + D — B, for x once fixed it becomes possible to take B so small that x shall lie also within (R' +I>- B). We have to show that Px has a domain (R + D), and that fx=Px for all points of this domain. This is done by a double use of Cauchy's inequality (§ 78). Take c on {R' — B) and use D^, where D^ is slightly less than D ; then if 7 be the maximum value of n\ i/'(^k)l,= + —i-{x-cf +. on the circle (c, Z>i), we have Now, remembering that P^c is itself a power series, namely QO 1 m(m— i) ...{m — n + O^m^"*"", we get, by a second application of Cauchy's inequality, mim-l)...(m-rt+i) , , „ ^ 7^1" Now by the binomial theorem, ^ , ^^ m{m-i)...ifn-n + i) fD.y 12- l8o SINGULAR POINTS OF ANALYTIC FUNCTIONS. [CH. XIV Hence, writing C for \c\, A.C'{..%-. ..(I)"-©"-- ^ /C\" . R' R'-C I + -„? J is the absolute value of the term amX" when X lies on the circle with centre o and radius C ('+^'): hence, by the theorem of § 76, the radius of convergence of Px is not less than C ( i + ^ ) . But the upper limit of the numbers Cil +-D-I) is .^'( I +-^j = /?' + /?; hence the radius of conver- gence is not less than R' + D. It remains for us to prove that the radius of convergence is exactly R' + D. Assume if possible that the radius of convergence is R' + D + ly where Z?' is a number greater than zero. For points c in the open region (R') we can assert that the radii of convergence of the series P(x\c) are all greater than (R' + D + D') — R', for the domain of a series about c extends at least to the circle {R' + D + D'). But this is impossible, since it means that the lower limit of these radii is Z>+ Z>' instead of Z?. Hence the auxiliary theorem is proved. There is not much difficulty now in proving the main theorem. Allowing c to range over the interior of (^R + D), the radii of convergence of P{x\c) have a lower limit zero. By dividing the circular region {R + D) into small squares (complete and incomplete) as in § 88, we see that for points c in at least one of these compartments the lower limit of the numbers is likewise zero. By increasing indefinitely the number of squares it follows that there is at least one point Xo, within or on (R' + D), such that in every neighbourhood of x^ the lower limit of the radii of convergence is zero. This point x„ cannot lie within {R' + D), for the radii in that case are at the worst very nearly equal to R' -^ D — X„ and therefore cannot have a lower limit zero. Hence the point ;r„ lies on {R' + D). Suppose now, if possible. I02-I03] SINGULAR POINTS OF ANALYTIC FUNCTIONS. l8l that there is a series Q(x- x„) which coincides with Px in the common domain. Then for points c near x^ we have series f{^\c) which can be deduced from Q{x-x^) by the use of Taylor's theorem ; as the radii of convergence of these series are nearly equal to that of Q{x-x^) it is impossible that they should have a lower limit zero. Thus the theorem is proved. A point jr„ of the kind considered serves as an obstacle to continuation ; we have proved that on the circle of convergence of any element of an analytic function tliere is at least one obstacle. These points are called singular points or singularities of the func- tion, and at each of them the function is said to have a singularity. Ex. I. Prove that i is a singular point, and the only one, of the analytic function i/(i - x). Ex. 2. When Qx/P^ is expressed as a power series, how far does the domain of this series extend ? 103. Non-essential and Essential Singular Points. Let ^ be a one-valu£d analytic function. We shall say that it is analytic about x„* or regular at x„ when it can be represented in the neighbourhood of ;r„ by a series P{x—x„). We shall also say that a function is analytic over a region (closed or open) when it is analytic about each point of that region. The word holomorphic is used in the same sense. Points about which ^ is not analytic are therefore singular points. For example il{x — cy^, e^i^-") are analytic about all points near x = c, but not ahontx = c. More generally the region constituted by points about which fx is analytic is bounded by a finite or infinite number of singular points. We shall say that the point x=c is a non-essential singtdar point or a pole of the analytic function fx when fx can be made analytic about c by multiplication by {x — c)™ where m is a positive integer. When m is the lowest integer which will serve, this integer is called the order of multiplicity, or simply the order of the non-essential singular point or pole. Inter- preting jr— 00 as \lx these definitions apply equally to ;tr= oo . * Adopting, with a slight modification, a suggestion of Professor Bocher, BtM. Am. Math. Soc. voL iii. p. 89, we use 'analytic about,' in preference to 'regular at.' 1 82 SINGULAR POINTS OF ANALYTIC FUNCTIONS. [CH. XIV Singular points which are not of the kind just described are called essential singular points. Suppose that x = c is a pole of fx, of order in. Then {x — cf^fx is expressible as P„(x — c). By division we get ■^ {x-cY'^ (x-cy-^ ■*"■■• + -^+ao + «i(-^-<^)+«2(^-^)'+--- (O- Since ^= oo when x = c, the pole is an infinity oi fx, and it is said (in agreement with § 6i) to be an infinity of order m. Later (see ch. xvii.) we shall show conversely that an isolated (§ 104) singular point which is an infinity ol fx must be a pole. The simple rule that singular points c of the kind (i) are to be called non-essential and all others essential is intended to apply only to one-valued functions. If applied to many-valued functions it would make ;r = o an essential singularity for -Jx, \lslx\ furthermore we can no longer say that every isolated singularity which is an infinity is necessarily a point of the kind (i); e.g. x=o is an infinity for il\lx. We see from (i) that we can remove the singularity by subtract- ing from fx the rational fraction composed of the first m terms. When <:= 00 , we must subtract a polynomial fl_m;tr'» + «_™^.^"^i -I- . . . + a_iJr. Let us consider, in the light of this remark, the ordinary resolution of a rational fraction into partial fractions. Let c be a zero of the denominator, of order m ; then c is an infinity of the fraction, of order m, and we have {x-cYfx^P,{x-c); whence by division >=(-^)--+-+^. + ^«(^-^> (x — cy^ +•••■' — ^ "^s no mfinity at ^ ; we have removed the infinity ji: = f. When in this way all infinities, which arise from zeros of the denominator, are removed, we examine •*=<». This is an infinity if the order of the numerator is greater than that of the denominator ; to remove it we subtract and fx — 103-104] SINGULAR POINTS OF ANALYTIC FUNCTIONS. 183 a polynomial in x which is determined by division. We have then a rational function without singular points; that is, a constant. Ex. Express as partial fractions x^^j{x^- if and x^''l{x^+ i)*. 104. Character of a One-valued Function determined by its Singularities. Throughout the theory of functions close attention has to be paid to the number and nature of the singular points. It is largely by such a study that the interior structure of a class of functions is best revealed. Let us start with simple cases in which essential singularities are absent. We know that the singularities of rational fractions are non- essential ; is it true conversely that the class of one-valued analytic functions with non-essential singularities only is the same as the class of rational fractions .■" This question suggests in turn two questions as to the possible distributions of the non-essential singularities. (i) Can a one- valued analytic function have no singu- larities .' (2) Can a one-valued analytic function have infinitely many singularities, all of which are non-essential .'' When these two questions have been answered in the negative, the primary question can be disposed of very readily. The answer to (i) is contained in the following theorem: — A one-valued analytic function Jias at least one singular point. Let us assume, if possible, the existence of a one-valued analytic function fx, not a constant, which has no singularities. Because fx is analytic about 00 we have an element d;„-|-ai/jr-l-^2/;tr-+ ... -f«„/;i:"-t- ... . Hence, by § 79, \fx\ differs very little from A^ for all values oi x exterior to some circle {X^ ; and therefore there is a circle {R) such that at every point x exterior to (R\ we have \fx\<1' where 7 is any number greater than A^. Now consider the element Px\ the radius of convergence must be infinite for otherwise there would be a singular point in 184 SINGULAR POINTS OF ANALYTIC FUNCTIONS. [CH. XIV the finite part of the plane, namely on the circle of convergence (§ 102). Because _/ir is a transcendental integral function there are points exterior to (R) for which IAI>7- Hence the initial assumption was false, and the first question is answered. Suppose next that fx has one but only one singular point and that this singular point is non-essential and situated at 00 . There must be an equation /X = «o-»^™ + «! J^™"' + . . . + «m_i^ + P(l jx), where «„ 4= o. By subtracting from fx the polynomial ^o^™ + «i^"^' + . . . + a,n-iX we remove the only singularity and get an analytic function which must reduce to a constant, say Um- Hence fx = ^o^"" + ffi^™"' + . . . + a„^^x + a^. That is, a one-valued analytic function fx which Jias no singularity in the finite part of the plane and a non-essential singularity at infinity is necessarily a rational integral function of x. The rational integral function has one singular point and that a non-essential singular point situated at x=' = ; ■ {X - Cj^' {X - C^y^ ...{X- fr)'"' ■ (2) Let fx have infinitely many non-essential singularities ; then these points must have no limit-points or in other words must be isolated in the finite part of the plane, otherwise there would be essential singularities other than ^= 00 . (3) Let fx have infinitely many non-essential singularities, each of order i. These can be named c,, c^, c^, ..., f„, .... where no c exceeds any subsequent c in absolute value and L f„ = 00 ; for within any circle {R) the number of singular points is finite. Assuming that there exists a function G^x whose zeros are simple and situated at tlte c's, then fx x G^x has at most one singular point and that situated at infinity, so that fx X G^ = GxX, and fx=G^\GiX. Here then we are confronted with the problem of the con- struction of G^x ; we shall consider this in the next chapter, and thereby complete the proof of the theorem. 106. Limit-points of Zeros. We have seen that a limit- point of non-essential singularities of a one-valued analytic function fx is an essential singularity of the function. We now show that an analogous theorem holds for the zeros of fx. Suppose that c is a limit-point of the zeros oi fx, and let us consider whether c can be either a point about which fx is analytic or a non-essential singular point of ^. In the former case we have fx = P(x-c) in the neighbourhood of c and therefore, by § 80, fx reduces to a constant. Excluding this case, we see that x = c must be 105-107] SINGULAR POINTS OF ANALYTIC FUNCTIONS. 1 89 a singular point. Assume, "f possible, that it is a non-essential singular point, and cons."aer the function ij/x. The zeros of /x are infinities of i//x, also c itself is non-singular for i//x. Hence i//x has infinitely many non-essential singularities at points that have x = c as a limit-point, without at the same time having an essential singular point a.\.x=c. This is impossible by § 104. We draw from what precedes the important inference that the zeros of a transcendental integral function Gx have no limit- points in the finite part of the plane: — in other words are isolated in the finite part of the plane. 107. Deformation of Paths. Let us understand by a circuit in the jr-plane any closed path which does not intersect itself A circuit divides the entire plane into two regions, — an inner and an outer ; each of these is said to be simply connected. The surfaces of (i) a sheet of paper without holes, (2) one with holes, provide examples of surfaces which are respectively simply and multiply connected. We shall have other examples later (§ 152). Suppose now that a region F in the ;tr-plane is simply con- nected and that P {x — x„) is any power series about some assigned point x^ of T. From this power series we can con- struct for the region T a portion of an analytic function fx, which we shall term a localised function o). This excludes the possibility of there being any obstacles to x in V ; such an obstacle would mean 8=0. We wish to prove that x is one-valued in T, and this will be done by showing that when x describes any two paths C, (or a/ii?) and C2 (or akd) which lie wholly within F, then ^x starting from an element at a will arrive by either route at the same final value at d. This theorem may be enunciated in the following manner: — T/ie difference between the final and initial values of x, or the change in ^x, when x describes any circuit ahbka in V, is zero. In proving this theorem we shall use paths instead of chains in order to simplify the exposition as much as possible (§ 91). The proof depends on the obvious propositions: — (i) The change of a power series when x describes a circuit in the domain of that series is zero. (2) When a path is described successively in opposite directions, for example akh and then immediately afterwards bha, the total change of x for the compound path is zero. Fig. 48. On Ci interpolate points c', c", ..., <:<"', <:<"+", as in fig. 48, where « = 3 ; and connect a, b hy b. path C which also lies 107] SINGULAR POINTS OF ANALYTIC FUNCTIONS. 19I within these circles and is intermediate in position between C, and Cj. On this new path interpolate points d' , d", ..., afw, x when x goes along C from a to ^ is equal to the sum of the changes from a to d', d' to d", d" to d'", etc., and these in turn can be replaced by the changes along ac'd', d'cV'd", d"c"c"'d"', etc., since the description by 4>x of circuits ac'd'a, d'c'c"d"d', d"c"c"'d"'d', ..., each of which lies wholly in the domain of one element, must restore the initial value. The paths c'd', c"d", . . . are described twice in opposite directions ; and therefore the corresponding changes destroy one another. The remaining parts make up d. Hence the change of i -Oi-Oj, (I -ai)(i -Oj) (i -as) > 1-0,-02-03, and so on. n Hence if the series 2a„ has a sum s, the products 11 (i —Om) 1 form a sequence of numbers which (i) do not increase, (2) remain greater than i —s. Hence they have a limit ; and the infinite operation 11 (i — «„) is convergent ; the limit is called the product and is itself often denoted by 11 (i — a„). Thus T/ie product 11 (i — a„) exists wJien 2«„ is convergent. When positive numbers ^^, ySj, ..., ySn, ••., less than unity, converge to /8, where /3 > o, then their real logarithms converge to Log ;8. For this is implied in the continuity of Log/3, and the continuity follows from the expression as a power series Log /3 = /8-I-i(y8- !)»+.... Hence, in our case, Logn(i-a„) has a limit, that is, 2 Log ( I -a„) has a limit, on the supposition that 2a„ has a limit. Ex. Prove L Log /3„= Log L /3„ from the formula Log \alb\ = Log I a I - Log | * |. 109] WEIERSTRASS'S FACTOR-THEOREM. 195 Next for any numbers a„, where | «:„ | =2„< i, we have I +«„ = exp Log(i + a„), where the logarithm is defined by the series If 2Log(i + fl„) is convergent, say to s, then the addition theorem of the exponential (§ 95) holds, and n L n (i -)- Urn) = exp s. 1 Now S Log ( I + «„) can be written as a double series, which, if we replace each term by its absolute value, becomes a, + aiV2 + a,V3 + -" + aj+(tjV2 + ct,V3+... + 03 + 0372 + 0373+ ... + , the «th row converging to — Log (i — a„). Now let the first column So,i converge. Then, as we have proved, 2Log(i— o,^) converges. Thus in our array of positive terms the sums of the rows converge. Hence: 2 Log{i + «„) is absolutely convergent wJien 2 1 13„ j is convergent. In this case we say that the product n(i+rt„) whose factors can now be arranged in any order, is unconditionally convergent. For example, n(i— jr^/w'') is unconditionally convergent, because 2 i/«' is convergent. But if 2a„ is not convergent while '%a.^ is convergent, then the above proof applies to 2 [Log ( i + «,j) — aj. For example let «„ = - i/(«+i). Then 2 ( + Log ) is convergent ; \«+ I " w+ 1/ ** that is, L[i/2 + 1/3+ ... + i/(«+ i)-Log(« + i)] is a finite number. Hence, adding i and changing « to « — i, we have Euler's result: L [i + 1/2 + ... + \\n— Log«] is a finite number, 7. 13—2 196 WEIERSTRASS'S FACTOR-THEOREM. [CH. XV For the calculation of this constant 7, which is "577215665 .., we may refer to Adams's Works, vol. i. p. 459. 110. Construction of Functions Gx yxrith. assigned Zeros. The well-known theorem that a polynomial of degree « can be expressed in the form C (x - a^) {x - a^) ...{x-aj, suggests naturally these two questions : — (i) How far is a function Gx determined by its system of zeros ? (2) Given a system of numbers a^, a^, a.^, ..., fl„, ... arranged in ascending or stationary order of absolute values and such that L fl„ = 00 , is it possible to construct a transcendental integral function Gx which shall be zero of order one at the a's and have no other zeros ? Note. The zeros are assumed to be of order one ; zeros of higher multiplicity are accounted for by the usual device of making several a's coincide. I. Functions Cj; with no Zeros. We have proved (§ 108) that a function Gx with no zeros is expressible in the form e^'^, wltere G^x is an integral function, rational or transcendental ; and accordingly we have the following theorem which contains a complete answer to the first question : — Tlte number of transcendental integral functions with tJie same system of zeros as an assigned transcendental integral function Gx is unlimited, but these functions are all of t lie form Gx^<^, wJiere GiX is a rational or transcendental integral function. II. Functions Gx with a Finite Number of Zeros. It follows from this theorem that the general form for a transcen- dental integral function with a finite number n of zeros, all simple, is {x - aj {x-a.^ ...{x- an) f "^ III. Functions Gx viWii infinitely many assigned Zeros. Let us now consider the second question. The answer is furnished by the following theorem of Weierstrass's : — I09-IU] WEIERSTRASS'S FACTOR- THEOREM. 197 Tliere exists a transcendental integral function Gx whose zeros are tJie points «„ and are sitnple. We proceed to the discussion of this theorem. 111. Weierstrass's Primary Factors. 11 (i-jir/«„), which n=l appears to satisfy the requirements, may diverge. Weierstrass replaced the factor i — xja^ by what he called a prhnary factor (i -.*:/«„) e*"W«»i, where ^^x denotes a polynomial in x. The primary factor is to be treated as a whole, and like i - ;tr/a„ it has one zero and one singular point (at 00 ). We shall now show how to choose ^nX. Because x = e^°^^, the typical term, say En, of the product n (I -.r/a„)^»w«") can be written ^Log(i-a/a„)+*„(«(i„)^ and this can be transformed in turn into exp-|-i-f^r"V-^ f^r^v ...1 {mn + I \aj nin + 2 \aj j mn if <^„;r be 2 xf^jn. Let X, An denote \x\, |«„| ; then to secure «=i convergence we must have X < A^. The series —{...} will be called the truncated series for Log ( i — x/an). Suppose that a,, is the last of the set a^, a^, ... that lies in any assigned closed region (R); then for all points of this region we can write Er+i ^r+2 ^r+3 • • • to infinity 00 = exp S [the truncated series for Log(i —xjan)] = exp-( i i xPjparA. Our first concern is with the double series i i xP/pa^P (I), for which we suppose X &R; the convergence of the product of the E's, depends on the convergence of the series. We shall prove that this double series can be converted (by summation by 1 98 WEIERSTRASS'S FACTOR-THEOREM. [CH. XV columns) into a power series in x whose radius of convergence is not less than R ; it will then follow that exp (— the double series) has no zero in the closed region {K). The following lemma will be needed in the discussion of the double series (i). Lemma. Given an infinite sequence of unequal numbers rti, ^2, ^3, . . . , «n, ■ • • ) arranged according to increasing or stationary absolute values and such that I^an^ ^ and | «i | > O, then it is always possible to associate with each number a^ of the system a positive integer m^ which will make (2) converge for every finite value of x. The values ;«„ = «—! satisfy the requirements of the lemma, for they make the ratio of the (« + i)th to the «th term tend to the limit o, and therefore make the series (2) converge. It is evident that many other choices could be made for the num- bers 7«„. We shall now use this lemma. Let us take w„ = « — i. The sum of the series of absolute values in the row containing ;»r/a„ is increased by the suppression of the coefficients i\p. The resulting series of absolute values 2 {XjAj^P is geometric and has for its sum (■-f-rXif-)'- <{■-#-!"(#)"• Hence the row in XjAn is convergent and has a sum less than A —7-^, where A, = R ^ (x/a^)} n=r+l converged ; we could then replace the product of the expo- nentials by the exponential of the sum (§ 95). Thus inserting * It is to be understood that « and m in this and the next two articles take negative as well as positive values, unless the contrary is explicitly stated. 112] WEIERSTRASS'S FACTOR-THEOREM. 20I again the factors corresponding to «,, «„, ..., a^ we have for one of the values of the logarithm on the left-hand side, log n (I - x\a^) ^-w«-) = i {Log (I - xla^) + 17', we have, for OS? 17'. II. Next, let ^ tend to 00 , | ?; | remaining less than or equal to T)'. Since G"x has the period i, all values taken by G"x- within the strip contained between the straight lines 7; = + 7;' are taken also in the rectangle bounded by 9; = + 17', f = 0, f = i. As G"x is finite for all finite values of x, its absolute value in this region must remain less than some definite number A ; therefore also | G"x \«= ± i, ± 3, ± 5 will be absolutely convergent, by the general theorem, and will, by pairing opposite values of m, give the preceding formula. Hence cos(7rjr/2) = n(l-;r/;«)«^"" (3), where m takes all odd integer values. By logarithmic differentiation we have ^ ta,n{Trxl2) = t\_i l{m — x) — i/m] (4), which may be deduced directly from (i) by the formula 2 cot TTX = cot {TrxJ2) — tan (7rxJ2). For the cosecant we have 2 CSC irx = cot (,'n-xJ2) + tan {'irx/2), whence tt csc ttx = - + !,' ( 1 1+2 ( ) X \x—2n 2nJ \in — x ml = i/^+ 2' (-)"[i/(;r- «)+!/«] (5). As a further illustration of factor-formulae, let us resolve sin TTX — sin nra into factors, where a is any non-integral number. The zeros are x = ni—a, and x=2n-\-a, where m is an odd in- teger. As 2 i/(« — of is absolutely convergent so are 2 i/(»« — of and 2 i/(2« + a)l Hence ■* \ 7M.-a TT / _ ■* sin 7r;tr- sin ira = £«^n i =— e™-" Hi =— /"+", \ m — a) \ 2n + a) and the extraneous factor is determined precisely as for sin irx 113] WEIERSTRASS'S FACTOR-THEOREM. 205 by showing that G"x is a constant and that this constant is zero. We have then G'x=b, -ircosTTX , , V / I , = * + 2 — +- sin irx — sin -jra \x-m + a m — a + 2f^i + ' \x — 2n — a 2« + a.l and, when x = o, b = - 7r/sin -n-a. Hence exp Gx = exp {c - Trxjsm ira) ; and since, when x = o, — sin ira = exp G^o = exp c, we have finally exp Gx = — sin ira . exp (- ttx/sit) va). In particular if a = - 1/2 the system m-a coincides with the system 2ti+a, each being of the form (4« - i)/2 ; and we have I + sin TTJr = e"* II [( I - 2xj(4n - i )) ^/(4n-i)]2_ or writing j:/2 for x I -l-sin (■7rx/2) = e'"'' U [(i -;t:/(4M - i))g»^/i*«-i'p. Hence from (3) l+sin(7r;.r/ 2) ^ ^,^ U (l - x/(4n - l)) e' i^-^^ cos (7r;tr/2) 11 ( I - .*-/(4« + I )) e^nm+i) ■ Hence, by taking the derivate of the logarithm, j7rsec(7rjr/2) = 7r/2 + X/™+'f-^+-) (6), ' \x— 7n fnj where m is any odd integer, the factor /"'+' being + i or — i as m is of the form 4« — i or 4« + i. The formulae (i, 4, 5, 6) effect the resolution of the elementary fractional circular functions into partial fractions. Instead of basing these on the factor-formula we might prove them independently. For example, in the case of cot ttx, there is a simple infinity when x—n, but 7rcot7r;i:— i/{x—n) has no infinity when x = n. We cannot use the expression Trcotirx—'^jKx—n) since the sum is not convergent ; we can however pair off opposite values oo of n and say that Trcot 7r;ir — 2 [i/(;f — «)-|- i/(jr+«)]— i/x has 1 no infinities in the finite part of the plane and is therefore Gx. 2o6 WEIERSTRASS'S FACTOR-THEOREM. [CH. XV Or we can render '%ij{x — n) absolutely convergent by adding to each term i/« where n is not zero. Thus TTCOtTTir— 2'[i/(;r— «)+ i/«] - ijx is at most a transcendental integral function; which is then shown to be zero. This point of view leads to a very important theorem known as Mittag-Leffler's theorem, by which we can substi- tute for the simple infinities a^, a^, ..., a^, ■■■, where L,a^ is 00 , isolated singular points of the most general kind which a one-valued function can have, and then write down an expres- sion for such a function. For this theorem we must refer to more advanced treatises, as we shall not require here more than can be deduced at once from the factor-theorem. Ex. I. Let /x=n{i -x^/m') where m is odd. Prove without appeal to known properties of the cosine thaX/{x+^)=/x. Ex. 2. Deduce (3) from (2) by using the formula sinTT (jr+l/2) = C0S7rjr. Ex. 3. Deduce (4) from (i) by the formula cot n {x+ 1/2)= —tan TTjr. Ex. 4. In (i) write for x, =^+217, the conjugate number x, =|-2ij, and then subtract the formula so obtained from (i). We obtain jr (cot nx- cot Trx) = {x~x) S, where ^—2 . _ w-_ ., = the sum of the squares of the reciprocals of the distances between ($, ij) and the points o, +1, ±2,.... Hence prove that „ TT Sinh 2377^ Tj cosh 2Jri; — C0S2»r|" Ex. 5. Comparing (2') with the known power series for sin nx, prove that 2 l/«2 = jr2/6, 2 2 ■lln^hl^ = n*liogr^=l + ^-i^, + ^^, + +^.+ Calling this function ■yjrx, we have tjr(x+ i) — i}rx= — i/x-, yfr (l -x) + i^X= 2 lj{x+ ny = 7r' CSc" ttX. From the properties of ifrx it is possible to evolve those of Tx; this has been done by Hermite. Ex. I. Prove that ylfx+ ylr{x+lln) + ...+yjf{x+(n- l)/n) = n^ {nx). Ex. 2. Prove that TaTx ^i _^ ^ (-)"(a-i)(a-2)...(a-w) i r(rt + jr) X .1=1 I.2...W jr+«' CHAPTER XVI. INTEOEATION. 115. Definitions of an Integral. There are two available definitions for a definite integral I f^d^, where /f is a function of a real variable f. One of these treats the definite integral as the limit of a sum of elements /frff, the other treats integration as a process inverse to differentiation. These definitions, which we shall call the first and second definitions, can be generalized so as to apply to analytic functions of ;r; we shall discuss both definitions and show that they lead to the same results. Let us begin with the simple example / af^dx, where « is a positive integer, and use the first definition. First we are to think of a path from jTo to x; we shall call x^, X end-values of the path, in preference to the misleading term limits. On this path we are to interpolate it, points x,, jtj x^ so that we have /i + 2 points x^,x-i,x^, ...,x^+i, where x^+, is merely the final value x itself The points are to be so inter- polated that by increasing /* sufficiently, all the strokes x^+i — x^, where X = o, I, 2, ...,/*, can be made as small as we please; the points may for example be at equal distances, or may divide the path, which we suppose to possess length, into equal arcs. By ;r**^we understand a typical expression for XiJ^(x^+i — Xx), and by the integral itself the limit (if there is one) of the sum XxK'^(xx+i—Xt,), when fi tends to infinity. A=0 M. H. 14 210 INTEGRATION. [CH. XVI The novelty lies in the infinity of possible paiAs of integration between x^ and x ; in the case of the real variable there are but two, namely along the real axis between x,, and x, or along the rest of the real axis via oo , Now we have <+l - ^a"+' = (^A + h^r^^ - x^+\ where /ix. = Xk+i - Xk- Hence ;ir^+i-A:x»+' = («+l)/i*(;rx» + eA) (O. where ex can be made as small as we please by making h^ small enough. Let e be the greatest of the numbers ex where \ = o, i, 2, . . . , /i ; then 6 is also as small as we please, if we take yx large enough and therefore all the numbers /«x small enough. Adding the /t + i equations (i) we have ^M+1 -^o"^' = (« + !) 2;«rx»Ax + («+!) Sex^x ; or «"+! _.i-n+i "^^^ -S;irx"/^x «+ I = |Sex^x! g2|ex||/^x| Se2|/^x|. But when fj. tends to oo , e tends to o and 2 | /«x | to the length of the path. Thus, for a path of finite length, ~n+i_ 71+1 lim2;rx"/^x= "^^ ^ ° ■ or f A^fl:»r = l^ ^._ (2). In this case the integral is independent of the path, but this is merely owing to the simplicity of the instance ; we have seen already that I dxjx does depend on the path from x„ to x, and this is a fact of cardinal importance. It will be observed that in this case when « is — i the right-hand side of (2) is meaningless; for all other integer values of n the equation (2) holds good, and the proof is the same, except that we appeal to the binomial IlS-n6J INTEGRATION. 211 theorem for a negative exponent ; but when « is a negative in- teger we must exclude paths which pass through the origin, for the origin is an infinity of;*;". Now let us consider the equation (2) in connexion with the second definition of integration. The function ;s*' will be replaced, for the sake of greater generality, by the power series P(x-c) and x„, X will be supposed to He in the domain of this power series. By the indefinite integral of P{x-c) = an->ra^{x — c)->ra^{x-cf-)r..., is to be understood the new power series P (x - c) defined by P {x — c) = jP {x — c) dx = const. +ao(x—c) +.. X, J Xa A is a constant and t/ie path from x^ to x is the same for both integrals. IV. The integral \ {f^ +fjc + . . . +/„;«r) dx, where the path of J L integratiofi is Z, is equal to the sum of the integrals \ f^dx, j f^dx,..., j f^dx. The natural generalization of IV. is to replace the sum of the nf's by an infinite series which is uniformly convergent over L. If we wish to change the variable x in the equation Fx=jfxdx 117] INTEGRATION. 217 by a substitution x=4>y, where y = jf^y . <^ydy as for a real variable. For by differentiation we get p'y • <^> =/#' • y, that is, F'y =fi>y, which is right since F'x =fx. An examination of the definite integral shows that the value of that integral may depend on the chain or path from x^ to X, since Fx was defined by means of a particular chain of series, and therefore Fx in Fx — Fx^ may be altered in value by the employment of a new chain or path originating from the element about f„. An illustration is afforded by the formula dx Jo r = tan~'jr. /. i+x^ Here the integral of a one-valued function is a many-valued function. The formula only receives its full meaning when account is taken of the infinite multiplicity of paths from o to x. The general question suggested is this : given an analytic function fx and two patJis L, L' from x^ to x, is the value of X fxdx for the path L equal to or connected by a simple relation with the value of the same integral for the path L'f Suppose that fx under the sign of integration is replaced by an element of an analytic function, and that xjix, xjix are two paths lying wholly within the domain of this element ; then denoting by (xjix), (xjix), {xkx^ the integrals from x^, to x along the paths xjix, xJix, and from x to x^ along xkx^, we have proved (§ 1 16) that (xjtx) = (xJix), an equation which can be transformed into {x^) + (xkxo) = o ; this means that the integral over the closed path xJixkXf, is zero. Cauchy discovered a general theorem relating to the vanishing of jfxdx when taken over a closed path ; or, — replacing such an equation as {x^x) + {xkx^) = o by {x^) = {x^x), — to the possi- bility of deforming one path from -r„ to x into another from Xo to X without altering the value of the associated integral. 2l8 LNTEGRATION. [CH. XVI The regions of which we shall speak in the subsequent articles of this chapter will be assumed not to contain the point 00 unless the contrary is explicitly stated. — T' 118. Cauchy's Theorem. Given that fx is analytic over a simply connected region V, then over any circuit C in V, L fxdx = o. c Consider the analytic function Fx = I fxdx, where Xg is some fixed point of F and the path from x^ to x lies wholly in T. Corresponding to each element of g, p'g', . . . is described twice in opposite directions. As /xdx+ I /xdx=o, Jp Jq with similar equations for p'g', ..., all the integrals due to the new lines disappear and the remaining integrals combine to give I fxdx+\ /xdx + ... + l fxdx — o. J C J Ci J c„ 119. Residues. W/ien the function fx is as before, except that r contains a finite i^umber of non-essential singular points Ci, Ci, ..., cx. of fx, the integral \ fxdx = 2m'^a-i, Jc 220 INTEGRATION. [CH. XVI where the integral is taken positively over the curves that constitute tlu boundary CofV, and a_i is the coefficient of il(x-c) in the expansion 2 an{x—cY at a singular point c. n= -m Since the integral of the sum of a series with a finite number of terms (say n+i) equals the sum of the integrals of those terms, it follows that if I denotes integration in a positive sense over He) the contour of a small circular region with centre c, then 2 an{x-crdx=a^l -—- + .. . J(e)n=-m Jl.e)\X — C) + a.J '^-+\ P{x-c)dx, i(c)X — C J(c) = a-i\ -^ = 27r«a_, (§30), because all the other terms in the series of integrals vanish. Here we make use of | :i^dx=o when the integer m^— i,and J (e) of fPxdx = o when the integration extends over a circuit in the domain of Px. Let all the points c^, c^, ...,C\ be made the centres of small circles and let V be bounded by the curves C and by these small circles. Since _/5: satisfies all the requirements of Cauchy's theorem for this region V which has arisen from V by cutting out the t's, it follows that I fxdx — I fxdx — I fxdx — ... — I Jxdx = o, where the circles of centres Ci, c^, ...,Ck are described positively with respect to their centres. Hence L fxdx = 2iri'^-i. ' c This theorem shows that the coefficient a-i and the term 00 a_i{x—c)-^ in an expansion 2 a„(jr — <:)" play a specially important rdle. Cauchy has given to «_, the name residue relative to c, or at c. At non-singular finite points there is of course no residue ; but the point 00 needs special examination. 119] INTEGRATION. 221 When CO is a non-essential singular point, the expansion — oc about 00 is 2 ^z^, where w is a positive integer; when oo is a non- n=m — 00 singular point we have the expansion 2 a,^ ; thus we have a n=0 term a^^\x even when oo is a non-singular point By the in- tegral round oo we understand the integral in the negative sense (that is, the positive sense with regard to oo ) round a circle so large as to include all singular points other than the possible singular point ;r = oo (supposed isolated) ; thus /, fxdx = — 2itia_-i . (00) Tlu residue at oo is defined to be — a_i. For one-valued analytic functions we have the following general theorems : — The integral of a one-valued analytic function round a circuit which contains only non-essential singular points, and of these only a finite number, is 2tri . {the sum of the residues at these points). Similarly we can prove that if the only singular points outside the circuit are non-essential, and are finite in number, the value of the same integral is — 2-n-i . {the sum of the residues at these points). Hence it follows that the sum. of all the residues of a rational fraction is zero. The inference is that an integral from x^ to x will alter its value, in general, wlien tJie path is deformed continuously from xjtx to xjix, if it pass over singular points in the process. When a function is one-valued and analytic, the value of ^ fxdx taken positively over a closed curve A is unaffected by any continuous deformation of A which does not involve a passage over singular points. For let A, B be two positions of the moving circuit. By Cauchy's theorem I fxdx -f- / fxdx = o, where the second integral is described negatively with respect to the interior of B. over 222 INTEGRATION. [CH. XVI In particular when a circuit can be made to vanish by a continuous contraction which does not require a passage over a singular point of the function, the value of the integral of the function taken over the circuit is zero. 120. General Applications of the Theory of Residues. Suppose XhaXfx is one- valued and analytic about all points of a region V bounded by one or several closed curves, as in fig. 5 1 , and that c is a point in V. The integral — : I , taken ^ *• Zirtj^x — c a circuit A in F, may be called Cauchy's integral, for it plays an essential part in the development of the theory of functions along Cauchy's lines. We shall assume that c is not a zero of fx, and that A can be contracted continuously until it vanishes, without passing out of T. Within the region bounded by A, the function fxl{x-c) has only one singularity, namely an infinity of the first order at c. The residue at this point is found at once by the use of Taylor's theorem ; for the element at c of the analytic function fc +f'c {X - c) +f"c {X - cy/2 !+..., and therefore the residue in question is^. But /, fxdxiix— c)= ZTt {the residue o{/xl(x—c) within A} = 2'iri/c ; hence we deduce this very useful theorem, due to Cauchy : If fx be an analytic f miction of x which is one-valued and analytic about the points on and interior to a circuit A of the x-plane and if c be a point interior to A , then fc^±-.\f^^ 2iniAX-c' t/te integral being taken positively over the circuit A. Ex. Prove that if x-c be replaced by {.x-Ci){x-Ci)...{x-c^, then J_/' fxdx ^ A fc^ I II9-I2I] INTEGRATION. 223 If about a point c,fx= (x — c)~^ P„ {x — c), then c is an infinity of order m. If w be a negative integer, then t is a zero of order m. Let then m be an integer positive or negative ; and let /x=(x-c)'»Po(x-c). Then ^og/x = m log {x — c) + log /•„ (^ — <:), = tn log (x-c) + Po {x - c), where the circle of convergence of the new power series extends to a zero or singular point of ^. Hence f'xjfx = ml{x -c) + P,{x- c). Let ^ be a function which is analytic over a region V which contains a finite number of zeros and isolated singular points of fx, the latter being all non-essential ; then gx=gc+g'c{x-c)-\-..., where t is a point of T ; hence when <: is a zero of ^ whose order is m, we have gxf'xjfx = mgclix -c)+P„{x-c); and the residue of gxf'xjfx at c is mgc. Similarly if cf be an infinity of order m' lying within F, the residue oi gxf'xjfx at c' is — m'g(^. Hence HA be a circuit lying in the region F, -^ / gx d log fx = 't {mgc -m'gd), 2Tr» J ^ the summation applying to all zeros c and infinities cf of fx which lie within A ; it being of course understood that when c is a zero its order is tn, but when c is an infinity its order is ;«'. When gx is a constant we have again a result already considered (§ 1 08). When gx = x vre have the following theorem : — — . I X -^ dx = tmc — tm'c, 2111 J A. fx that is, the integral is equal to the sum of the values for zvhich fx vanishes diminished by the sum of the values for which fx is infinite, multiple zeros and infinities being counted as often as is indicated by their orders of multiplicity. 121. Special Applications to Real Definite Integrals. We shall illustrate, by examples, how the theory of residues 224 INTEGRATION. [CH. XVI can be applied to determine integrals of real functions for special intervals. Ex. I. Consider the integral J/'dxl(x — tfi). Let /9 be positive, and let the path be the real axis from — p to p and a semicircle on this as diameter. The integral along the base is I ^d^li^—ifi)', and along J -p the semicircle it is I e^''iddj{i — i^jx), where x = pcis 6. Jo When p tends to infinity this second integral tends to the limit o, for ^^ = eipcos9^-(>sin«^ and while the first factor is i in absolute value, the second tends to zero when o < ^ < tt. Also by the theory of § 119 the integral round the closed path is 27r?. (the sum of the residues inside). There is only one residue, namely when x = j/3, and this is er^ ; thus the integral round the path is 2Trie~^ ; and we have or whence If we replace the positive number $ by the negative number — /3, then we have no residue inside the path, and therefore — /3 cos f + f sinf Hence or ^' + /3» -//f=o. cos^ cos f f sinf d^ = j°_ d^ = ire-l>, I2lJ INTEGRATION. 225 /. Too fc sin f Ex. 2. The equation |^ — m^^ = '^^~^ is proved only for , positive values of /3 ; it suggests that when /3 =0 we shall have ^— ^f = TT. But an integral found on the supposition that 2/3 is within the path must not be assumed to hold when 2/9 is on the path. When j8 = o we alter the path by describing a small semicircle about O, say of radius e as in fig. 53, and replacing the interval (— e to e) of the real axis by this semi- circle. Then the integral is zero since there are no residues inside, and it reduces to J — zo J IT J e The second integral tends to the limit I idd or — Trt when e tends to zero. Hence f e'^d^l^ + T e'^d^/^ = 7ri, J -a> Jo or I sin ^d^/^ = tt. Ex. 3. The function ^"/(i +e^), where a is real, has poles at x = (2n + i}m. Integrate along the rectangle whose horizontal and vertical sides are given by r/ = o, 27r, and ^= ± p. Ziri ■ P 3 "- Fig. 54- There is but one residue inside this rectangle, namely when x=ivi; and this residue is - e''". The integrals along the vertical sides are The limit of the first when p tends to 00 will be zero if a < i ; M. H. '5 5< 226 INTEGRATION. [CH. XVI the limit of the second will be zero if a > o. Let then o < a < i ; there remain the integrals along the sides, that is, /•"= g-f ,j. 27r? , . . at = — ^ „• = TT/sin aTT. j_„i+e^ ^ ^'"- £-"" ' \ ' Ex. 4. If in the preceding example we replace a by a + 2/3 or a, the argument is not altered provided o< a< i. Hence J ^-fq7^^^ = Wsin(a + 2/3)7r, r g "-*'^cis/8g sin(a-z/3) 7r I .^ 2 cosh ^/2 ^ ~ sin (a + ?/8) tt sin (o — «/S) tt sin OTT cosh /Stt — / cos ott sinh /Stt or or or = 27r cosh 2/37r — cos 2a7r Hence, writing a — i = fi, so that — i/2 from which, equating imaginary parts, we deduce /. _oc I — ^ d^ = 'Tr cot OTT. By equating real parts we obtain again the result of Ex. 3, so that the rectangle of this example gives both integrals. Ex. 6. When we replace a by a = a + i^ we have still I — -. dP = nr cot air ; for we have still I . d^ = ■„ „ ; and as this value led to ir cot air when a was real, it must do so when a is complex. Hence, proceeding as in Ex. 4, we have £(»-»« cis/3? /, _^ 2sinhf/2 " e*"^ cis /3^ d^=--K cot (O + /^) TT, J_„^^isn;l'^^='^*""^'^+^^^"- Hence T ^-^^^ rf| = tt tan (>t + ^/8) tt + tt tan (/. - e^) tt j_„o smhf/2 27r sm 2yjK cos 2/i7r + cosh 2/37r ' 15—2 228 INTEGRATION. [CH. XVI whence, changing the sign of /a and subtracting, /. sinh /^? cos BfdB = ZL^ill^'^ sinh f/2 ^^ ^ cos 2fnr + cosh 2/37r Similarly we have . f" fll^ j8| ^fc ^ ^ ^^^ (^ ^ ^-^^ TT - TT tan (;ii - 2/3) tt j-„ sinhf/2 _ 27r? sinh 2^Tr__ ~ cos 2/u,7r + cosh 2/377 ' , pcoshuf . ^^.j. TT sinh 2/377 whence I . , /y sin/3?flf= v ^o ■ Jo sinhf/2 ^ ^ cos 2/177 + cosh 2/377 The integrals in Examples 4 and 6 are required in discussing Fourier's Integral (see Byerly's Fourier's Series and Spherical Harmonics^ Ex. 7. Finally let us take ^e-^dx over the rectangle whose horizontal and vertical sides are t; = o, /8 and f = ± a. a -1-2/3 Within the rectangle e~^'^ has no singular points. Hence the integral is zero. Along two sides it becomes I e~^ci^ and I ^-(f+'WVf; along the others it is 1 e-^^+^^y idr/ and I ^-(-''+^^)'/^77. These last two tend to zero when a tends to infinity. Hence J — QO J GO or £:^'[°° ^-f^cis(-2;8f)=r ^-fW^; I2l] INTEGRATION. 229 and the integral on the right is known to be •n-"^. Hence e-^ cos 2^^d^ = nf-'-e-^-. 1: Lest it should be supposed that the method of evaluating definite integrals, as described above, is in all cases the most suitable, we remark that it is difficult to determine | e^^Wf itself by this method. CHAPTER XVII. LAURENT'S THEOREM AND THE THETA FUNCTIONS. 122. Laurent's Theorem. Lety5r= i/(x — a) {x — b), where o<|ai<|^|. The function has two simple infinities, namely x = a and x = b. About the origin there is an element Px whose domain extends to a, and about oo there is an element P{ijx) which converges at all points exterior to (| b |). But there is also a series Px + Q(i/x) which converges when |«|<|jr| I ^ I we must use expansions in ajx and bjx ; thus , _ i_ -.^ (J. i_\ „ ■^^~ a- b „r-i U"+' ^"+V "^ ' (3) For the ring we must use «=0 n=-l ij(x~d) = - S x»/b IJl+l n=0 Hence /x = a — b 2 ;«:»/«»+■ + 2 j:»/3»+' »=-l n=0 Laurent's theorem is a generalization of this example for a one-valued analytic function fx which is analytic over an open region bounded by two concentric circles {R) and {R"), where R> R'. Let C, C be any two circles about o which have radii slightly less than R and greater than R' respectively. Applying Cauchy's integral to the ring bounded by C and C , we have fxdx fc= ^ A f^ - ' f -^ 'X — c where ^ is a point within the ring and the circles are described positively with respect to the regions interior to them. For points X oi C and points ;ir of C" we use respectively \\{x-c)= ilx + c/x'+ ...+ <:'V;tr»+' + c"+'/x''+'- (x - c), ll{x-c) = -llc-xjd- ...- A-"/c"+i - ;tr"+Vc»+' (x - c). Multiply the two series by /x; the resulting series converge uniformly and can be integrated term by term. Hence /c = ao + aiC + a2C^-\- ... +a„c"+ ... + a_i/c + a^jc^ + . . . + «_,i/c" + . . . , where a„ = - . f /xdx/x''+\ R2, and 2 a„ (;ir — c)" converges uniformly along (c, R) because n=0 Rz (cos nz + i sin nz) dz, ^7r J 1 24-125] THE THETA FUNCTIONS. 237 that is, — I ^s (cos ny cos nj: + sin ?ij/ sin «^^) a'.:-, IT J or ~ N>'^ cos n{y—z) dz. When « = o we have the single term — - ^zds. Combining these results ^y=-\ 1+22 cos M (j — z) (bzdz. This is the ordinary form for Fourier's series; it should be remarked however that it has been established under conditions that differ from those that usually occur ; the theorem in the present form holds for complex as well as for real values, but this gain in generality is accompanied by a loss in another direction, due to the restrictions on ). Now xf{q'^x) = —fx; hence — exp 27ri (z; + w/2) ^3 {v + o)) = %v. Changing v into i; + J, we deduce exp 2'7r2 (v + * cos irv + 2^*'* cos 3'jri' + 2q'^'* cos Svrt' + . . . , a series which will be denoted by '^xV. For convenience of reference we repeat the formulae for the four ^-functions : they are ^jj; =1+2^ cos 2Trv + 2q* cos 4777/ + 2^ cos Sttv + . . . , ^2^^ =1- 2q cos 27rz; + 2^ cos ^ttv — 2q^ cos tirv + . . . , ^if = 2^"'' cos -KV + 2^«'* cos lirV + 2^^'* COS StTI^ + . . . , ^7/ = 2q^^^ sin TTZ^ + 2^"^ sin Stt^- + 2$''"'^ sin ^ivv - ... ; of these functions the first three are even, the fourth odd. Detailed information on these functions will be found in works on elliptic functions. We now take up a more symmetric function with the same arrangement of zeros as the ^-functions. * The reader must be warned that in the literature of the subject there is a diversity of notations. The notation adopted here is that used by Jordan in his Cours d' Analyse, vol. ii. (1894) ; it differs from that employed in our Treatise on the Theory of Fiitutions, (1893). CHAPTER XVIII. FUNCTIONS ABISma FROM A NETWORK. 127. The Network. All the points of a plane which are given by the formula where zm^ and 20)3 are given constant complex numbers and wz, and Wj are any integers, positive, zero, or negative, are said to • • • • 2Uj 2u,+2u, 4u,+ 2u, • • • • -2u, Su, 40 • • • • -2,-2&r, -2fc)j 2c.;,- 2w, 4<«',-2Wj Fig- 57- form a network (fig. 57). We suppose that the ratio Wa/o), is not real ; thus avoiding the degenerate case in which all the points lie on a line. We can call w a multiple of 2,m^ and 2w^. Any two numbers which differ by a number w are said to be congruent one with the other. The sign = means ' is congruent with.' Now we can select from among the w's pairs of numbers other than 2t<)j and 2&)2 which will equally give the same network ; for example 2w^ and 2(«i)i-f-<»2)- For clearly 2Wia)i + 2;«2(Q)i+(»2) 16 — 2 244 FUNCTIONS ARISING FROM A NETWORK. [CH. XVIII gives the same points as 27«iO)j + 2ni^a.^, though of course not in the same order. We call any pair of the numbers w, with which the same network can be built, a primitive pair. The con- dition that a selected pair, say 2w,2; the summation being for all points w = 2w/,ft>i-l-2»«2W2, except the origin. It is to be remarked that these convergence theorems are true for all real values of \, > i in the first case, and > 2 in the second case, but we are concerned only with integer values. The points w (fig. 57) lie on the sides of the following parallelograms with centres at the origin. The first parallelo- gram has for its corners + 2a)i + 20)3 and there are 8 points on it. Let h be the least distance from the origin to this parallelogram ; then for the 8 points 2i/|«'h<8/S\ The second parallelogram has corners at + 4«i + 4a>2, and there are on it 8 . 2 points. Its least distance from the origin is 28, hence for these 8 . 2 points 2i/|ze'|^<8.2/(2S)\ 246 FUNCTIONS ARISING FROM A NETWORK. [CH. XVIII The wth parallelogram has corners at + 2« i, that is, if X> 2. Hence the series S'i/| w |* is convergent if \ > 2. Similar reasoning, when 8 is replaced by the greatest distance from o to the first parallelogram, shows that 1!\\\w ^ is divergent if \=2. 129. The Functions au, %u, f)«. We have found in sin u a simple example of a transcendental integral function of grade i ; we proceed to the construction of a transcendental integral function of grade 2. Since 2'i/|z£'|^ diverges and 2'i/|«'|^ converges, the trans- cendental integral function which is o' at, and only at, the points w must be of the second grade (§ iii); the typical factor is therefore ( I — ulw) exp (w/zf + it'jziv'^), and the infinite product till' {(l — «/z£/) exp {u/w + It' 1 2211/')], where tJie accent signifies the exclusion ofw — o, is a transcendental iiitegral function which answers the requirements. This function is denoted by + ic-l2ufi)\ ^u = ilti + l,' {i/{u — w)+i/w + ujw'^] ^u=l/te'' + l'{il{u- Tvf -ijw-] show that an, §>< are odd functions, fti an even function, that the residues of i^it at its infinities are all equal to i, — or, as it is often stated, l^ii is infinite at m = o like ijti and at 11 = w like I l{ti — w), — and that the infinities of i^u are all of order 2 with zero residues. Further they show that ^u = D log a-M, fti = — D^ log ait. When ate, ^u, fu are regarded as functions of three variables u, 2a)i, 2(1)2, the functions become homogeneous of degrees i, — 1,-2. For example every factor in aujti is homogeneous of degree O, and therefore ati is homogeneous and of degree i. The derivate of fu is P'« = -.7-^'(^73^' = -"^(^W ^^^' an odd function with a triple infinity at each point w and no other infinities. We have derived i^u, 2, where m-^, m^ are integers. For since the equa- tion p' {u -I- rt) = f'u holds for all values of ii, a is an infinity of (gin, as is seen by putting « = o. But the only infinities of Iglu are the points w ; hence a is one of the points w. For this reason 2<»i, 2a>^ are called a primitive pair of periods of (glu : — not tJie primitive pair because (§ 1 27) they can be replaced by others which would give the same network of points w. * We refer to other works for the proof that a one-valued analytic function cannot have more than two periods. 130-132] FUNCTIONS ARISING FROM A NETWORK. 251 The property of (glu that its periods form a network is one possessed by all elliptic functions fu. This means that elliptic functions have two and not more than two independent periods of which all other periods are multiples. An elliptic function takes the same value at all congruent points. Thus if we know one point (say u^ for which (g!u has a given value (say o), we know further that (glu is o at all the points «„ + w. This of course does not imply conversely that all the points at which ^'u is o are included among the set u^-^w\ what it means is that the zeros arrange themselves in one or more networks. This peculiarity of elliptic functions, — that the points at which they take an assigned value fall into a finite number of networks, — is characteristic. It suffices for many purposes to consider such functions in a cell (§ 127) instead of over the whole plane. For the behaviour of the function in the cell whose corners are «o. «i, «„ + 20), + 20)2, «'(ii'-«) = p' (-?/); and therefore, since f' is an odd function, p' (w — zi) = — iglu. Let u = w/2 ; then i^'(wlz) = -(gl{w\z), whence f' {w\z) is either o or 00 . The points k'/2 include all points 7«,cb, + vi.p^ ; when m^ and m^ are both even this is simply the old network w, which we can call the network of periods ; when w, and m^ are not both even. 252 FUNCTIONS ARISING FROM A NETWORK. [CH. XVIII we have three new networks (fig. 59)* which can be called the networks of half-periods. At these latter points {glu is zero, since it was 00 at the network of periods and there only. ^ ^; < / '' / / / >v -^■. ^ Zeros and infinities of <^'u. Fig. 59- 133. Are p, f, a periodic ? Having found that ,. Thus ?(?< + 2<»x)=?'« + 2jJx (lO). Hence f (2< + 2W, + 20), + 20)3) = ?(?/ + 20), + 20)3) + 27;, = ?•(« + 20)3) + 27;, + 27;, = 5i^ + 27;, + 27?, + 27/3. But by definition 2^]. Let u = — w\2 ; then since cr is odd — I = exp [— w^(wl2) + k], and we find, by division, that <7 (zi: + w)ja-u = — exp [(2a + w) f (a//2)]. In particular C7(?/ + 2&)a) = -£21x(»+'»x)o-?< (12). Thus o-?/ is not periodic; it is said to be qtiasi-periodic, because the function is reproduced save as to a simple exponential factor. CHAPTER XIX. ELLIPTIC FUNCTIONS. 134. Case of Iiiouville's Theorem. The existence of elliptic functions has been established by the actual construction of three functions of this kind, namely %u/^^e, p«, p'u ; we can at once extend the number. For clearly any rational algebraic function of fpu, f'li is a doubly periodic function and also a transcendental fractional function ; or again the derivate of an elliptic function is another with the same periods. The most important of all the general propositions employed in the discussion of elliptic functions is to this effect : — No integral function, whetlur rational or transcendental, can be doubly periodic unless it reduce to a constant. Let Gu be such a function ; then j Gu \ is less than some fixed positive number -y when u is confined to a definite part of the z^-plane, say within the parallelogram of periods (assumed to exist). The periodicity of Gu carries with it the inequality I Gti I < 7 for all finite values of u. This result being at variance with the theorems of § 56 and § 93, the theorem is proved. 135. Integration round a Parallelogram. The structure of an elliptic function fii is discoverable from the behaviour of the function in a cell T (§ 131); Cauchy employed for this purpose the method of integration round the parallelogram 256 ELLIPTIC FUNCTIONS. [CH. XIX which bounds the cell. The spirit of this method will be apparent after a study of the following applications. I. We suppose T so chosen that no side passes through a w. The integral I fiidu taken round T is zero. For at con- i T gruent points on opposite sides of the parallelogram fu has the same value ; so that I fudu= fudii^ whence I fudu-^\ ficdii — Q. J U„ J H„+2Ui+2Uj These are the integrals along the first and third sides. Similarly for the second and fourth sides I /udu + I /udu = o ; thus the whole result is zero. Hence, by § 119, the sum of the residues in T is zero; and it follows as an immediate con- sequence that if /u has in T only one infinity, tJu infinity cannot he of tJie first order. But fu can have in T a single infinity, if tJu infinity is of the second or higher order ; this was the case with p«. II. If gu be any polynomial, then from Cauchy's theory of residues (§ 120), gu d \ogfu = 2iri \%mgc — ^.m'gc'l, L T where c is a zero oi fu of order m, c' an infinity of order m', and the summation is for all zeros and infinities in T. If gu be merely i, then the formula is / f'udulftc = 2'iri(Iim — Xm'). T But the same argument as before shows that the integral vanishes. Hence Xm = 2>«' ; that is, in T the number of the zeros offu is equal to tlie number of infinities, each zero or infinity being counted as often as its order indicates. Further the function fu — k has in T'the same number of zeros as infinities. But an infinity oifu — k is an infinity offu; hence 135] ELLIPTIC FUNCTIONS. 257 the number of points in T at which fu takes any assigned value k is independent of k. This number is tlie order of the elliptic function. Thus i^it is of the second order and has two zeros in a cell. III. If instead oi gu = i we take gu = it., we have tif'tidiilfii = 2-Tri {Xmc — 1,in'c'). ' T But nf'ujfu takes at the points u and 21 + 2(0^ values whose difference is 2m«f'ii\fii. Hence the integral along the first and third sides of T is 2ai2f'tidicjfu L -I' J u = - 2&)„ [logfiUo + 20)2) - logfu^] = a multiple of 47r/wo. So for the second and fourth sides ; whence "^mc —'Zm'c' = some multiple of 2coi and 20)2, or Xmc = '2m'c' ; or, in words, tlie sum of the zeros is cotigruent with the sum of the infinities. More generally, applying the same argument to ft — k, we can say that the sum of tJie arguments for which an elliptic function takes an assigned value is cotigruent with the sum of its infinities. For the function (§u the theorem tells us nothing new; for we can infer from the even character of the function that the solutions of fii - k pair off into opposite values u and — ti. IV. Weierstrass's form of Legendre's Relation*. Let us integrate i,u round T. We have along the first and third sides t,udu-\ t;{u-ir2(o^)du, rMo+2i 0)0 j and similarly % T/l j = ■^272- «D3 eoi I CO2 0)3 In our choice of T we have made the chief amplitude of a)o/o)j lie between o and -n- ; that is, the coefficient of i in a./co^ is positive. Had we taken the chief amplitude between o and — tt this coefficient would have been negative. In the one case the description of the parallelogram is positive, in the other negative; and therefore this second case would have given — 7ri/2, not ■7ri/2. We shall give an instance of the way in which this relation can be used to simplify formulae in elliptic functions. By successive applications of the formulae all, we get -i this becomes o- (« + z£/) = (-)'".+»»!+™'i"'» exp 27] {u + wl2) . all (14), where »? = Wjiji + m^^. ■^ 136. Comparison of Elliptic Functions. Consider two elliptic functions _;^?< and_/^?^ with the same primitive periods. I. If they have the same zeros and infinities, in each case to the same order, — that is, if f^ is 00 "■ at c, so is f^u and so on, — then f^ujf^u has no infinities. Each infinity of fyi is cancelled by an infinity of fji and each zero of f^u by a zero of f^u. Hence (§ 134) the ratio is a constant. 135-137] ELLIPTIC FUNCTIONS. 259 Thus an elliptic function is determined save as to a constant factor when its zeros and infinities are assigned. II. Again if the two elliptic functions fyt, and f^u have the same infinities and the negative powers of the Laurent series about each common infinity are the same for both, their difference is an integral elliptic function ; that is, by Liouville's theorem, a constant. III. Let f^u and f^ic be of the second order, with common infinities c and c' but different residues r, and r^ at c. Then near c fyi = rJ{u-c) + P{2i-c), fyi = rJ{H-c)+Q{7i-c), and therefore r^fyi - rj^u is finite at c. That is, it is an elliptic function with not more than one infinity in T, and that simple ; it is therefore a constant. 137. Algebraic Equation connecting the Functions (^u, ^'tt. Taking the second method of comparison of § 136, — that by infinities alone, — we know that near it = o fu = I jii" + yy +i,s,u*+..., f'li = — 2/«' + 2 . 3^4?/ + 4 . 5 v'3 + . _ . Thus ^'Hi is of the sixth order ; and we have ^'-U = 4///* — 24s Ju- — SoSe + FiU', ^ti = I /tt^ + gsjtc- + 1 5 Js + P^u-. Hence p'-u — 4^ti — — 6osJu^ — 140^5 + /'l^<^ and p''^2( - 4^hi + 6os^ <^i = - I40i's + P-^ic''. The function on the left is an elliptic function which has no longer an infinity at ri, = o\ for on the right there are no negative powers of zi. Hence it is a constant; and the constant, by letting u be zero, is seen to be — 140^6- Hence (^-n = Ofi^ii — 6osjpii — 14OJS. If g„ = 6ost = 6o2' I /t(/, the equation is i^'''n = ^(^hi - gjffu — g^ (16). The numbers ^2 and ^3 are called the invariants. 17 — 2 26o ELLIPTIC FUNCTIONS. [CH. XIX They are invariants of the network because they are not altered when we replace SW] and aa,, — the original given constants, — by another primitive pair; they are also invariants of the expression 4Jf>^-^2^-.?"3 ''^g^'^'i^d as a quartic in ^ one of whose zeros is p = oo. .— — "' 138. The Addition Theorem for ^ic. Let us apply the theorem that the sum of the zeros of an elliptic function is congruent with the sum of the infinities, to express f (?/, + u^ in terms of functions of ?/, and ic^ alone. Consider ^'u — c^tc — c\ where c and c' are constants. This is an elliptic function with the same periods as fit ; it is oo in 7" only at u = o, and is co there like — 2Jii^ — clu-. It is therefore of the third order and has three zeros ?/,, ?/,, ^h in T. The theorem states then that n^ + u^ + u^ = 0. Now since for each of these points {c^?( + cj = (^'uy = 4ft( -g^, ; thus it is, in factors, 4 (g»« - «i) (F« - ^ii) (&>« - ^3). \\'hence ^i + £2 + ^3 = o, ^ «2^3-l-e3^i + ^l^o = -^2/4.[ (18). ^1^2^3=^3/4 I And we have of course f'hi = 4 (p?< - ^i) {fu -e.^{i^u-e^ (19). 137-138] ELLIPTIC KUNCTIONS. 26 1 From (16) we derive 2i^'it(§l'ii= 12(^11 (gi'ti.—g„(§!u, or fu = 6i^Hi-g„\2, (§"'u=\2fU(^'tl, and so on ; the 2«th derivate is a polynomial in 1^21 of degree «+ I, and the (2ii-\- i)th derivate is the product of (^ii into a polynomial in i§ni, of degree n. If in any of these, — the last written is the most convenient, — we substitute the series (7) for (§u and the derived series for i^ic, (^"u, we infer, by equating coefficients, a recurrence formula for jjii in terms of s^, s^, ..,, s^^-i- Hence it follows that j^j is a rational function of ^o and ^3; that is, all the numbers ^'\jw"\ where « > i, can be expressed rationally by means of the first two. Ex. Prove 2'i/a/8=^2V2''. 3- 5^- 7, 2'i V=^2^3/2^ 3. 5 . 7 . 11. Some special cases of the addition theorem must be noted. First let 7(2 = tox ; then p'toi, = O, gJojx = e/, ; and the formula be- comes, on replacing ti^ by ii, . I / p'u \- p (« + ».) = - (^ J_ J -p«-.. _ e^e^ + 2ex- or [^ {u + (Ok) - ^k] {ipii - Sk] = e^e^ + 2^a- = («^ - ^a) (e, - Sa) (20). Second let 7u = ?/, = u. Then e>2« + 2QU = lim - ^ ^ ^^\ 262 ELLIPTIC FUNCTIONS. [CH. XIX or F"= .^3 If again we put in the addition theorem ii^ = 2ii, 2h = ti, we can obtain a formula which when reduced gives flu. But the proper method for obtaining i^nii in terms of (^u is by means of the function (Tmija^^''u. In the addition theorem there occurs the square of the expression I (p'«i — i^'iii This expression is to be remarked, as it is often met with in the study of elliptic functions. To view it as a function of one variable only let us write it 1 f'u — i§!a 2 (§11 — (^a It is an elliptic function with the same periods as i^u ; it is oo ^ at li = O, with the residue — i ; it is not oo at u = a, since both numerator and denominator have a simple zero there. But it is CO ' at ?< = — « ; and the residue there is + i. It is then an elliptic function of order 2 with one infinity at it = — a but the other at « = O and independent of a. Changing the sign of a the function 1 p'zi + (§la 2 fu — i^a has infinities at a, o with residues i, — i. 139. Expression of an Elliptic Function by means of i^ti. Taking now any elliptic function fu with the same periods as fu let it be oo at an infinity a like c_^\{u - a) + c_^l(ji - fl)2 + . . . + c_,„/(?^ - ay The functions g) {ti — a), f' (u — a),... are oo at a like ll{u-d)\ -2l{u-a)\... Hence by subtracting 3! c^.(^{u-a\ --c_,f'{n-a), -|<:^f)"(?/ -a), etc., 138-140] ELLIPTIC FUNCTIONS. 263 we remove all the infinite terms except the first. To remove this one, we subtract 2 ^ ^u — ^a ' But in so doing we introduce a simple infinity at the origin. When however we remove in this way all the infinite terms at all the infinities of fit, there remains an elliptic function with at most one network of simple infinities ; that is, a constant k. Thus we have, for any elliptic function, an expression 2 fu — ^a + l,c_^p(7e — a) -^'Zc-sp'iti-a)- (i). From the addition theorem, ^(u-a) and its derivates are rational functions of pu and ^'ti ; hence we can express /u itself as a rational function of ^u and ^'ti. If /u is even the odd function ^'u will disappear, but if /le is odd p'te will occur as a factor, all higher powers of p'u being removed by the funda- mental formula (16). And generally we can say that /u = R, (pu) + f'uR^ (pO. where R^ and R^ mean ' rational function of Hence any two elliptic functions fyi and fyi with the same periods are connected by an algebraic equation. For we have the three equations fu = R, (p7t) + p'uR^ (^u), /,u = R,{^u) + ^'tiR,(^n), from which we can eliminate pit and p'lt. In particular an elliptic function ^< and its derivate/X having the same periods, are connected by an algebraic relation; an instance of this is the formula (16). 140. The Addition Theorem for ^u. We have said that f is not elliptic. But such an expression as ^(u + a)- ^u is elliptic; for ^u + 2cO), + a)-^{u + 2Q»,) = ^{u +a) + 277a - ^U - 2i}K = ^(u + a)-^it. H--g)'fl+... [l ■\-ii'^^a + ...], 264 ELLIPTIC FUNCTIONS. [CH. XIX The infinities are 2c = o,if^ — a; and the residues are respectively — I and + I. The function - ^ — 2 ^11 — fa has the same infinities and residues. Hence Kin + a)-tti-k=- ^ ^ 2 fti — ^a To determine the constant we expand near the origin. The expression on the left is fa + 7ii;'a -iju + Pjc - k, and that on the right is I ^ 2 — 11^ f' a + Pm- 2u 1 —u'^^a + PoU- ' _ I = — i/ti — tipa+ Comparing the constant terms we have k=^a. Hence the addition tlieoreni : i;(,, + ,)_^«_f, = i£iii:£:f (31). 141. Integn^ation of an Elliptic Function. Integrating the formula (i) of § 139 we have \fudu = k' + ku + l,- c_, \ ^ — ^ du 2 ] fu-fa We have then, for the integral of any elliptic function, (i) a term kit, (2) terms of the form i^{u — d), (3) integrals of the form irg;«+^ 2 J fu — ^a (4) elliptic functions. By the addition theorem for ^ic, ^ ^ ^ * 2 fu-fa 140-142] ELLIPTIC FUNCTIONS. 265 SO that the terms (2) contribute a single term in t,u. Also integrating the last equation we have ~ , ^^—du — log — ^^ ' + uKa. Thus on the whole the integral is ifudu = k^u - 2f_, r« + %c_, log '^^^^^ +Au, where fyi is an elliptic function! i?"^'tclt{ = i{2g^r)^-^g^a>y), u being arbitrary. 142. Expression of an Elliptic Function by means of an. We shall next show how the theory of elliptic functions can be deduced directly from the o--function without the intervention of the j?-function, and without the use of integration. We shall thus have alternative proofs of some of the preceding theorems. I. A n elliptic fimction fit has a finite number of zeros and a finite number of infinities in a cell. The reasoning of § 104 excludes the possibility of there being infinitely many non-essential singularities ; for such a system of points has limit-points and these limit-points are essential singu- larities, whereas fu has no essential singularity except at ?^ = 00 . And similarly by § 106 fit cannot have infinitely many zeros within the parallelogram. II. Every elliptic f miction fu can be expressed in tlie form ^^^ (T(u-a^)a{u-a.} ... a{u-ar) a {u — bi) a-{u — b^ ... a- {u — b^) ' where the cis are the zeros and the b's are tlie infinities of fee in the cell Each factor in the numerator of a {u — «]) a {u — a^ a (« — a.^^ . a {u — a^) a {u — bi) a (u — bo) a-(i(— bs) ... cr (u — bg) vanishes at the corresponding a and at all points congruent to a ; hence the zeros of the numerator are the same as the zeros of fu. Similarly the infinities of the quotient, or the zeros of the denominator, are the infinities of fu. Hence the quotient of fu 266 ELLIPTIC FUNCTIONS. [CH. XIX by the expression in the a's is a one-valued analytic function which has neither zeros nor infinities ; this quotient must there- fore be of the form e'^'^ (§ 93) and the theorem is proved. It is not true conversely that every expression of the form Gn °" (" ~ ^') '^ (" - ^2) q"(?< —a,) ...a- (u—ar) a {u — bi) a- {u — b.^ a {u — b^ . . . a- {u — b^ represents an elliptic function. We shall prove, for example, that the number of the a's must be equal to the number of the i^'s, and that Gu must be a special polynomial which reduces, under certain conditions, to a constant. III. The number of zeros of an elliptic function fit in the cell is equal to tJie number of infinities in the same region. We suppose that the zeros and infinities are all simple. This does not interfere with the generality of the results, for the case in which a zero (or infinity) is multiple is merely a limiting case in which several points previously distinct have moved into coincidence. Let «i, «2i ■•■, rtr and b^, b.,, ..., bg be, as before, the zeros and infinities that are situated within the parallelogram of periods. Then fJ^ = ^G« <^ (." - ^1) o- (?^ - ^2) • •■ 0- (u - a r) a {u — di) cr{u — b^) ... o- (u — bg) ' The expression on the right-hand side is to be reproduced when u is changed into tt + 2cux ; hence since a-(u-a + 2cok) = - ^^\"'-«+"\'cr (u - a), we must have I = (-)>-» exp [G (u + 20)0 - Gu] . exp\2{r-s)7]^{u+Q)^)-2T}J't «„- 2 bn) . L \n = \ n = l /_ This requires that the expression 2(r-j)7?x(« + a)x)-277A( i an- 2 bn\ + G{u + 2a)x)-Gu, \n=l n=l I shall be a multiple of -kL Hence G {u + 2v>t^ — Gu = 2?7x( S fl„- 2 bA-2(r-s)f)K{u + (ji^-^ht.'iri (i). \»=i 11=1 / 142] ELLIPTIC FUNCTIONS. 267 This integer h^, must be the same for all values of Ji. ; for the expressions in ti change continuously, whereas h^ if it change at all can only change discontinuously. By differentiating (i) twice we get G" {n + 2^0 = G"n ; a relation which shows that the transcendental integral function G"u is doubly periodic. It can have no infinity within the parallelogram of periods and must reduce therefore to a constant*. Hence Gu = AtjU- + A^ii + A^. Substituting this value in (i) and putting X= i, 2 successively, = 2r)i[l rtn- 2 dn]-2r),{r- s){n + o),) + /i^vi ...{ii), 4.A0O32 {u + W2) + 2Ai(Oo = 27)i(i, an— 2 dn)-2rj^(r-s){?( + o).^ + /i.,7ri...{ni}. Vn=l 11 = 1 / As these equations are to be satisfied by all values of ?/, we can equate the terms in 71 on the two sides of (ii) and (iii) ; hence 4^o&>i = - 2% {r - s), 4^oQ>2 = - 2772 (;- - j) (iv). Suppose that in the equations (iv), r — s=^o ; then 77iO)„ — 7)20)1 = o, a result at variance with the formula (13) of § 135. Hence r must be equal to s, a theorem established already by means of integration round the parallelogram. The preceding equations give a new proof of the following theorem : — IV. Tke sum of the zeros of an elliptic function in a cell is eqzial to the sum of the infiftities in the same region increased by a suitable period. By equating those terms of (ii) and (iii) that are independent of u, we get ^iWi = % ( 2 an - 2 bn] + h-cKi, A^o)o = r}Jl a, I- S bn) + h.Tri. 268 ELLIPTIC FUNCTIONS. [CH. XIX Hence (vj'^o— V2f^i){ - «»— 2 ^,j) = (//.,cbi — //i(Uo)7r/, V)i=i ?i=i ' (7?i(Uo — TjjfOi) A, = (/l.,r]^ - /hrj.) Tri. But r]yCL>., — r).,Q)j = 771/2 ; hence r r n=l n=l and ^1 = 2 (/^27;i — /^i?72)- Hence the theorem is proved ; and further it is proved that i/ie most general form for an elliptic fimction is ^ a- (u - a,) a (ii-a,)... a ju - a, ) ^^„^,„_^,,,,^ __. 2), a {h — ^1) a {ic — d.,) ... cr {ti — b,^ where k^ , Ju are integers given by r V 2 rt„- 2 ^„ = 2 (A,a)i - /^ift),). This general formula shows that two elliptic functions with the same zeros and the same infinities in the parallelogram of periods differ merely by a constant factor (§ 1 36). V. The equation fn = k admits r non-congriient roots, and the sum of these roots is equal to the sum of the infi7iities offu increased by a suitable period. This is proved at once by applying Theorems III., IV. to the elliptic function /z/ — k. -4L 143. Relation connecting « — e,^. Writing in (23) 71 = eu^ we get pti — €>, = a (tUA + 11) a {aiK — it)la^aix,). Hence the function an exp (— rj^u^lzcoi) merely changes sign when u becomes u + 2coi ; therefore it has the period 40),. When tt becomes « + 20)2, the ratio is multiplied 144-146] ELLIPTIC FUNCTIONS. 27 1 by - exp [2j?i, (« + Q)j) - 2r;,iV- = I + 2r}-^WiV- + ... ; so that equating coefficients 20)1 = /^I^'O, 277iO)i5f'0 + ^'"0/6 = o. Hence finally, eliminating k^, we get (72(i>iV = 2o)i exp 2r]iWiV- . 'isvl^'o (26), and i27;,a), = -^"'o/^'o (27). 272 ELLIPTIC FUNCTIONS. [CH. XIX These equations with those of the preceding article are of great use in numerical applications of elliptic functions ; and it is for this end that we have selected them as illustrations. When the network is given, that primitive pair of periods is to be selected which makes q, = exp 103.2/(01, as small as possible ; the ^-series then converge with remarkable rapidity. The formulas given as examples in § 145, together with ^1 + ^0 + ^3 = 0, suffice for the calculation of e^, e„, e^ in terms of &>i ; formula (27) gives 1J1, (13) gives 772, and (26) gives , — and further that a positive description of a circle which includes the origin 18—2 276 SIMPLE ALGEBRAIC FUNCTIONS [CH. XX can be treated as a negative description of a circle which includes ;tr = 00 , and therefore the branches permute round jr = 00 . The difficulty which arises from having two points in the j'-plane corresponding to one in the jr-plane was obviated by Riemann. He supposed the ;r-plane covered by two sheets, parallel and infinitely near to one another. Thus to a given point X correspond two places : one in the upper sheet, one in the lower. One of these places corresponds to one j/-point, the other to the other. A place is named by its x and the corresponding y ; thus, supposing the sheets horizontal, two places in the same vertical will be {x, ji) and (x, —y). We shall call them co-vertical. At the origin jr = o we have only one place since the j's are equal ; so we regard the sheets as stuck together (or in contact) at the origin. But a circle round the origin is to lead from the place (;tr, y) to the place {x, —y). Therefore there must be a bridge between the sheets, which may run from o to 00 along an arbitrary line, say along the ray of positive real numbers. Whenever a moving place crosses this bridge it passes from the upper to the lower or from the lower to the upper sheet. Along the bridge the surface intersects itself. Fig. 61 shows a vertical section of the surface, perpendicular to and intersecting the bridge ; and fig. 60 gives an idea of the appearance of the surface as a whole*. X X^J, X.-J, Fig. 60. Fig. 61. This Riemann surface answers all the requirements: — these are that two geometric points shall correspond to a given x and * Fig. 60 is reproduced from a photograph taken from the model in Brill's collection. 147-148] ON RIEMANN SURFACES. 277 that a path which passes round the origin an odd number of times shall lead from one _y-point to the other, while a path which passes round the origin an even number of times shall lead from aj-point to itself. Thus on our surface a circle is a closed path where it does not include the origin ; for if it cross the bridge once, say into the lower sheet, it recrosses it into the upper sheet. But a circle which includes the origin is no longer a closed path on the surface ; it becomes closed when described twice. The j/-plane and the x-surface are 50 related, point-to- point, that any closed path on the one corresponds necessarily to a closed path on the other. What we have called (provisionally) the bridge will serve as a cut in the j;-plane which determines two branches of the function ; in this case the two branches are assigned to the upper and lower sheets respectively. When, conversely, a cut has been employed to create branches, it is often convenient to use that cut as a bridge on the Riemann surface, and to call it a branch-cut. For instance when we choose x^'-, — x^'" for the two branches, the axis of real negative numbers is a branch-cut for the corresponding Riemann surface. 148. Corresponding Paths in the x-, /-planes when j''=x. Before leaving this simple case let us determine some corresponding paths in the planes of x and jf. We have seen that if p, 6 and p', ff be the polar coordinates of x and y, then p=p'^, e=2d'. Hence the passage from a given polar equation of a .y-curve to the polar equation of the corresponding jr-curve, and vice versa, is immediate. Thus the circle />'= constant maps into the repeated circle p= constant; two rays ff = a and ff=a + jr into the repeated ray 6 = 2a; the lines p cos {ff - a) == ±k map into the repeated curve p cos^ (612— a)=K^, which is a parabola. Ex. Draw the maps in the ^--plane of the lines ^'=0, ±1, ±2, and ij' = o, +1, ±2, where j'=^'-l-zy. The repeated line pcos{6-a)=K maps into the curve p'^cos{2ff -a) = K, which is a rectangular hyperbola with centre at _)'=o ; and so on. Ex. Draw the maps in the /-plane of the lines $ = 0, ±1, ±2, and i/=o, ±1, ±2, where x=^+ti]. These examples show how we can identify, by means of the polar equation, the curve that corresponds to a given curve. But it should be observed that the mapping itself gives a clear idea of the form of the new curves. This process deserves some illustration. 278 SIMPLE ALGEBRAIC FUNCTIONS [CH. XX Thus when x=y^ we have (Ixji-J x=dy. Now when,)' describes a straight line am (/c is a constant, say a. Hence the jr-curve is such that am «&■ — ^ am :r = a. That is, if <^ be the amplitude of the tangent at a point, 6 the amplitude of the stroke from x—o to that point, 20-fl=2a, or (^ -6=2a — <^ ; and this expresses the well-known fact that rays issuing from the focus of a parabola are reflected at the curve so as to be parallel. So again when x describes a straight line through a point a, we have am (x-d) = constant. If ^^ = a, then y''' — b'^=x — a; therefore, for the corresponding curve, am (y^ - b^) = constant, or amO'-^) + am(_K+*) = constant ; or the sum of the angles made with a given line by the lines from the ends of a given diameter to any point is constant. And this is a characteristic property of the rectangular hyperbola. Next let the point x describe a circle, with centre a. Then, if l^=a, we have y'^-b'^=x—a, and |^-(5| l^z + ^l^lr-rt | = a constant. Thus the product of the distances of any point of the >'-curve from two fixed points b, -bis constant. Such a curve is called a cassinian. Fig. 62 Fig. 6j. shows the map in the j-plane of a system of concentric jr-circles. It can be shown that when the jr-circle does not include the origin, the cassinian consists of two ovals ; when it does include the origin, the two ovals unite into a unifolium. If we use the Riemann surface we have in the former case two separated circles, one in each sheet, corresponding to the two ovals ; but in the latter case a single repeated circle corresponding to the unifoUum. 148] ON RIEMANN SURFACES. 279 In the separating case the cassinian becomes a figure-of-eight ; this particular cassinian is known as the lemniscate. The diameters of the concentric system of circles are given by am (jr- a) = constant ; these straight lines map into the curves 3.xci(y-b)\Axa.{yA-b) = constant. By the property of isogonality these curves cut at right angles the system of cassinians. It may be left to the rea.ier to show that these curves are rectangular hyperbolas which have b, —b for ends of a common diameter. The above orthogonal curves admit of an easy generalization. Let any rational function R{y) of y be put equal to x. Let x describe a circle, say about the origin as centre ; then the corresponding path of j is given by I ^(y) I = constant, o' P1P2P3 = «p'lP'2P'3 > where the p's and p"s are the distances of y from the zeros and infinities of fiiy). The orthogonal system of curves will be the maps of straight lines through the origin, and will therefore have the equation 6^ + 0^ + 63+. .. = a + ffi + e'2+ff3 + ..., where 6^, ^2, ... are the amplitudes of the strokes to _j' from the zeros, ff^, ff^,... are the amplitudes of the strokes to y from the infinities of R {y), and a is a parametric constant. When j/ passes to 00 the amplitudes oi y — a and_j'-^, where a and b are given, tend to become equal. Hence for the real points at 00 on a curve of this system, we have flj = 52= ... =ff^ = ffi== ... =^ say ; so that, if there are n zeros and n' infinities in the finite part of the plane, we have {n — n')^ = a (mod. 2it), showing that the curve has n-n' asymptotes inclined at equal angles 2 «■/(« — «')• Ex. Prove that these asymptotes meet at a point. Returning to the equation _y2=jf, let us see what corresponds to a ^-circle. Let the centre be b, and write/=*-fy, x=l^+x'. Then y=2*y+y^. Regard y as a point describing its circle with a constant angular velocity ; then y2 describes a circle with double that angular velocity. The motion of x' arises then from a superposition of two circular motions ; 2*y describes a circle with a certain constant angular velocity, and y^ describes a circle about iby with double that angular velocity. The composition of these two motions is a famihar question in Kinematics, and the resulting curve is called a limaqon. This curve has different shapes according as the ^-circle does or does not include the point y=o. When it does not, the curve is a unifolium (fig. 63, i) ; but when it does, the curve subtends an angle 4»r at the branch- 28o SIMPLE ALGEBRAIC FUNCTIONS [CH. XX point x=o, winding twice round it. The curve in this case (fig. 63, 3) crosses itself at a point Xf^ which can be determined as follows. Fig. 63. While _y describes the above circle, say C, —y describes another equal circle C. On the x-plane these map into the same limagon, for the points y, -y map into a single point x on the jr-plane ; but on the x-surface they map into two limagons L and L, a vertical cylinder through L cutting out on the surface the other lima^on L ; for the points y and —y map into covertical places {x, y), {x, -y) of the jr-surface. When C and C do not intersect, L and L' do not intersect ; but when C and C do intersect at points j„, —yf„ then L and L' also intersect and the places of intersection are (Xf„y^ and {Xf„ -y^, where Xg=y^. Hence to find Xf, we take the map of either point of intersection of C and C Ex. When C is the circle ( i , p), the point jtq 'S i - p^. In the separating case when the circle passes through the point y=^o, the limagon has a cusp ; for to the angle n zX. y—o corresponds the angle 2jr at x=o, so that when j/ passes through its origin, x reverses its direction. This special limagon is called the cardioid (fig. 63, 2). 149. Example II. y'^ = (x — a)j{x — b). The correspondence of X and y is i, 2 ; accordingly we suppose (when the matter is to be discussed fully) a two-sheeted Riemann surface spread over the jr-plane. The values of y are equal when x=a and when x=b. The point a is a branch-point, for a circle about a which does not include b increases am (x — a) by 27r and leaves am {x — b) unaltered, so that the amplitude of a selected y is increased by tt, not 27r, and one branch passes into the other; for a similar reason b is also a branch-point. 148-149] ON RIEMANN SURFACES. 28 1 Let Xo, j/o be a pair of values satisfying the equation. When X describes any path starting from and returning to x^, either j/ returns to y^, or the final value is -jf^. To distinguish we must examine the angle made by the path at both a and d. When the whole changes of am {x - a) and of am (x - b) are 20Tr, 2/37r, then that of ^ is (a - /8) tt. The ^-path is therefore closed when a — y8 is even, not closed when o — /3 is odd. The effect of a y/'^T^^T^ /<^^ path depends Jt7/i?/)/ on these angles OTT, (/' • \\y/'' '\| ySTT ; therefore we can continuously de- ;^-~~__— rT /' ,' / form the path provided we do not let • \ \ J J it cross a branch-point. The path can \ ; / j // be resolved into a succession of circuits \ ; / / .y beginning and ending at x^ and not \ .7/'^/ intersecting themselves (fig. 64). A "n>^ circuit which does not include a °, branch-point is one for which a = o, y8 = o, and may be left out, since it has no effect on j„. A circuit which includes both branch-points is one for which a = /8= I, and may also be omitted. The only circuits which affect the question are those which include a alone and b alone. Now a bridge between the sheets from the branch-point a to the branch-point b will ensure that a moving place on the Riemann surface shall change from one sheet to another when it makes a circuit round a or ^ ; that is, it will lead from the place (jTj, Jo) to the place {x^, —y^. Thus by supposing such a bridge our Riemann surface is completed. The bridge may be along any curve provided its ends are a and b and it does not cut itself. Using this bridge as a branch-cut the two sheets determine two branches, as in § 147. Observe that the example y'^ = x\s, closely connected with the present example. For if we write z = (x — a)l{x — b), then y'^ = z becomes f = {x- a)/(x - b), z = {x- a)l{x - b). The transformation s = (x - d)l{x - b) is to be regarded as applied to both sheets of the surfaces spread over the x- and 282 SIMPLE ALGEBRAIC FUNCTIONS [CH. XX ^-planes ; so that two covertical places of the ;r-surface become two covertical places of the ^r-surface. When the covertical places of the ;r-surface become one at a branch-place, so do those of the ^^-surface. Thus the branch-places a, b of the one surface map into the branch-places o, oo of the other surface. 150. Example III. Rational Functions of x, y where y^-={x — a)l{x — b). On the Riemann surface just described, y is a one-valued function ; that is, for a given place on the surface y has one value only. In fact the raison d'itre of a Riemann surface is to have one value of a dependent variable y at each point of an x-surface, instead of several values of y at each point of an x-plane. But also a rational function z,= R {x, y), of x and y has one value only, when x and y are both given, and therefore is one- valued on the surface. If the function be even in y it is of course one-valued in x alone, and the x-plane will suffice for our purposes. When, however, z contains odd powers of ^ it has two values for a given x, differing only in the sign of the square root "Jipc — a)l{x — b). These become equal when the square root is o or oo ; that is, when x = a or b. Moreover they are inter- changed when the square roots are interchanged. Thus the surface will serve for the representation of the pairs {x, z), equally with the pairs {x, y) ; but it must be observed that now the pairs {x, z\ (x, — z) do not occur at covertical places. As an example let us consider the rational function z=y(x-b). Then z^=y^(x- by = (x- a){x- b). We need no new surface ; we have still the branch-places a, b, and the necessary bridge between a and b. But the relation of jr to ^: is quite different from the relation of x to y. Whereas, corresponding to a given y there was one x, to a given z there are two x's. Thus in a complete discussion of the 2, 2 corre- spondence given by z^ = (x-a){x-b), we should be led to consider two Riemann surfaces: — a two- 149-ISO] ON RIEMANN SURFACES. 283 sheeted surface spread over the jr-plane and a two-sheeted surface spread over the ^-plane. There would be a i, i correspondence between the places (x, z) of the jr-surface and the places {z, x) of the ^•-surface. In this example we have, by eliminating y, passed from the simple to the more complex; and the reverse is the proper order. To consider the relation 2^ = {x— a) {x — b), we can write 2==y{x — b), and then consider f={x- a)/{x - b), so that ;tr is a rational function of y, and therefore z also is a rational function oi y. The reader is familiar already with this order of ideas in connexion with integration, for the above is one of the ways, in elementary Integral Calculus, of reducing jV(jr- a) (3r -*)(&• to the integral of a rational function. The transformation 'J {,x- a) {x- b)=y {x - b) \cz.As to \ 2 {a - bf ^ dy. The equation z^ = {x — a) (x — b) presents a new feature. As before two values of z are equal when x = a, b ; but now, in addition, two values of z are equal when x=co. There is however no change of branches round jr = 00 ; for a path round both a and b adds 27r to each of the quantities aim{x—a), am (x—b) and therefore 477 to am z, thus restoring the initial z. Analytically we have z=±x{i - a/xyi' ( I - b/xy- ; whence expanding by the binomial theorem we have for large values of x, not a series containing fractional powers, but a Laurent series with one positive power, namely a term in x. This answers to the case of a node in the plane curve and will be called the nodal case. The sheets of the jr-surface touch at ;r = 00 and so do those of the 2^-surface at z=oo; but there is no other connexion between the sheets in the neighbour- hoods of those places. We return to this in § 1 58 ; before passing on observe a distinction between this phenomenon and the one which presented itself in the case of branch-points. Consider the two equations {y-bf=x- a, and (jy-bf = (x- af + {x- af. In the former case two values ofy are equal to b when x=a, but only one value of x is equal to a when y = b. In the latter case 284 SIMPLE ALGEBRAIC FUNCTIONS [CH. XX (the nodal case) not only are two values of y equal to b when x=a, but also conversely two values of x are equal to a when y=b. 151. Example IV. f — iy=2x. The correspondence of X and J is I, 3; we require a three-sheeted jr-surface. The values of y corresponding to a few values of x are shown by the table x = o I — I 00 1 o - I I 00 y=-\ 31/2 - 3^'^ Qiy I 00 -f -2 00 Here we are careful to include all cases where values ofy become equal. These are sure to lead to branch-points because x is one-valued in y and therefore there are not two values of x to give rise to the nodal case. It simplifies the explanation to draw the curve which re- presents the correspondence of the real values of x and y, (fig. 6s). Fig. 65. Taking the three sheets, we assign the values 3^'^ o, — 3^'" to the three places ;tr = o in any order, say 3^"' to the upper sheet, o to 1 50-151] ON RIEMANN SURFACES. 285 the middle one, — 3"^ to the lower sheet. When a place s on the surface starting from (o, o) moves to the right along the real axis, a glance at the curve shows that the corresponding value of j decreases from o to — i ; while if the place start from (o, — 3"^) and move in the same way, the values of j/ increase from — 3"^ to — i. Thus the middle and lower sheets unite at ;ir = i ; on the upper sheet ( I, 2) is an ordinary place. So if we allow jr to pass through real values from o to — i, the curve shows that the places which unite &tx = —i are those which start from (o, o) and (o, 3"^). Thus the upper and middle sheets unite atx=— 1. We draw a branch-cut from each branch-place to 00 con- necting the sheets which unite at that branch-place; the two branch-cuts are arbitrary in form, but must not intersect them- selves or one another. We will suppose them drawn along the real axis (fig. 66). In the figure the paths in the three sheets are represented respectively by a continuous line, a line with dots, and a broken line. Fig. 66. The construction of the surface is now effected. To each place on it corresponds a definite value not only of x but also of J'. Notice how the requirements at 00 are satisfied. When x is large the values of jy are approximately (2Ary'S v(2J^)"^ v^{2xyi^ where v = exp (27r//3) ; thus a large circle round 00 is to inter- change these three values cyclically. Now a place s which starts 286 SIMPLE ALGEBRAIC FUNCTIONS [CH. XX at a place ^i in the upper sheet and north of the real axis and describes about the origin a large circle positively, will pass successively into the middle and lower sheets before reaching a place S3 in the same vertical with the initial place. That is, it returns after one revolution to a place S3 in the lower sheet. In another revolution it passes under the branch-cut (12) and passes at the branch-cut (23) into the second sheet; thus after the middle revolution it is at J2 in the middle sheet, where S2 is vertically below s^. A third revolution brings it back to Jj and its path is closed. Ex. Draw on the jr-plane and also on the ;ir-surface the maps of the circles |_j'|=i,|^| = 2. 152. Simply connected Riemann Surface. Starting with a given equation Rijy) =x, where R{y) is a rational function of degree n in y, we have for the adequate geometric representation a _y-plane and an «-sheeted ;jr-surface with certain branch-places; of course the nodal case does not occur since there are not two values of x to become equal. These two surfaces are in place- to-place correspondence. On the jr-surface y is one-valued and so is any arbitrary rational function z oi x and y. Eliminating X we have ^ as a rational function of y, and we can by means of this relation map the _y-plane on a ^-surface. The places of the .ar-surface and the 2-surface are in i, i correspondence with those of the _y-plane and therefore with one another. There are of course, according to the assumed rational function z, different ^-surfaces ; but all have one pro- perty in common. This we proceed to explain. If we draw a circuit on the ^v-plane we divide the plane into two regions, — an inside and an outside, — such that it is not possible to pass from the one region to the other without crossing the circuit. To this circuit corresponds a circuit on any ^r-surface which must also divide that surface in the same way. For on account of the correspondence a path which crosses the circuit on the plane maps into a path which crosses the corresponding circuit at the corresponding place on the surface. If the plane be cut along the circuit it separates into two JSI-IS3] ON RIEMANN SURFACES. 287 parts ; if then the surface be cut along the corresponding circuit it also falls apart. This is a fact characteristic of surfaces which can be mapped with i, i correspondence on a plane. For it can be proved that only surfaces which fall apart when cut along any circuit can be so mapped. Such surfaces when cut along any circuit form two simply connected surfaces (§ 107). Given the possibility of mapping there still remains the further and important question as to how it is to be done. The next example will yield a surface which is not simply connected. 153. Example V. y'^ = {x — a^{x — a^{x—a^(x—a^. Here we require a two-sheeted .j:-surface with branch-places at a-^, a^, a^, Ui. When jr= 00 the values oi y are equal, but a large circuit round a^, a^, a^, ai increases am^ by 47r, so that there is no interchange of values round 00 ; that is, 00 is a nodal place. When X describes in the x-plane positively (or negatively) a path C which starts from ;r„ (say) and passes once round one of the points a, say a^, but does not include any other, the two values Fig. 67. of^ permute (fig. 6j) ; for am (x — a) is unaltered for a = a-^a-^, a^, and increases by 2ir for a = ai. When x describes a path C round two branch-points, say a^, a^, similar reasoning shows that the two values of y are restored at the end of the path. It is easy and useful to construct paths that are more com- plicated and to take account of their effects on the two values o{y. 288 SIMPLE ALGEBRAIC FUNCTIONS [CH. XX Fig. 68. All these effects are provided for by the construction of two branch-cuts on the Riemann surface, from «! to a^ and from a^ to ^4 respec- tively. There is an essential distinction between this surface and those previ- ously considered. The present surface is not simply connected. Draw a circuit A (fig. 68) in one sheet round the bridge a^a„. Then the path B leads from a place p on the one side oi A to a place q on the other without crossing A. Thus the surface if cut along A will not fall apart; it is still one surface bounded by the two rims of the cut ; having been to begin with a surface with no boundaries at all. If the cut surface be cut again from p X.O q along B it still does not fall apart, for it has one continuous boun- dary as shown in fig. 69. But in point of fact any further cut from the boundary to the boundary, or any cut along a closed circuit, will sever the surface into two surfaces ; so that when cut along A and B the region is simply connected. And (assuming this) it can then be shown that any circuit on the surface is deformable into repe- titions of the circuits A' &nd B. This discussion of the connectivity of a Riemann surface is fundamental in Riemann's theory of the integrals of algebraic functions, or Abelian integrals as they are called in general. There are two special cases which might find a place here; namely the integrals jR{x,y)dx, where R stands for a rational function of its arguments and {\) y is a rational function of x, (2) y is given in terms of x by the equation of Ex. V. ; these cases are often called the rational and elliptic cases, because they bring in (i) integrals of rational functions of x, (2) elliptic inte- grals. A discussion of these two special cases would illustrate Fig. 69. 153-154] ON RIEMANN SURFACES. 289 the advantages of Riemann's methods ; but we shall not under- take any such discussion here, as the methods referred to belong properly to the whole theory of Abelian integrals and not merely to these special cases. 154 Fundamental Regions. A system of operations is said to be a group when every operation compounded of any number of operations taken from the assigned system is itself an operation of the system. For example, translations in the ;»r-plane form a group, for a translation combined with a trans- lation is a translation ; again the bilinear substitutions x ={ax-\-V)\{cx-^d) or {x ,{ax + b)\(cx ■\- d)) form a group, for if x' , x and x", x' are connected by bilinear relations, so also are x" , x. We can impose in this latter case the restriction that a, b, c, d are to be integers which satisfy the equation ad —bc—i and still we get a group. To take a much simpler example, the bilinear substitutions j-'—gznri/ji^ r = o, I, 2, ..., w— I form a group. In this group every substitution can be expressed as a power of the substitution ^ = {x, e'^'^x) ; in fact the group is I, S, S"^, ..., S"~^, where 5"= i, i standing for (x, x). Such a group is called a cjc/ic group and 5 is called the generating substitution of the group. The substitution S is a special elliptic substitution (§ 37) for which the fixed points are o, 00 ; it arises from the substitution X —X^ X — Xi In place of the factor ^'" we may take e*' and still have a cyclic group ; when Bjir is not a rational number the group contains infinitely many distinct substitutions S*", and there is no longer a relation 5"=i. That there may be important connexions between groups and functions is indicated by what we know of the elementary functions ;r", ^ and of the elliptic function p«. The first of these is invariant with respect to the group of substitutions {x, ^^"•■x), the second with respect to the group of substitutions {x,x+2rTri), M. H. 19 290 SIMPLE ALGEBRAIC FUNCTIONS [CH. XX and the third with respect to the group of substitutions (u, ±11+ w) where w^ 2m^^ + 2m^^, m^ and m^ being integers. Given any one-valued analytic function fx and all the bilinear substitutions {x, (ax + b^l^cx + d)^ which do not affect the value of the function, it is clear that we get a group, for the combination of two sub- stitutions which do not affect ^ is a substitution of the same character. Conversely it is important to connect, if possible, with an assigned group of bilinear substitutions a one-valued analytic function fx such that iax->rb\ when [x, -%] is a member of the group. The discussion of this problem belongs to the subject of automorphic functions and lies beyond the range of this book ; but we can gain from the special examples x'^, e*, p« a fairly good insight into the meaning of what are known as fundamental regions and a perception of their importance in the study of functions. I. Equivalent points. Taking any point x of the plane and applying to it all the substitutions of a group, we get a set of equivaknt points. For example the substitutions (ti, u + 2;«ift)i -f- 2m^m^, or {u, u + w), form a group, and the points ti-irw are equivalent points. I I. Fundamental region of a group of bilinear substitutions. In the case of jt" we divide the plane into n regions by the rays Q = — irjn, 7r/«, Stt/w, . . . ; each region is to contain one but not both rays, for example the first region — tt/w to tt/w contains the ray tt/w, the second region tt/w to 37r/« contains the ray yirjn and so on. Select one of these regions, say the first ; we shall call this region a fundatnental region of the group because it contains one point and not more than one point which is equivalent with respect to the group of substitutions (x, e^n^x) to an arbitrarily selected point x of the jr-plane. In the cases of e^, fu we are led to consider bands and parallelograms. With respect to the group of substitutions 1 54] ON RIEMANN SURFACES. 291 {x, X 4- zr^ri), bands (§ 100) of breadth ztri are fundamental regions ; with respect to the groups of substitutions {u, u + w) and («, ±u + w) the fundamental regions are the parallelogram of periods and half the parallelogram of periods respectively. III. One-valued functions associated with fundamental regions. When we take the substitution {x, e^"'x) which converts the ray — 7r/« into ir/n, and apply it successively to the fundamental region (— tt/m to w/n) we get in the jr-plane n sheets laid side by side ; the plane is covered once without gaps. Now let us consider the equation j = x" ; instead of spreading n sheets over the j/-plane we can equally well take these n regions lying side by side in the jr-plane. That there are n values of x for a given jf can be inferred from knowing that Jtr" takes all its values in each of the regions: that the function jr" which passes once through all its values in the fundamental region is one-valued is a consequence of the non-overlapping of the aggregate of regions, and so on. Suppose that we had started with the region (— tt/w to ir/n) we could have evolved the group from it by observing that the substitution (x, eP^'^x) converts the one edge into the other, and therefore will, on successive applications, rotate the initial region into the remaining regions. Also we could have inferred that the one-valued function (if any) which takes all its values once within the region must satisfy the relation /(i?^'" jr) =fx, other- wise it would not have the same set of values along the two rays — «■/«, 7r/«. If we begin with a band bounded by two parallel straight or curved lines which are coordinated by the substitution {x, x + w), we are led to the theory of one-valued functions fx which satisfy the relation f{x + w) =fx, and take their values once and only once in the fundamental region belonging to the group of substitutions (x, x + ra). The infinitely many bands for ^ cover the plane once and without overlapping ; this indicates that when_j/ = ^,^ is one-valued and defined for all values of;ir, and that x is an infinitely many-valued function of j/. Instead of the infinitely many bands in the ;ir-plane we can use an 19 — 2 292 SIMPLE ALGEBRAIC FUNCTIONS. [CH. XX infinitely many-sheeted surface in the j/-plane. When y is not allowed to cross the negative half of the axis of real numbers, X is restricted to a band ; if we allow x to move into adjoining bands we must let y cross the barrier. To the complete jr-plane then corresponds the infinitely many-sheeted ^/-surface with all the sheets hanging together at o and 00 and connected by a branch-cut which replaces the barrier. The infinitely many bands in the ;r-plane serve equally with the infinitely many sheets of the surface spread over the jc-plane to indicate a suitable separation of log x into branches. Since sin ;»r = cos (7r/2 — ;ir) the fundamental region for cos.ir will serve also for sin x. Now cos x is unaltered by the substitutions (x, ±x+ 2;Tr) ; hence the fundamental region is that given by this group of substitutions and not the band of breadth 27r given by the group of substitutions {x, x + inr). Half, not the whole, of the latter band must be taken (§ 100). A similar remark applies to fii ; here the group is (w, ±u + w), and hence a funda- mental region for i^u is half of the cell Ua, u^ -1- 2a)i, «(, + 2a)i -1- 2&J2, «(| -f 2&).2. The fundamental regions which we have been considering can be deformed in many ways. It will be sufficient if we point out that the dotted curvilinear parallelogram of fig. 70 can replace the rectilinear parallelogram without affecting the properties of the associated elliptic function (^ii. Fig. 70. CHAPTER XXI. ALGEBRAIC FUNCTIONS. 155. The Algebraic Functioii. Let F{x, y) be a poly- nomial formed with positive integral powers of x and y ; we suppose this polynomial to be irreducible, which means that it cannot be decomposed into the product of several factors of a similar kind but of lower degrees in the variables. The function y defined by the equation F{x, y) = o is called an algebraic function of x. Evidently x must also be called an algebraic function oi y. When F{x, y) is reducible, the equation F=o defines as many distinct algebraic functions as F{x, y) admits distinct irreducible factors. Let m, n be the highest powers of x and y. Then for a given X there are in general « finite values oi y (§ 59); let us agree that there shall be in all cases « values of ^ finite or infinite. Similarly when y is given we shall suppose that there are in all cases in values oix. Thus the equation determines a correspondence of the x- and ^-planes such that to an ;r-point correspond n j'-points, and to a j'-point m jr-points. There is an important difference between the usual point of view of projective geometry and that of the theory of functions. In the former the straight line is fundamental ; because a certain straight line meets a conic in two points we arrange that all straight lines shall meet the conic in two points. For example the curve xy= i is met by the line x=i not merely 294 ALGEBRAIC FUNCTIONS. [CH. XXI in J/ = I but also in jy = oo . In the theory of functions on the other hand it is the correspondence of x and jy that is insisted on ; the equation xy=i is a i, i correspondence between x and J, and therefore to j;= i corresponds just one value j/ = i. The algebraic equation F{x,y) = o may be written j/»/„;ir+jj/"-i/i;t:+...+^/^,;tr+/„;jr=0 (l), where /„,^, ... , fn are rational integral functions of x of degrees not greater than m. The points at which a value of y becomes infinite are given by f^ — o. This equation has wz roots ; for to J = 00 are to correspond m values of ;r. If then it is only of degree mi it has ;« — m^ infinite roots. It may happen that a value a of x which satisfies /^ = o will also make /iX = 6, f^^o. In this case the point oo figures twice among the points in the jf-plane which correspond to x=a. If f^ = o, y"i« = o, /^ = o, /3a 4 o, there are three infinite values of y ; and so on for higher cases. Ex. If j'^jr+yx+jf-i=o, y is a two-valued function of jr. For what values of x are the values of y equal.' What is x when j/ is co ? And what is V when jr is 00 ? '' O,-/ 156. Proof that an Algebraic Function is Analytic. The theory of functions rests very largely on the basis of power series. The transition, then, from the above definition of an algebraic function to the series for j/ in powers of x is an essential preliminary to the knowledge of this function. Suppose for a given finite x, say x,,, we have found a finite j, say j/o ; and that the other values of j/ are all distinct from jo. Writing x — x„ for a new x and y —y^ for a new y we can re- arrange the equation (i) in the form y = c-,^x + c^x'' + Ciixy + c^y'^ + . . . + c,„„;ir™7" (2) ; for by hypothesis when x \s o there is to be one and only one y equal to o, so that in the terms which contain y alone, y itself must occur. Let us consider more generally that the series on the right is infinite ; and let no coefficient be greater than a given positive number 7, so that the infinite double series is convergent when \x\< i, \y\< i. 155-156] ALGEBRAIC FUNCTIONS. 295 Then we can prove that there is one and only one power series in x without a constant term, P-,x = a-,x ->(■ a^x"" ■\- a^i^ ->r (3), which, when substituted for y in the above equation (2), satisfies it identically ; and that this series converges within an assignable region. For first if we assume that there is a series for y of the form (3), convergent when|jr| (5), that is, let «!, aj, ••• become a,, a^,... when for every f we write 7. Then evidently the numbers Oi, etj, ... are all positive; and a„ is not less than | «„ |, for | «i | = ] Coi I = 7 = «!. I «2 I = kzo + ^11 «i + ^oa^i" 1 ^ I ^20 I + I Ai«i I + I c^.a^ I £ 7 + 701 + 701" and so on. Therefore the series (3) converges in a circle {R) if a^x + a^x"- + a^ + . . . 296 ALGEBRAIC FUNCTIONS. [CH. XXI does. Thus the question is reduced to this : can we assert that a power series P-^x for y exists when y = r^x-^ rfX^ + r^xy + 7^" + . . . + Jx-a), which difiFers from the former only as regards the sign of the square root. When x describes a circle about a, — this circle lying in the region of convergence of the series, — then the square root changes sign and y becomes y' while y' becomes y. And so generally if by ^x — a we mean a selected root, a series y — b = PiVx— a is accompanied by r - i other such series differing only in having another rth root ; denoting exp {2'iri/r) by o, the set of r series have no constant terms and are y — b = Py/x — a, y-b = P[a v^F-^], y"-b = P [a^ '^x~^a\ i_y(r-ii -b = P [a*-' ^/x^'a\. 298 ALGEBRAIC FUNCTIONS. [CH. XXI When X describes a small circle about a, the amplitude of x-a increases by 2-k and therefore that of s/x — a by 27r/r. Thus ^x — a becomes a\lx—a and so on; and this means that, when x describes tlie small circle round a as centre, t/ie values y, y', y", ..., _j'i''~" permute cyclically. Ex. The function y — >fx+Jx is six-valued. Mark the six points for which x= I and follow their changes when x describes positively a circle about x=o returning to the point i. 158. Double Points on the Curve F{x,y) = o. So far we have considered (i) the ordinary case where a single value b corresponds to a single value a and conversely; (2) a special case in which r equal values b correspond to a single value a, while conversely to this value b corresponds only one value a ; here x — a = Pr{y — b), y-b = P (^lx-a\ where the notation P^ indicates, as in § 75, that the series begins with {y — by. There remains for consideration another special case: that namely in which each of a selected pair of values a, b occurs more than once : — that is, when x=a, y takes r equal values b, and when y — b,x takes s equal values a. This question is too difficult to be treated here in all its generality ; but we shall discuss the case r = s = 2. Taking a, b as new origins, then when y is zero iifl is to be a factor and when x is zero y^ is to be a factor ; so that the algebraic equation is c^'^ + Ci^xy + c^y^ + higher powers o{x,y = o. We suppose here c^^o, Cf^^o; as otherwise more than two values of x (or _;') will be zero when y (or x) is zero. Thus our equation is {y — ax){y — /3-r) + higher powers = o, where neither a nor yS is o. Notice that the two values of y which become equal to o when jr is o satisfy not only F=o, dF/dy = o, but also dFldx = o. IS7-I59] ALGEBRAIC FUNCTIONS. 299 Let y — a.x=:sx; then on substituting {ol + s)x for y the equation becomes z{a-^ + s) + xP {x, s) = o, where P (x, z) is a terminating series of positive integral powers of jr and z. Hence if a4=y9. we have (§ 156) z-= PrX where r>o, and y = ax + Pr+iX, where the suffixes indicate that the series begin with the rth, {r + I )th terms respectively. Similarly the factor y — ^x contributes a series y = ^x + P^+,x. Hence we have power series in x for the two values of y which vanish when x = o, and x==o is not a branch-point although two values ofy are equal when x = o. But if a = /8 we have z'^-\-xP{x, ^) = o, and it makes a difference whether or not P{x, s) contains a constant term. If it does, then x=P,{z), z^P4x, y = ax + xP '/x = Pi ^Ix, and x = o is a branch-point ; and similarly y = o is a branch- point. The case in which P (x, z) does not contain a constant term, — that is, when the terms of the second and third orders in the original equation have a common factor, — belongs to the more general question to which reference was made above ; this is discussed in our larger treatise (ch. IV.). 159. Infinite Values of the Variables. There remains for consideration the case in which one or both of the selected pair of values a, b are 00 . First when a alone is 00 , we determine the series for^ — ^ by writing :t:= I /;r'; points near 00 in the x-plane become points 300 ALGEBRAIC FUNCTIONS. [CH. XXI near o in the jr'-plane and we can determine from the algebraic equation between y and x', in the normal case, a series ^-^=.'A.'=i/'o(i; {ox y—b. This is equivalent to replacing j:— oo by \\x. Special cases can be treated as before. Similarly when b alone is oo we write _y=i/y and discuss the algebraic equation between y' and x for the pair of values x = a, y' = o. Lastly when both a and b are oo we write x= \\xf,y— ijy' and discuss the algebraic equation between x and y' for the values x' = 0, y' — o. If for example F(x,y) — o take the form y'^x — x!^ —y"- — i = o, then when x=vi both values of y are oo , and when j = oo either 4:= 00 or x=-\. Writing ;i:= xjx', y= ijy, the equation becomes x'-y'^-x'^-xy* = o; whence x' = P^ y' and y' = Pi (V^')- The actual coefficients in the series are readily determinable in this simple case, for we can solve the equation at once for either x' or y'. Near _j/ = 00 , jr = i ; we have, writing j = i/y, jt — i = y, I +x' -y"{i +xy - I -y'^ = o, or y — 2y'^ — 2y'*x' — j/' V" = o ; whence x' = P^y"^, or x-i=Pi{ily''), and ily-Pi{'Jx-i), so that 3/= -j=^+C, + C.,slx-l+C^{^X-lf-\-...-\-Cr,{'^X-lY-\-.... NX ^ I This is a case in which an infinity oiy, — that is, a value of x lS9-l6o] ALGEBRAIC FUNCTIONS. 301 which makes y infinite, — leads to a Puiseux series ; in such a case the infinity is also a branch-point. 160. The Singular Points of an Alg^ebraic Function. The general expression for the values of j/ which are infinite when x=ai% obtained from \ly = Pr'Jx — a, giving y=;i{x- ay P^ r), 1 1 r=l,*=2 etc. have no terms in {x-af^ (x-a)^\ (x-a)*'\ ..., since the coefficient is in all cases i+v + v^ and this vanishes. As there are no fractional exponents the corresponding series are of the form P{x — a). The argument applies generally and shows that the symmetric functions of ji.ja, ...,yn are equal to power series in x—a, preceded it may be by a _fimU number of powers (x — a)'^ for which the exponent is a negative integer, and a similar result holds for the other branch-points (which, it will be recalled, are finite in number). Hence all symmetric functions oiyi,y-2, ■■•,yn are one-valued functions of ;tr with no singularities other than a finite number of non-essential singular points (oo inclusive). It follows from the theorem of § 104 that these symmetric functions are rational functions of x; and as the coefficients /ij/s, ...,/„ in y" -t- A y'~' + A.;'""' + ...+/>n = iy -yi) {y -y^) ...{y -yn), are equal to — Sji, +'^yiy2, •■■, (—Tyiyi---yn, y satisfies an algebraic equation of degree n in y whose coefficients are rational functions of x. The theorem that we have proved is a good illustration of the remark in § 104 that the character of a function is often determined best by observing the behaviour of the function at its singular points. 161. An Algebraic Equation in x, y defines a Single Function. We shall now show that it is possible to start from any non-singular point x^ with any one of the n values of y at that point and by describing a suitable path arrive at an l6o-l6l] ALGEBRAIC FUNCTIONS. 303 arbitrary non-singular point x^ with any one of the n values J'1.^21 ■■■,yn that satisfy the equation in^ for this value oi x^. To fix ideas let us start with a definite series y-y^ = {x-x„)P{x- x^) at x^ ; this can be continued so that one value of y shall be determined for any value x^' so long as the path employed does not pass through a branch-point (and we shall suppose this to be the case for all paths employed). Suppose that one of the values 3.x. x^, say y—y^ = {x — x^)P{x—x^), can by no means be attained ; then starting from x^ with this value and passing along any path to x^ the value oi y at x^, say Q{x — x^, must be distinct from y^-\-(x — x^P{x — x^. This means that when x describes any circuit the values _y,,^2, ...,yn separate into two classes. The first class includes all those values, — sayj'i.j^'a. ■■■■,yr, — for which it is possible to find a path that will convert y-i into any other of the set; the second class includes the remaining roots which cannot be derived in this way from y^. The r values y^, y^, ...,yr may be permuted by the description of a circuit ; this cannot happen unless the circuit includes one or more branch-points. Suppose, to fix ideas, that the circuit includes one branch-point, b say. The cir- cuit may be contracted into a loop, that is, into a path from x^ to a point b followed by a small circle round b and the same path as before reversed in direction (fig. 71); and the final value oi y will be the same after as before the contraction. Now if y take near b one of a system of cyclic values yi, y^, ■••,yr (§ i6o), these values will permute cyclically. But if it take near b a value which is not one of a cyclic system, the final value is the same as the initial value. Thus in any case the symmetric combinations of y^, y^, ..., yr are unchanged and therefore they are one-valued functions of x which must be rational functions since the singular points are finite in number and non-essential. It follows that jj/,, y^, ..., yr satisfy an algebraic equation Fj (x, y) = 0; and this implies that F 304 ALGEBRAIC FUNCTIONS. [CH. XXI is divisible by Fi contrary to the hypothesis that F is irre- ducible. TAe equation F{x,j/) = o defines, t/iejt, a single function whose n values can be interchanged by choosing suitable circuits. Lastly any closed path whatever may be contracted into a series of loops taken in a determinate order and described in a determinate sense ; and the final value of y for that path may be inferred from the effect of the several loops taken in that order. Fig. 72 illustrates this deformation of closed paths; in each case the larger curve is deformed into the smaller (see also fig. 64). 162. Riemann Surface for an Algebraic Function. Without entering into details we shall indicate in a few words the nature of the Riemann surface that is used for the general algebraic function of x. We construct the surface on the ;r-plane. The surface contains w sheets because y is «-valued, and the n values of y are attached to the n points of the surface which lie vertically above the assigned point in the jr-plane (supposed horizontal). These n sheets are connected by lines of passage which permit interchanges among the n values oi y. If in the ;ir-plane r values jj.j/j y^ permute cyclically when x describes a small circle («), then on the surface the path cut out by a vertical cylinder standing on («) is a spiral 161-162] ALGEBRAIC FUNCTIONS. 305 which winds r times round (a) before returning to its initial place. As x passes r times from x^ to x^, the moving place (x,y) starts from {x^,y^ and passes successively to The Riemann surface gives a clear idea of the extent of the circles of convergences of the series P{x—x^ at a non-singular point. These circles extend to the nearest singular places in t/ieir respective sheets ; and here it is to be observed that a value x = c which is an infinity or a branch-point of some of the n values oi y is an ordinary point for other of these values, and that on the Riemann surface the places that correspond to these latter values offer no obstacles to the expansion of the cor- responding circle of convergence. 20 M. H. CHAPTER XXII. CAUCHY'S THEORY AND THE POTENTIAL. 163. Cauchy'B Definition of a Monogenic Function. In this final chapter we shall point out some of the more salient features of Cauchy's definition of an analytic function, indicate the point of junction of the respective methods of Cauchy and Weierstrass, and then discuss a few simple cases in the theory of the potential for the plane. Suppose that, following the plan of § 51, we assign in any way we please one real value to each point of a region T in the jr-plane ; in this way we construct for the region T a function of f , T). Let u, V be two functions so constructed ; then u + iv, =y, is a function of x for the region T in the sense that when x is given, f and 1} and therefore u and v are determined. When u and V are left completely arbitrary, the combination of u, v into the single expression u + iv offers no advantages ; for, ultimately, u and V have to be considered separately as functions (in the sense of § 5 1) of the two independent variables f and r). Cauchy saw clearly the absolute necessity of sorting out from the total mass of functions of x those which may, in a useful sense, be regarded as functions not merely of ^ and 77, but also of ^ + ir). He discarded such functions of jt as f — ir), ^7), and retained such functions as (^ + irjY, sin (|+ «i;), etc. It is necessary to draw a line of division between the two classes ; this can be done with the help of certain partial differential equations. 163] cauchy's theory and the potential. 307 Take for u + iv such expressions as (f + ir)Y, sin (f + zi;), ^+*''; then g|(M + zz;)=-Zg^(«+zz;) (i); and, on equating real and imaginary parts, (i) leads to d^~d^' a^~~aj ^^^• On the other hand when u + iv is equal to such an expression as f — if}, — one of the functions which Cauchy discarded, — these differential equations are not satisfied. We have arrived at these equations by starting from definite expressions ; we shall now show how they can be reached by more general reasoning. Let u, V be defined as in the beginning of this article, and let y = u + iv =fx, where jt is a selected point interior to T. When x increases by Ax, = A^ + zAti, x + Ax lying within F, let y increase by Ay, = Azi + iAv. Now let Ax tend to zero ; the question arises as to what conditions are necessary and sufficient in order that — may tend to a unique finite limit when Ax tends to zero ; when this limit exists at x it is denoted by -^ or f'x as in the case of the real variable, and called the derivate of x. Selecting an X in the finite part of the plane, sin ;r has a derivate cos;ir which does not depend on the way in which Ax tends to zero ; on the other hand if we ^\xt y^^-ir) and allow ;r + A;ir to approach x along a ray through x with the chief amplitude o, we have for this mode of approach, ^y A^ — lAr) _i —i tan a ii?o A^ = .f-o!^,=o AfTiAi, - i+itana ' a quantity which varies with a, and therefore with the ray. Here there is a definite limit for the assigned mode of approach, but this limit depends on a and therefore there is not a unique limit for all modes of approach of Ax to zero. We shall find that the differential equations (2) are necessary conditions for the uniqueness of the limit at the selected point x. 20 — 2 308 CAUCHY'S theory [CH. XXII I. Necessary conditions for the uniqueness of tlie limit. Evi- dently if there is to be a unique finite limit for -^ , this limit must be equal to the limit under the special circumstances, (i) ^ constantly real and equal to A|^, (ii) Ax constantly purely imaginary and equal to lAr). In case (i) we have Ay _ u(^+A^,v)-uilv) v{^ + A^, v)-v(^,v) . Ax~ Af ^ Af dv it is therefore necessary that -j- shall satisfy dj _du . dv , - ^=9| + '8| ^^^' while in the second case we have Aj' ^ M (g, 7? + A i;) - u (g, 7?) _^ ^ v(^,v + Av)-v{^,v ) Ax iArj iAri ' ^, ^ dy .du dv , . ^°*^^* £=-% + a^ (4). Equating (3) and (4) we get the differential equations (2) ; these differential equations require, by implication, that u, v shall have partial derivates with respect to f and ij at the selected point x. Hitherto we have considered only two special modes of approach of Ax to zero. If there is to be a unique and finite Ay limit for -^ for all modes of approach, then we must have (replacing x + Ax by x" for shortness), dy .. fx'-fx du _ .dv . . at the selected point x. It is only another way of stating (5) if we assert that there is a derivate at the point x provided a positive number 8 can be found such that fx'-fx -f'x <« (6), X —X for all values of x' that satisfy the inequality \x' — x\7, Az/ = ^ A? + ^ At; + ^3 Af + rA'n, where lim r^ = o (\ = i, 2, 3, 4) when Af, At; fe«^ independently to zero. Before passing on to consider whether these two conditions are also sufficient, we wish to point out a consequence of the f^ — fx equation (5) ; namely that •^— ; — — converges uniformly to its X — X limit fx when xf tends to x (§ 55), whatever be the mode of approach olx' to x; in particular this is true when the approach is along rays through x. II. Sufficient conditions for the uniqueness of the limit of -^ . The two conditions that we have just shown to be necessary if there is to be a unique and finite limit, are also sufficient. For, 310 CAUCHY'S theory [CH. XXII using the second of these conditions and putting h = p cos a, k= p^m. a, we see that du du . . /dv dv . \ . ^t; cos a + 5- sin a + ? 57; cos a + 5- sin a -~ = -2 ■ ^-^2 i h i/f (p, a), Axr cos a+ z sin a where ■yfr (p, a) is a function of p and a such that lim tfr {p, a) = o. p=0 The expression on the right-hand side tends to the limit /du .dv\ ((lu .dv\ . cos a + 2 sin a ox TTT.-^ i ;-r: , since cos a + ? sin a is a factor of the numerator by reason of equation (i). Thus -^ exists and is equal to xj+ «^. Suppose now that it is given that ri, v admit at x continuous first derivates with respect to f, r\, then the second of the two conditions that we have been considering is necessarily satisfied and may be omitted. Hence when the partial derivates of u, v of the first order are continuous functions of ^, t) at a selected point X, the differential equations (2) constitute the necessary and sufficient conditions for the existence of dyjdx at this point. The continuity oi fx is implied in the existence oi fx, as is seen by inspection of formula (7) ; the new conditions that have just been imposed makey^tr also continuous. To indicate that the function y, =fx, has the property that Ay -r— tends, in general, to a unique finite limit Cauchy employed the term monogenic \ in this way he excluded such functions as ^—it) as non-monogenic. Riemann dispensed with the adjective; in his terminology "a complex variable w is called a function of another complex variable z, when the former varies with the latter in such a way that the value of the derivate dwjds is independent of the value of the differential dz.*" * Riemann, "Grundlagen fiir eine allgemeine Theorie der Functionen einer ver- anderlichen complexen Grosse," Ges. Werke, p. 5. 163-164] AND THE POTENTIAL. 311 164. DlfflcultieB underlying Cauchy's definition. We shall now point out some of the difficulties that underlie Cauchy's definition of a monogenic function. I. Is monogenic to mean monogenic at a point, or monogenic over a region } It is the usual, though not the invariable, custom to consider monogenic as an adjective applying to a region. Given that a function is monogenic over a region, is this to mean that the function has a finite derivate at all points of the region, or is allowance to be made for exceptional points, lines, regions .' To take a simple example, is i/x to be regarded as monogenic in a region which contains jr = o.'' It is customary to make allowance for such points : e.g. e^'" may be treated as monogenic over the whole plane although it has no derivate at x=o. To a considerable extent it is merely a matter of nomen- clature whether we do or do not admit singular points into the regions over which functions are monogenic. But there are necessary restrictions to this practice. In § 94 we discussed an arithmetic expression which defines in different regions parts of distinct analytic functions : Weierstrass, in view of this fact, regarded such an expression not as defining a single monogenic function, but as defining parts of two distinct monogenic functions. In fact one of the lessons to be drawn from a study of analytic functions is the importance of keeping clearly distinct the idea of a function defined by properties and the idea of the depen- dence implied in an arithmetic expression ; the two ideas are by no means necessarily coextensive. An expression may represent QO an analytic function only partially, e.g. 2 ;ir« ; or again it may »=o represent different analytic functions completely in different regions ; or it may represent certain analytic functions partially in certain regions and other analytic functions completely in other regions. We shall not stop to justify these statements, though it is not difficult to do so with the aid of § 94. The point which we desire to emphasize is that Cauchy's definition implies in various ways a considerable preliminary grasp of the logical possibilities attached to the study of singular points. 312 CAUCHY'S theory [CH. XXII fxf —fx II. The inequality (6) shows that "^ , J is to converge uniformly to its limit (§ 55). Hence if xf tends to x along rays through X, it is necessary for the existence of f'x that the convergence for these modes of approach shall be uniform. We shall show below that it is possible to construct a function fx which shall have a unique limit for -^ for all modes of approach of x' to X along rays and yet not have a derivate f'x. From fx —fx ■ . what has been said the convergence of" ^ ,_ must, in this case, be non-uniform for the system of rays. Here then we have a difficulty connected not with the singular but with the ordinary points of a function : the recognition, namely, that the conver- gence to the limit must be uniform. Riemann's definition of a function (§ 163) must not, therefore, be so interpreted as to restrict the paths of approach of Lx to zero to straight lines. Stolz* illustrates the case of non-uniform convergence to the limit /V along rays through o, by the function fx which is equal to ^x ^' ^ for values of x different from o, and is equal to o for x=o. If we put f = p cos a, 7; = psina, where a is the chief amplitude of a ray through o, we have V fx -fo lim =^— = 0, p=o X Ay so that for each ray the limit of ~ exists and is equal to o ; but the convergence to this limit o is not uniform. To convince ourselves of this fact let us put e = a proper fraction, and take the system of rays from a = o to a = n-/2. Along each ray we shall suppose that a stroke of length ^ is measured off, where h' is the greatest positive number such that we have, for all points x situated on the ray at a distance from o less than d', \ fx-f0 \ __2$jf_ I X I' -^+7' As a approaches nearer and nearer to nJ2, d' gets smaller and smaller ; for the stroke cannot extend as far as the parabola ij^=f, because on this parabola the expression on the left-hand side of the inequality has the value i ; hence the lower limit of d" is o, and there is no single value 8 of the kind considered in connexion with the inequality (6) which will serve for all rays of the system. In other words — does not converge uniformly to o, * Stolz, Gmridzuge der Differential- und Integralrechnung, vol. ii. p. 80. 164-165] AND THE POTENTIAL. 313 o is not the limit (in the sense of equation (5)) of this ratio, and the function fx is not a monogenic function of x. III. There are difficulties which relate to the postulation of continuity for the derivates ol fx. Cauchy's theory of functions has in view the same functions as those considered by Weierstrass, namely, functions fx which are analytic about an ordinary point X. This being so, the derivates /';«:, /";r,... must all exist and be continuous at x; but it is evidently undesirable to postulate explicitly the existence and continuity of f"x, f"'x,..., if these facts are consequences of the existence and continuity of fx and fx. Among Cauchy's many important contributions to mathe- matical knowledge, a high rank will always be assigned to his proof that a theory of functions can be constructed on the basis of monogeneity coupled with the continuity oif'x (Cauchy himself included the continuity of fx in the definition of a monogenic function). To appreciate properly the remarkable nature of this discovery of Cauchy's it is necessary to bear in mind that when the variable is real a function (in the sense of § 48) may be continuous and yet not have a derivate, so that the existence of fx, f"'x,... is by no means an evident consequence of the existence and continuity of fx and fx. Granting that at an ordinary point a monogenic function must have a continuous derivate if it is to be admitted into a theory of functions of a complex variable, the question at once arises: — does there exist a class of monogenic functions fx for which fx is discontinuous 1 To present the matter in a somewhat different way: — what is the irreducible minimum of conditions to be imposed upon fx, if we desire to have fx, in general, analytic about x ? It may very possibly be true that in order that fx may be analytic about a point x it is necessary and sufficient that fx shall be one-valued, finite, and admit a finite derivate for all points of a neighbourhood of x*. 165. Extended form of Taylor's Theorem. Cauchy proved his theorem on integration (§ 118) for every function * Pringsheim seems to assert this in MatA. Ann. vol. xliv. p. 80 (1893), but see a foot-note in his paper on Cauchy's theorem, Sitzber. d. k. bay. Ak. d. JViss. (1895), vol. XXV. 314 CAUCHY'S theory [CH. XXII fx which is one-valued and admits a one-valued continuous derivate over a closed region F. The same consequences follow as before as regards residues, etc.; the only difference is that_/ir is defined in a new way. But that fx is analytic about each point of r now requires proof, whereas, before, this property was postulated for fx from the start. The process by which Cauchy established the theorem that when fx is one-valued and admits a one-valued and continuous derivate at each point of a closed region (c, R), it can he expanded as a power series P{x—c) whose radius of convergence is not less than R, is precisely that used in § 122, the ring being supposed reduced to {c, R) by the vanishing of R'. The coefficients in the expansion fx = a^ + a-i{x-c) + a^{x-cf + ...■\-an{x -cY -{■..., I r fxdx are the definite integrals — ; I -j^ — —-- , so that _^/-A^ ^,„)^^^r fxdx 2.iri]cX — c' ■" Ziri] cix-cY""^ It is evident from what has been said that the theorem which we have called Cauchy's theorem (§ 118) occupies a central position in the theory of analytic functions as developed by means of integration. We shall give at the end of this chapter a second proof of this theorem which involves the transformation of a double into a simple integral by means of Green's theorem. This proof was discovered by Cauchy and was used later by Riemann. By proving that a function fx which is one-valued, continuous, and admits a continuous derivate over a neighbourhood of x, is analytic about x we have arrived at the point of junction of the theories of Cauchy and Weierstrass. The analytic function is always monogenic ; but to be able to say conversely that Cauchy's monogenic function is always analytic the idea ' mono- genic' must be made more precise in that part which Cauchy left vague. One analytic expression must be allowed in some cases to represent more than one function completely or partially. This is more an addition to than a modification of Cauchy's idea. 165-166] AND THE POTENTIAL. 315 Cauchy's theorem was contained in genn in a memoir on definite integrals (1814); explicitly in a supplement to this memoir (1825). Gauss enunciated the theorem in 1812 in a private letter to Bessel. Green's theorem is contained in his Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, 1828; that proof of Cauchy's theorem (§ 170) which depends on Green's theorem was published by Cauchy in 1846. It is possible, but not probable, that Cauchy was acquainted with Green's memoir, for it was not till 1850 that Green's memoir became readily accessible to mathematicians. Riemann's proof, which applies also to many-valued functions, was published in his inaugural dissertation (1851), Ges. Werke, p. 12; it agrees essentially with Cauchy's. 166. The Potential. When u + iv is an analytic function of X, say then df^^^df^J (^+^'')' the second derivate f" being definite at any point about which /"is analytic. Hence adding we have Thus u and v satisfy the same differential equation, which is Laplace's equation for two dimensions. Thus our theory applies to an important physical concept, namely the potential in a plane. By this term we mean any function of ^, ri which can be used as the u or v of an analytic function u + iv oi ^^ it). It partakes of the properties of the associated analytic function ; its singular points are to be sought among the singular points of the analytic function, and it may be many-valued. We shall confine our attention however to a region of the plane, and within that region we shall suppose the potential to be one-valued and continuous; then at all points of the region it will have continuous derivates of all orders. Historically the developments of the potential, as a measurable physical concept (like a distance or an angle) and as a function 3l6 CAUCHY'S theory [CH. XXII of the position in the plane (that is, ultimately, as a number), are closely connected. An introductory discussion of the potential in a plane will now be given ; for its appropriate application and extension in the various departments of physics reference must of course be made to the physical treatises. When we write down an analytic function of ^+iv or x, and separate the real part and the imaginary part so as to express the function in the form u + iv, where u and v are real, we obtain a pair of potentials u and v. A convenient way to think of such a function as m or i' is to conceive an ordinate to the ;ir-plane erected at the point (^, t)) or x, the length of the ordinate representing the value of the function u ox v. A characteristic property of the surface generated by the ex- tremity of the ordinate is that the projection on the jr-plane of the indicatrix at a point about which u + iv is analytic, is a rectangular hyperbola. For we have, by Taylor's theorem, the projection of the indicatrix is obtained by keeping u —u constant and omitting powers above the second ; and the condition that the cylinder so obtained shall have as cross- section a rectangular hyperbola is ;r^ + ;r— =o. of ^ cif Potentials occur in pairs, u and v, where u + iv is an analytic function of ^-\-it). The two are often said to be conjugate. It is to be observed that if v is the conjugate of u, then not u but - « is the conjugate of v, so that the choice of the adjective conjugate is not entirely happy. When the one potential is given the other is determined, save as to a constant, by the equations 'bu _ dv du dv _ but it is usually more important to know the form of the analytic function than to know « or z/ separately. 166-167] AND THE POTENTIAL. 317 Some simple potentials are : (i) The distance from a point to a fixed line; for evidently f and ij are themselves potentials and so is a^+zS^j^ + y where a, /3, y are constant. It must be noticed that because Laplace's equation is linear, the sum of any given number of potentials is itself a potential. The representative surface is here any plane. (2) The amplitude of x. For log jt = Log p + ?0, whence Logp and 6 are potentials. Thus the angle made by the line from any fixed point to X with any selected zero of direction is a potential. The representative surface is a helicoid. The potential includes then two of the fundamental quanti- ties of geometry, distance of point and line, and angle. But it does not include the distance p of two points, nor any power of this distance; p" is not a potential, but Logp is; whereas in three dimensions i/p is a potential. Hence the present potential is often called logarithmic. (3) Let ^1, dt, ..., dn be the amplitudes of strokes from fixed points to x. Then ai^i + fla^a-l- ... +a„^„ is a potential. 167. The Equipotential Problem. The problem of potentials is : given a system of real values continuous, in general, along a circuit, to find a potential « which shall be one-valued and continuous within the region bounded by this circuit and shall take the assigned values on the contour. A case of special simplicity and importance is when the boundary is made up of equipotential lines, that is to say, lines along each of which « is a constant. We may call this the equipotential problem; and we shall assume here that the solution is unique. The uniqueness of the solution is strongly suggested by physical considerations; and is readily proved by means of Green's theorem (§ 169). 3l8 CAUCHY'S THEORY [CH. XXII But even when the boundary consists of equipotentials the problem in its generality is quite beyond our scope. The inverse process, however, of determining the region when we start with a known analytic function and therefore with a known potential, is merely a matter of mapping. For example, as a case of (3) of § 166, let the region be the upper half of the ;r-plane, and let 0,, 6^ be the amplitudes of strokes from fixed points of the real axis, then 61 — 6^ is a potential ; it takes along the real axis the values 0,77, o. Hence, conversely, if the real axis is divided into three parts which are kept at potentials o, tt, o, the potential at any point of the upper half-plane is d^— d^. The equipotential lines are 0^ — 0^ = constant, and are therefore arcs of circles. The conjugate potential is Log pi — Log /Og ; it is constant along the circles p^jp^ = constant. That these circles cut the arcs at right angles appeared in § 21 ; but it is an illustration of the general property of isogonality, by which the orthogonal straight lines u = constant, v = constant, in the {u + «Z')-plane map into orthogonal lines, provided we have at their intersection u -\- iv — u^ — ivo = P^{x — x„). If we have P„(;jr — ;r„) instead of Pi{x — Xo), then (§ 108) the angle in the (u + ?V)-plane is « x (the angle in the jr-plane). When u is the potential considered, the lines v = constant in the jr-plane are the lines of flow (lines of force in Electrostatics) ; conversely, when v is the potential considered, the lines of flow become equipotentials, and vice versa. The present example solves the equipotential problem for a crescent, or region bounded by two circular arcs when these arcs are kept at given constant potentials. For, measuring the amplitudes ^1, 6, from the intersections of the arcs, the equations of the arcs are ^1 — ^2 = 72 ; and the function a{0i — 6^) + ^ is a potential which takes 167] AND THE POTENTIAL. 319 constant values «i, «j along these arcs. The constants a and /8 are found at once when these constant values are given. Thus we have for any coaxial arcs for which ^1-^2 = 7. 7i. 72. « = 07 + j8, «i = a7i + )8. th = a7a + ^, and 7 7i 72 I I I = 0. The potential a„ + Sam^m applies similarly to the case when a straight line, say the real axis, is divided into « + i intervals, each kept at constant potential. Let the first interval be that for which all the 0's are zero, the second that for which 0, = tt and the remaining ^'s are zero, and so on ; also \eX.v^,v^,v^, ..., ■z;,, be the «+ i values of the potential v along the ist, 2nd, 3rd, ..., (« + i)th intervals. For simplicity let the straight line be the real axis and let the region bounded by this straight line be the upper half of the .ar-plane. We shall have O0+ 7r(ai + a2) = z'2, and so on, « + i equations to determine the constants. Let us suppose (fig. 73) that the potentials v^, v-^, v^, ..., z^„ are in decreasing order; it follows that Oj, Oj, ..., a„ are negative. The figure formed in the plane of a + iv must be noticed. It is of course formed by a series of lines v = constant ; but of these lines how much is to be taken ? The analytic function u + iv is here ^ = /a„ + /3„ + 2am log {x - ^m), where f„ is a dividing point of the straight line, which is taken as the real axis in the ;ir-plane. And the conjugate potential to V is « = /3„ + 2a,„Logpm, where /Sm = I ? — ?m I- 320 CAUCHY'S theory [CH. XXII Hence since Om, is negative, as f describes the real axis from right to left n passes from — oo to + oo at fi, from +00 at fi to + 00 at fa. and so on ; finally passing back to — 00 only when f has passed f„. In fact ?/ attains a minimum when an equation with n - i roots separating the points fm- Thus the map in the jz-plane is as shown in fig. 73 (which is drawn for the case n = 4), and contains a system of straight lines which are parallel to the real axis and which extend to 00 on the right, but not to — 00 on the left except in the case of the first and last lines. Each of the lines intermediate between the first and last is traversed twice, and the terminal points of these lines can be found from the roots of the above equation. The straight lines /oA' AA> AA'/sA' are the maps of the small semi-circles (assumed extremely small); the dotted line on the left is the map of the dotted semi-circle (assumed extremely large) in the jT-plane by which the .*r-circuit is completed. Ex. Consider the case when the given potentials t/q, v-^fV^, ..., v^ are not in order of magnitude. We can solve in the same way the equipotential problem for the case of a circular boundary divided into n+i intervals, or we can infer this from the straight line by the principle of inversion. For if y = u-'riv =fx, and we write x = {ax' -(- b)j{cx' + d), then x and x' describe inverse curves X and X' ; and if the function fx has been 167-168] AND THE POTENTIAL. 32 1 determined which maps the parallel half-lines u = constant on X, then the function _/"|(ax' + b)l(cx' + d)} will map the same half- lines on X'. With regard to equipotential lines in general it must be noticed that at a point of the ;r-plane at which v is infinite u may have any value, so that all the lines u = constant will lead to the infinity. 168. Schwarz'B and Christoffel's mapping of a straight Line on a Polygon. Suppose that we have a rectilinear polygon in the jr-plane with exterior angles Bitt, o^tt, .... a„7r, measured in the negative sense so that Som = — 2. The polygon is supposed convex so that o>am>— i. If with these conditions we write z = expj, then 2 = exp (/S„ + ij.^) U{x- ^m)'", X and^ being connected by the equation (i) of the last article. The parallel half-lines in the jz-plane become rays in the 2-plane from 00 towards the origin, but only the first and last reach the origin. Moreover since for large values olx,z^P^{\ \x) s-plane M. H. 322 CAUCHY'S THEORY [CH. XXII since Sa^ = — 2, the semi-circle of fig. 73 becomes a complete circle ; that is, the first and last lines of the .sr-plane coincide. Now let w = Jsdx; then as x moves along its real axis from + 00 to o, we have dw = zdx, and am dw = am z + am dz = am z + ir. Hence the straight lines in the ^-plane of fig. 74 (drawn for « = 4, to fix ideas) remain straight lines in the w-plane ; and since, near |„, tV=j{x- ?„)•» Po {X -^„ddx={x- ^™y+'- (2o (X - fm), there is no abrupt change in w when x describes the small semi- circle round ^m', in the limit the map in the zc-plane is a polygon ; and since near x=^', g'q are positive arcs, we see that k = dr) at / and — drj aX, q and hence the part due to the strip is the sum of elements u^drj a.t p and g, and the part due to all the strips, that is to r, is I u;^ di], which is what we wished to prove. Applying similar reasoning to \ \ ^ ^ d^ dt] we get ILm'^i-'H^r/^-t^'^ «■ By adding (i) and (2), [[ fdu dv du dv\ ,^ , [ fdv , dv ,^\ -f[ uV^vd^dv (3), 92 92 JJt where V' denotes ^jj + 5— ^ • This formula (3) is one of the forms of Green's theorem; another form can be found by interchanging zt, v in (3) and equating the expressions on the right-hand sides of (3) and of the new formula. 324 CAUCHY'S THEORY [CH. XXII Let s be the length of the arc measured along C in the positive sense from a point of reference to the variable point/, and at / let a normal be drawn inwards of length «. The coordinates x, y of the extremity of this normal can be expressed (at least theoretically) in terms of s, n. Then ^ = cosVr, 9^ = sm^, s7 = ^'"^' 37 = '=°'^' where -v/r is the angle made by the tangent at / with the positive direction of the axis of ^. Hence 8| _ 9j7 dri _ 9f ds dn' ds dn' and (3) becomes -JJruV^vd^dv (4)- Interchanging u and v and subtracting the result from (4) we have the standard form for Green's theorem : — j^{u^£-v^£jds + //p (uV'v - vV'u) d^dr, = o (S). 170. Cauchy'B Theorem. A similar method of transfor- mation of a double into a single integral when applied to converts these into /c («^? - ■vd'n), jc {vd^ + udTJ), and hence /c (w + iv) dx — Jc {ud^ — vdti) + ijc {"vd^ + udi)) If the potentials u, v are conjugate, the double integrals vanish by reason of the equations satisfied by the conjugate potentials u, v ; and hence Cauchy's theorem is proved for the circuit C. Cauchy's integral .Jfxdxj{x—c), see § 120, can also be 169-170] AND THE POTENTIAL. 325 connected with the theory of the potential. The formula (5) was proved without making any use of the hypothesis that u and V are conjugate potentials; the theorems (i) to (5) are true if we merely postulate that «, v are to be one-valued and continuous within C together with their derivates of the first and second orders. Hence it is not necessary, even when u, v are solutions of Laplace's equation, that they should be con- jugate. We may, in particular, write « = i in (5) and deduce the equation /ea-^^ = -/L^^^''^'''' = ° ^'>- The restriction on C in equations (i) to (6) can be re- moved ; moreover the boundary of V can, if we choose, be supposed to consist of several circuits instead of one ; the simple integral is then taken positively with respect to T over all these circuits. As a special application take a region F bounded by a circuit C and a small circle (c, e) about a point c within C; and let u — Log p where p is the distance of c from a variable point on C. Then V»« = o, V=z/ = o, and (5) becomes But - °^^ , = -^, has the value --along {c, e); hence dn pan e = - I vds, by (6) ; e J (c, t) {c, e) being now described positively with respect to c. As the expression on the left does not depend on the value of e, for neither the integrand nor C depends on e, — the integral on the right must be independent of e; hence to evaluate it we make e tend to zero and observe that - vds * J (e, t) can be made to differ by as little as we please from - v{c) ds, € J (c, e) 326 CAUCHY'S theory and the potential. [CH. XXII that is from zirv (c). As the integral is known to be indepen- dent of e, 271^^ {c) must be its exact value. Thus 'W=.UH'I-"^}* <«>• The formula (8) is a particular case of the theorem 2injcx — c To prove this it is necessary to resolve both sides into the real and imaginary parts and then equate the imaginary parts ; for the details of the verification we must refer the reader to Picard's Traits d' Analyse, vol. ii. pp. 109, no. Even without this verification the dependence of fc, — the value of fx at an interior point, — on the values of the same function along the rim C suggests strongly a connexion between this formula of Cauchy's and the solution of the physical problem of the deter- mination of the state of a body at an interior point when the boundary conditions are given. LIST OF BOOKS. General Treatises : Among elementary books dealing generally with the Theory of Functions may be mentioned Burkhardt, Ein- fuhrung in die Theorie der analytischen Functionen, and ThomjE, Theorie der analytischen Functionen. Students who wish for a fuller treatment should read Forsyth's admirably written book, Theory of Functions of a complex variable. Copious references are there given to the original sources. Other general accounts are contained in our Treatise on the Theory of Functions, in Picard's Traits d' Analyse, vols. i. and ii., and in Jordan's Cours d' Analyse. The following short list of books dealing with special depart- ments of the subject may also be useful. The notions of number, limit, etc. : Chrystal, Algebra ; Stolz, Vorlesungen iiber allgemeine Arithmetik; Tannery, Thdorie des Fonctions d'une Variable r^elle. Trigonometry : Hobson, Plane Trigonometry. Elliptic Functions; Appell and Lacour, Principes de la Theorie des Fonctions elliptiques ; Tannery and Molk, Elements de la Theorie des Fonctions elliptiques ; Enneper-Muller, Elliptische Functionen. Algebraic Functions and Abelian Integrals : Appell and Goursat, Throne des Fonctions algebriques et de leurs Int^grales ; Baker, On Abel's Theorem and the allied Theory ; Brill and Nother, Die Entwickelung der Theorie der algebraischen Functionen, in the Jahresbericht der Deutschen Mathematiker-Vereinigung, 1894 ; Klein, On Riemann's Theory of algebraic Functions and their Integrals (translated by Miss Hardcastle) ; Neumann, Vorlesungen iiber Riemann's Theorie der Abel'schen Integralen ; Stahl, Theorie der Abel'schen Functionen. INDEX. The numbers refer to the pages. Abelian integral, 388 ; elliptic case, 288 ; rational case, 288 Absolute convergence, see Series Absolute value, defined, 17; notation for, 17 Adams, 196 Addition of complex numbers, 18; of ordinal numbers, 6; of strokes, 12, 15, 17, 61 Addition theorem, 157; of circular functions sinj:, cosjr, 21, 171 ; of ^. 54; of iP«. «6o, 261; of f«, 263, 269 Algebraic equation in x, y (irreducible), 293; defines a correspondence, 294; defines a single analytic function, 302 Alcrebraic function of x, 293; branch- points of, 297 ; coefficients in equation giving, 294; discussion of x=a:> , 299; infinities of, 300; infinities of due to the vanishing of coefficients, 294; is analytic, 294; properties of, 301 ; properties of, are distinctive, 302; Puiseux series, 297; Puiseux series, cyclic arrangement of, 297 ; rational, 84 et seq. ; Riemann surface for, 304 Amplitude of a ratio of strokes, 20 Amplitude of x, defined, 17; notation for, 17 Amplitude of jt, chief, defined, 17; no- tation for, 17 Analytic about a point, 181; over a region, 181 Analytic function of x, 73, 84; algebraic equation defines an, 303; alge- braic function is an, 294 ; character of indicated by the singularities, 183; compared with monogenic, 314; defined (Weierstrass), 15461 seq. ; element of, 154; element of, domain of, 1 78 et seq. ; general remarks on, 156; limit-point of infinities of, 185; limit-point of zeros of, 188; singularities of, 157, 294 et seq. Angle, postulate for, 10 Anliarmonic ratio, see Double ratio Anticycllc pairs of points, 33 Appell, 327 Argand, 17 Aritlunetic expression may define distinct analytic functions, 160; Tannery's example, 160, 161 Association of the terms of a series, loi ; when permissible, 105 Automorpbic function, 290 AzlB of imaginary numbers, 14; real numbers, 12 Baker, 327 Banks of a branch-cut, 274 Bilinear transformation, 28; converts circles into circles, 31 ; converts coaxial systems of circles into like systems, 32; converts inverse points into inverse points, 31 ; equivalent to two absolute in- versions, 42 et seq. ; fixed points INDEX. 329 of a, 58; of a plane into itself, 57 Btnomlal theorem, for general exponent, 169; for negative integer, 111 {ex.), IS* BAcber, 181 Branch, 274 Branch, chief, of log^c, 49; of sin'^j/, 173; oftan-V, 175 Branch-cut, 277; right and left banks of a, 274 Branch-point, 275 Bridge for a Riemann surface, 276 Brill, 123, 327 Borkhamt, 327 Byerly, 228 Cardloid, 280 Casey, 66 Casslnian, 278; different cases of, 279; square root of x used in study of, 278 Canchy, 129, 306-315 Canchy's inequality for Px, 129; for the Laurent series, 13 1 Canchy's integral, 222; monogenic func- tion, 306; theorem, 218, 324; theory of residues, 224 et seq. Cell, defined, 244, 251 Centroid, of a system of points, 25 Chain of elements, 154; standard, 156, '57 Change of a stroke, 19; of /x round a circuit, 193; of log Gac round a circuit, 193; of variable in in- tegration, 217 Christoffel, 321 Chrystal, 160, 237 Circle about a pair of points, 30; of con- vergence, 125 Circuit, contraction to zero of a, 222; defined, 189; integration round a, 217 et seq.; positive and negative description of, 192 Circular functions, 53, 170, 171; inverse, 172 et seq.; mapping virith, 175 et seq. Closed system of numbers, 1 1 Coaxial circles, defined, .31; various systems of, 31; see Elliptic, Hy- perbolic, Parabolic coaxial sys- tems Complex numbers, defined, 1 1 ; elementary operations on, 18 et seq.; form a closed system, 1 1 ; representation of, by points, 17 Complex power series, see Power series Conditional convergence, see Series Confocal ellipses and hyperbolas, 1 76 Congruence, relation of, for ordinal numbers, 5 Congruent in reference to two periods, 243 Conic sections, special properties es- tablished by mapping, 277, 278 Conjugate potentials, 316 Conjugate straight lines as to circle, 34 continuation of Px, 151 ; theorems re- lating to the, 151 et seq. Continuity and uniform convergence, 1 17; of a rational integral function, 85, 86; of/?, 73; of /I, when uniform, 81; of /(|, 1;), 79; of /(i, 1)1 when uniform, 82 ; of /x, 85 Continuous functions yj, /(J, 17) attain their upper and lower limits, 74, 79 Continuum of real numbers, 5 Convergence, circle of, 1 25 ; of a product, see Product; of a sequence, 72; of a series, see Series; radius of, 126; to a limit, when uniform, 82, 116, 312 Correspondence between two variables, 27 Cubic equation, solution of, 65 Cyclic tetrad of points, 32 Decimal, infinite, 7; as a sequence, 72; power series a generalization of, 123 Decimal sequence, 6 ; idea of limit con- nected with, 7 Decimal system of ordinal numbers, 7 Definite integral, see Integral Deformation of circuit, 193, 222, 304; of path, 189 Demoivre's theorem, 2 1 Derlvate of /x, monogenic function has 330 INDEX. unique, 307; necessary and suf- ficient condition for uniqueness of, 308 ; not o or 00 , isogonality when, 89; notation for, 89, 146; of Px, 146; of Px has same domain as Px, 146; of JT", 88 Differential, definition and notation for, 89; quotient, 89 Differentiation of an infinite series term by term, 147, 166 Discontinuity of a function of |, 73, 74; of a non-uniformly convergent series, 117 Distance, defined, 9 Dixon, 257 Domain of a power series, 146 Double integral used by Cauchy and Riemann, 314, 324 Double limit, see Limit Double points of a bilinear transformation, see Fixed points; of a curve F(^x, y)—o, 298, 299; of an in- volution, 63 Double ratio of four points, 35 ; is six- valued, 35; special cases of, 36; when real, 35 Doable series, see Series Doable spiral, 60; as a line of flow (level), 60 Element of an analytic function, any, can be used to generate the function, 156; primary, of an analytic function, 154 ; see Analytic func- tion, Cbaln Elliptic function, defined, 250; derivate of, is an elliptic function, 255; expressed in terms of ^a, p«, 262, 263 ; expressed in terms of au, 265, 268; integration of, 264; numerical processes, 274; order of, 257; partial fractions formula for, 263; product formula for, 265; theorems on the numbers and sums of zeros, infinities, and points at which the function has an assigned value, 257, 266 Elliptic functions, comparison of, 158 et seq. Elliptic substitution, see Substitation; system of coaxial circles, 31 End-values of a path of integration, 209 ; which belong to the domains of different elements, 214, 215 Enneper, 327 Equably convergent (equiconvergent) series, 116 Equlangnilar spiral, as a line of flow (level), 52; connexion with loga- rithm, 52 ; construction for, 47 Eqoipotential problem, 317 Equivalent points with respect to a group of operations, 290 Essential singular points, defined for one- valued analytic function, 182; at jr = oo for transcendental integral function, 184; each limit-point of zeros (or singular points) is, 185, 188; isolated, behaviour near, 186; no value at (unless path is specified), 187; oi fx is essential of \\fx, 187 Enler's constant, 196, 206; gamma func- tion, 207 Exponential function, addition theorem, 54, 162 ; addition theorem, genera- lized form of, 164; defined as inverse of logarithm, 5 2 ; geometric definition of, 47 ; group of opera- tions associated vrith, 289; has no zeros, 163, t93; has 00 for its sole singularity, 163; period of, 52 Factor-formula for sinjr, 199; for cosjr, 204 ; for au, 246 Factorlelle of x, defined, 206 Fine, 10 Fixed points of a bilinear transformation, 58; when coincident, 60; when distinct, 58, 59 Forsytb, 327 Four points, canonic system, 64 ; special configurations for, 35, 36 Fourier's integral, 228 Fourier series, derived from Laurent setles, 235; discussed, 235 et seq. Fractions, how defined independently of magnitude, 2 Function of {, continuous, 73; general INDEX. 331 definition, 72, 73; uniformly con- tinuous, 80; when continuous, attains its upper and lower limits, 72; when one-valued, 72 Function of i, 17, 77 ; admits upper and lower limits, 77 ; when continuous, attains its upper and lower limits,79 Function of x, can be defined gradually, 84 ; limit of, at a point, see Limit ; Riemann's definition of, 310; when analytic, see Analytic function; when analytic about or regular at a point, 181; when analytic over a region, 181; when continuous, 85; when holomorphic, 181 ; when localized, 189; when monogenic, 310; with no singular point except <» and no zeros, expression for, '93 Function of x, circular, see Circular Function <*, see Exponential function Function Fc x, 207 ; properties of, 207, 208 Function Tx, see Oanuna function Function Gx, see Transcendental and Rational integral function Function log Gx, see Logaritlim Function IIx (Gauss), 207 Function ^«, addition theorem, 260; algebraic relation between, and P'», 259 ; defined, 247 ; double periodicity of, 253; formula con- necting tru with, 268; group of operations associated with, 290; in terms of theta-functions, 270; invariants g^, g^, 259; relations connecting, with au, fa, 248; series in powers of a for, 249; zeros of derivate, 252 Function ^/jf^«-«;^ is one-valued, 269; satisfies a differential equation of the first order, 270 Function au, infinite product for, 246; in terms of theta-functions, 270; is transcendental integral of grade two, 246; quasi-periodicity of, 254; see Function jJa Function fa, addition theorem, 263, 269; defined, 247 ; quasi-periodicity of, 253 ; series in powers of, 253 Functional equation, 238, 271 Fundamental region, for a. group of bilinear transformations, 291 ; for a special group of bilinear trans- formations, 289; for sin~'x, 173; for singly (doubly) periodic func- tion, 291; for tan~'jr, 175 Fundamental theorem of algebra, 90 et seq., 193; associated theorem, 90 Gamma function, 207 ; reciprocal of, 207 ; some properties of, 208 GausB, 1 7 ; discovered primary factors for Fc X, 207 ; the function Iljr, 207 Goursat, 327 Grade (or class) of Gx, defined, 99; of sin a is one, 200; of au is two, 246 Group, cyclic, 289 Groups of operations for x", e', ^a, sin x, 290, 292; fundamental regions connected with these, 289 Hardcastie, 327 Harmonic pairs of points, 24, 33 Helicoid, 317 Hermite, 208 Hobson, 327 Holomorphic function, 181 Hyperbolic functions, 171; substitution, see Substitution ; system of coaxial circles, 31 Imaginary numbers, as operators, 13; axis of, 13; defined, 1 1 Increment of a stroke, defined, 19 Infinite decimal, see Decimal; product, see Product; series, see Series Infinitely slow convergence, of series, "5 Infinity, algebraic, defined, 301 ; circular points at, 34 (foot-note); of a function, 94, 182; of a rational algebraic function, 94; order of, 94; point at, 27 ; see Non-essential singular point Integral, definite, and theory of residues, 224etseq. ; definition of logarithm by, 48 ; value of, depends in general 332 INDEX. on path of integration, 210; value of, when unaffected by deformation of path, 217, 218 Integration, Cauchy's theorem on, 218; change of variable, 217; effect of variation of path, 217; end- values of path of, 209; of elliptic func- tion, 264; path of, 209, 210; round a circuit, 217 et seq.; round a parallelogram of periods, 256; special theorems on, 216; two definitions of and how these are reconciled, 211 et seq. Integration term by term of an in- finite series of analytic functions, 214 Interval, uniform (non-uniform) con- vergence in an, 115; when open, closed, partially open, 113 Invariant properties of a configuration of points, 32 et seq. Invariants g^, g^ of binary quartic, 260 ; of P«, 259 Inverse points, circle about a pair of, 30 ; defined, 30, 38, 42 Inversion, absolute, defined, 38; geome- try of, 38 et seq. Involution, double points, 63; construc- tion for partner of oo , 6y^ points in, 62 ; two points, when partners in, 63 Irrational numbers, introduction of, 4; with rational, form a continuum, 5 iBOgonality, property of, 37, 89; fails at exceptional points, 38; for bi- linear transformation, 51 ; for logarithm, 5 1 ; geometric appli- cation, 279 Isolated points, 151 Isolated singularity, non-essential singu- larity must be an, 185; behaviour near an, 234 JacoM, 257 Jordan, 242, 327 Klein, 327 Lacour, 327 Laplace's equation, 315 --taurent series, 125, 159, 271; annular region of convergence, 131, 234; Cauchy's inequality for a, 131 ; discussed, 232 et seq.; Fourier series derived from, 235; frac- tional, 159 Laurent's theorem on the expansion of a fiinction in a Laurent series, 230 ; proved for the general case, 232; special example of, 230; use of, in discussing singularities, 233 Legendre's relation, Jacobi's form of, 257 (foot-note) ; Weierstrass's form of, 257, 258 Lenmiscate, 279 Lima9on, 279 Limit, concept of, 67, 87, 88, 96; dis- tinct from 'value at,' 68, 73, 87, 93, 116, 233, 234; idea of, con- nected with infinite decimal, 7 ; uniform convergence to a, 82; when infinite, 69 Limit, double, examples of a, 114, 147, 166, 172 Limit, lower, see Limit, upper Limit of a sequence, necessary and suffi- cient condition for, 71 ; notation for, 68; whose numbers increase (decrease) constantly always exists, 69 Limit of fx when x = a defined, 87; when = 00, 88; when infinite, 88 Limit-point, defined, 133; distinction between isolated point and, 151 ; of non-essential singularities of an analytic function, 185 ; of zeros of an analytic function, 188; system of zeros of Px with o as, 133 Limit, upper, 68; attained by a con- tinuous function f%, 74 ; attained by a continuous function /(J, 1;), 79, 80; attained by a continuous function fx, 93; of a sequence, 68, 70 Lines of flow and level, 55; for elliptic bilinear transformation, 59; for hyperbolic bilinear transformation, 59; for loxodromic bilinear trans- INDEX. 333 formation, 60; for parabolic bi- linear transformation, 61 Llouvllle, theorem of, 159; application to elliptic functions, 2J5 Localized function, 189; change of, after description of a circuit, 1 90 et seq. Logailtlun, analytic theory of the, 164 et seq.; as a limit, 53, 169; as a limit, geometric verification, 53; computation of, 50, 167; effect of circuit on, 193; of Cx, igj, 193 Logaritlun of f, defined geometrically, 47 Logailtluii of X, bands for, 29 1 ; chief, 49; chief, notation for, 49; con- nects stretch and turn, 50; con- nexion with Mercator's projection, 54; defined by an integral, 48; infinitely many values, 49, 164 ; is an analytic function, 164; map- ping with, 165 ; Riemann surface for, 291; separation of, into branches, 291 ; special series asso- ciated with, 165 et seq. Logaritlimic series, 165 ; behaviour of, on its circle of convergence, 117; is conditionally convergent when jr = i, 103; is uniformly conver- gent in the closed interval (-1,1); rearrangements of terms of, dis- cussed, 104, 122 IiOgailtlunle singularity, 177 Loop, defined, 303 ; effect of the descrip- tion of a, 303 Loxodromic substitution, see SubsUtn- Uon Haclaiiiln'8 theorem, 146 Many- valued function of f, 72; of x, 84 ; of J, selection of a branch of a, 177 Happing applied to the theory of plane curves, 278 et seq. ; by means of log X, 50 ; by means of the circular functions, 176, 177; of a straight line on a polygon, 321; of one plane on another, 27; of unit circle on a plane, 167 Mercator's projection, connexion with log X, 54 Hlttag-Leffler's theorem, reference to, 206 Hoik, 327 Monogenic function, defined, 310; difii- culties underlying Cauchy's defini- tion, 311 Motion of » plane under bilinear trans- formation, 58 et seq. Miiller, 327 Multiple connexion of a Riemann surface, 288 Napierian motion, defined, 56 ; for bi- linear transformation, 57, j8 Natural boundary of an analytic function, 160 Neighbourhood of a point, defined, 78 Network of periods, 243 Neumann, 327 Newman, 207 Newton, 123 Nodal case for algebraic function, 283, 299 Non-esaentlal singular point, 181; ana- lytic fiinction (one-valued) with, at 00 and no essential singularity is rational integral and conversely, 184 et seq. ; behaviour at a, 186 ; order of multiplicity of a, 186 ; value at a, is 00 , 187 Non-essential singular points, are isolated in any finite region, 185 ; no function with no essential and in- finitely many, 185 ; see Besldues, Singular points Non-monogenic function, 310 Non-uniform continuity, see Continuity Nother, 123, 327 Obstacles to an analytic function, 157 Operations, elementary, applied to strokes, 18, 19, 20 Order of an elliptic function, 257 ; of a rational algebraic function, 94 ; of infinity of a rational algebraic function, 95 ; (or order of multi- plicity) of an infinity of a one- valued analytic function, 181 Ordinal number, discussion of, i et seq. Orthogonal systems of circles, see Co- 334 INDEX. axial drcles ; of curves arising from Ry=x, 279; of spirals, 55 OscUlaUon of an infinite series, 102 ; of ji, 80 Osgood, 118 Parabolic substitution, see Snbstitation ; system of coaxial circles, 31, 52 FaraUelogram law for addition of strokes, 19 Partial fractions, formulae in, for colx, etc. , 203 et seq. ; formula in, for elliptic function, 263 ; for rational algebraic function, 94 ; Mittag- LefBer's theorem, 206 Partition-function, 237 ; theta-function derived from, 240 Partners in involution, 62 Path of integration, 209 ; for log x, 49 ; of X when am ' = const., 29 ; of X when = const. , ^o Paths, corresponding, in x-, ^-planes, 277 Period of an elliptic substitution, 6 1 ; of cos x, 172; of exponential function, 52; ofsinjc, 172; oftanjr, 172 Periodic function, doubly, see Elliptic function, Function ^u ; singly, if, 52; sinj;, cosjr, tanjr, 172 Periodic n-ad, 66 (foot-note) Picard, 326, 327 Plcard's theorem on the behaviour of a function near an essential singu- larity, 187 Place on a Riemann surface, 277 Point at 00, 27; postulate with respect to correspondence between number and, 9 ; see Argand Pole, 181 ; see Mon-esaential singolar point Potential, a physical concept, 315; in a plane, discussion of, 315 et seq. ; logarithmic, 317 Power a', two meanings of, 168 Power series, defined, 119, 123; gene- ralization of infinite decimal, 1 23 ; in i/jr, 'Jx, etc., 129 Power series, complex, about a point, 124; behaviour near jc=o, 132; Cauchy's theorem on coefficients of, 129; criteria of identity of two, 132, 150; derivates of, 146; domain of, 146; formula for co- efficients of reverted, 143 ; frac- tional, 125 ; Maclaurin's theorem, 146; product of two, 139; re- version of a, 142, 165 ; Taylor's theorem, 144 ; theorems on the continuation of, 149 ; zeros of, are isolated, 149; see Logaritlunic series Power series, real, absolute convergence in an interval, 1 19 ; logarithmic, 121, 122; properties of, 120 et seq. ; uniform convergence in an interval, 1 20 ; see Loerarlttunic series Power series, series of, Cauchy's theorem on, in; differentiation term by term, 147; integration term by term, 140; sufficient condition for expressibility as Px, 140 ; Weier- strass's theorem on, 134, 137, 139 Primary element of an analytic fiinction, 154 Primary factors for cos x, 204 ; for Fc x, 207; for Gx, 199; for sigma- function, 246 ; for sin x, 200 ; for sin irx - sin iro, 204 ; for theta- functions, 272; Weierstrass's theo- rem on, 197 PrlmltiTe pair of periods, 244, 271 Principle of permanent forms, 168 PringBheim, 313 Product, absolute value of, 21 ; ampli- tude of, 21 ; of two power series, 139; of two strokes, 139 Product, infinite, as a limit, 194 ; con- vergence of, special case, 195 ; convergent, when absolutely, 195 convergent, when unconditionally, 195 ProJectiTe geometry, transition from, to geometry of inversion, 34 Property of isogonality, see Isog^onality Pulseux series, 297 Quasi-perlodicity of sa, 254 ; of f», 253 INDEX. 335 Radius of convergence, 125 Batlo, double or anharmonic, see Double Batlo of division of a stroke, 14 BaUo of strokes, its absolute value, 20 ; amplitude, 20; as an operator, 20; geometric construction, 21 Batlonal algebraic function, 93 ; con- verted into partial (iractions, 94 ; infinities of, 94 ; value when x=a>, 93; zeros of, 95 Batlonal integral function, 84 ; approxi- mate values when x is small or great, 85, 86; continuity of, 84; fundamental theorem for, 90, 193; Taylor's theorem for, 89 national numbers, defined as ordinal numbers, 2 Real function, see Function of £ Real numbers, axis of, 12 Region, closed, open, partially open, 124 Regular at a point, 181 Remainder residue of an infinite series, 99 Residues applied to elliptic function, 2j6 et seq. ; defined, 220 ; general applications of theory of, 222, 223; theorems on, 221 Reversion of power series, 142, 165 Riemann'B definition of function, 310, 312 ; proof of Cauchy's theorem, 314 Riemann surface, branch-cuts on, 288; circuit on, 287 ; covertical places on, 276 ; for algebraic function, 304 ; for logarithm, 291 ; for special functions, 177, 276 et seq.; rational functions on a, 282 Schwaiz, 321 Sequence, defined, 97 Series, double, obtained from single series, 107 ; converted into single, 1 10 ; summation of by rows, columns, diagonals, in Series, infinite, association of terms in, loi ; convergence of, 98 ; definition of, 97 ; differentiation and integra- tion of, 166, 214; oscillating, 104; remainder of, 99 ; simple tests of convergence, 99 et seq. ; when convergence is absolute, 104 ; when convergence is conditional (or unconditional), 107 ; when convergence is infinitely slow, 115; when convergence is uni- form, 113, 128, 129; see Double series, Fourier series, Laurent series, Logailtlunic series. Power series, Puiseuz series Series, uniformly convergent, 113; con- dition for, 115; defines a con- tinuous function, 117 Sieve of Eratosthenes, 107 Sign, minus, as an operator, 1 2 Simple connexion of a region, 189 ; of a Riemann surface, 286 Singularity, 181; see Essential singular point, Logailtlunic singularity, Non-essential singular point, Singular point Sing^ilaT point, acts as an obstacle, 157, 158; at least one, must exist, 183; at least one, on a circle of con- vergence, 178 et seq.; Laurent series at an isolated, 232 ; precise definition, 181 ; preliminary dis- cussion, 157 Singular points, character of « function determined by the, 183 et seq. ; distribution of, theorems on the, 185 et seq. ; essential, see Essential singular point; existence of, in all cases, 183 ; natural boundary, 160 ; non-essential, see Non-es- sential singular point; removal of, by subtraction of suitable functions, 182, 259; Picard's theorem on, 187 ; Weierstrass's theorem on, 187 Spiral, see Double spiral, Equiangular spiral Square root of x, analytic and geometric behaviour at x=o, 275 ; branches and branch-points, 275 ; chief, and notation for chief, 4 ; effect on, due to a circuit, 273 ; Riemann surface for, 276 Stahl, 327 Btolz, 312, 327 Stretch connected with turn by log 4;, 336 INDEX. 50 ; relation of, to the complex number, 13 Stroke, defined, 12, 15 ; logarithm of a, 23 ; «th power of a, 23 ; »th root of a, 23 StTOkeB, elementary operations on, 6, 13, IS, 17, 20, 21 Substitutioii, bilinear, elliptic, 59, 61 ; hyperbolic, 59 ; loxodromic, 60 ; of period two, 61 ; three, 65 ; parabolic, 61 Subtraction of strokes, 16, 61 Surface, multiply connected, 189; simply connected, 189 System of points, limit-points and isolated points in an infinite, see Isolated points. Limit-point Table of double entry, 108 Tannery, 160, 161, 327 Taylor's theorem, Cauchy's integral form for, 314; for rational integral function, 89 ; for power series, 1 44 Tbeta-fanctions, defined, 240 et seq. ; primary factors for, 272 ; §?« in terms of, 270; 258; primary factor, 197; theorem on behaviour of a one- valued analytic function near an essential singular point, 187, 234; theorem on power series, 134 et seq. Wessel, 17 Zero, defined, 94 ; «* has no, 163 Zeros, factor-theorem of Weierstrass, 197 et seq. ; of a function Gx are isolated, 189; of a one- valued analytic function, 188 ; of a power series, 132 ; of a power series are isolated, 149; of a rational alge- braic function, 95 ; of a rational integral function, 90 et seq. ; possible distributions of, for a power series, 133; theorem on sum of orders of, for a rational algebraic function, 95 CAMBRIDGE: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.