I 14 COKNELl UNIVERSITY LIBRARIES MethemetJcs Library Whi^e Hail CORNELL UNIVERSITY LIBRARY 924 059 413 520 DATE Z DUE i 4 {\[\ 1 01995 UVJL- J WOV ^ 2004 IHAR (] 7 7111 ir U "111 ^1 ^ /lllf.t ,i GAVLOnO PniNTEDIN U.S A. Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059413520 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 500 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. a^%^ ^, ip^- LECTURES ON THE THEORY OF ELLIPTIC FUNCTIONS BY HARKIS HANCOCK Ph.D. (BBBiaif), Db. Sc. (Paris), Professor op Mathematics IN THE UNIVEBSITY OF ClBOINlfATI Volume I ANALYSIS FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS London: CHAPMAJN & HALL, Limited 1910 Copyright, 1910, BT HABRIS HANCOCK Stmbope iprees r. H. GILSON COUPANV BOSTON, U.S. A GENERAL PREFACE In the publication of these lectures, it is proposed to present the Theory of Elliptic Functions in three volumes, which are to include in general the following three phases of the subject: I. Analysis; II. Applications to Problems in Geometry and Mechanics; III. General Arithmetic and Higher Algebra. In Volume I an attempt is made to give the essential principles of the theory. The elliptic functions considered as the inverse of the elliptic integrals have their origin in the immortal works of Abel and Jacobi. I have wished to treat from a philosophic, as well as from a formal stand- point, the existence, and as far as possible, the ultimate meaning of the functions introduced by these mathematicians, to discuss the theories which originated with them, to follow their development, and to extend as far as possible the principles which they established. In this develop- ment great assistance has been rendered by the works of Hermite, who contributed so much not only to the theory of elliptic functions but also to almost every form of mathematical thought. The theory of Weierstrass is studied side by side with the older theory, and the beautiful formulas which we owe to him are contrasted with the corresponding formulas of the earlier writers. Riemann introduced certain surfaces upon which he represented algebraic integrals, and by thus expressing his conceptions of analytic functions he revealed a clearer insight into their meaning. Instead of generalizing either the theory of Jacobi or that of Weierstrass so as to embrace the whole subject, it is thought better to make these theories specializations of a more general theory. This general theory is treated by means of the Riemann surface, which at the same time shows the intimate relation between the two theories just mentioned. In Volume II a treatment of elliptic integrals is given. Here much attention is paid to the work of Legendre, whom we may rightly regard as the founder of the elliptic functions, for upon his investigations were established the theories of Abel and Jacobi, and indeed, in the very form given by Legendre. Abe! in a published letter to Legendre wrote: "Si je suis assez heureux pour faire quelques decouvertes, je les attribuerai k vous plutdt qu' k moi "; and Jacobi wrote as follows to the genial Legendre: "Quelle satisfaction pour moi que I'homme que j'admirais tant en iv THEORY OF ELLIPTIC FUXCTIONS. d6vorant ses ecrits a bien voulu accueillir mes travaux avec une bont6 si rare et si precieuse! Tout en manquant de paroles qui soient de dignes interpretes de mes sentiments, je n'y saurai reprondre qu'en redoublant mes efforts a pousser plus loin les belles theories dont vous etes le createur." True Fagnano, Euler, Landen, Lagrange, and possibly others had dis- covered certain theorems which proved fundamental in the future develop- ment of the elliptic functions; but by the patient devotion of a long life to these functions, Legendre systematized an independent theory in that he reduced all integrals which contain no other irrationality than the square root of an expression of degree not higher than the fourth into three canonical forms of essentially different character. Thus he was enabled to discover many of their most important properties and to overcome great difficulties, which with the means then at hand appear almost insurmount- able. Methods were devised which furnished immediate results and which, extended by subsequent investigations, enriched the science of mathematics and the fields of knowledge. In this direction the great English mathematician Cayley has done much work, and to him a con- siderable portion of this volume is due. The admirable work of Greenhill has also been of great assistance. Much space is given in Volume II to the applications of the theory. These applications are usually in the form of integrals and the results required are real quantities, and for the most part the variables must be taken real. Thus the complex variable of Volume I must be limited to some extent in the second volume. The problems selected serve to illustrate the different phases treated in the previous theory; sometimes preference, as the occasion warrants, is given to Legendre's formulas, sometimes to those of Weierstrass. While the most of these problems are taken from geometry, physics, and mechanics, there are some which have to do with algebra and the theory of numbers. All true students of applied mathematics, engineers, and physicists should have some knowledge of elliptic functions; at the same time it must be recognized that one cannot do all things, and it is not expected that such students should be as well versed in the theoretical side of this subject as are pure mathematicians. For this reason Volume II has been so pre- pared that without dwelling too long up)on the intrinsic meaning of the subject, one may obtain a practical idea of the formulas. Much of the theory of Volume I is therefore not presupposed, and many of the results that have hitherto been derived are again deduced in Volume II by other methods, which, without emphasizing the theoretical significance, are often more direct. This is especially true of the addition-theorems. A table of elliptic integrals of the first and second kinds will be found at the end of this volume, which may consequently, for the reasons stated, be regarded as an advanced calculus. Volume III will be of interest especially to the lovers of pure mathe- GENERAL PEEFACE. V matics. In this volume the theory Vjecomes more abstract. Many problems of higher algebra occur which lie within the realms of general arithmetic. This includes the theories of complex multiplication; of the division and transformation of the elliptic fimctions; a study of the modular equations and the solution of the algebraic equation of the fifth degree, etc. The discoveries of Kronecker in the theory of the complex multiplica- tion not only prove the theorems left in fragmentary form by Abel and give a clear insight into them, but they show the close relationship of this theory with algebra and the theory of numbers. The problem of division resolves itself into the solution of algebraic equations, and the introduc- tion of the roots of these equations Lato the ordinary realm of rationality forms a "realm of algebraic numbers "; the same is true of the modular equations. Kronecker, Dedekind, Hermite, Weber, Joubert, Brioschi. and other mathematicians have develop)ed these lines of thought into an independent branch of mathematics which in its further growth is sus- ceptible of extension in many directions, notably to the treatment of the Abelian transcendents on the one hand and of the modular systems on the other. Jacobi in a letter to Crelle wrote: "You see the theory [of elUptic func- tions] is a vast subject of research, which in the course of its development embraces almost all algebra, the theory of definite integrals, and the science of numbers." It is also true that when a discovery is made in any one of these fields the domains of the others are also thereby extended. INTRODUCTION TO VOLUME I Every one-valued analytic function which has an algebraic addition- theorem is an elliptic function or a limiting case of one. The existence, formation, and treatment of the elliptic functions as thus defined are given in Chapters I-VII of the present volume. An algebraic equation connecting the function and its derivative, which we have called the eliminant equation, is emphasized. This differential equation due to Meray is first used as a latent test to ascertain whether or not a function in reality has an algebraic addition-theorem, and, sec- ondly, as shown by Hermite, its integrals when restricted to one-valued functions are one or the other of the three classes of functions: rational functions, simply periodic functions, or doubly periodic functions. We regard the first two types as limiting cases of the third, the three types forming the general subject of elliptic functions. All three types of functions are shown to have algebraic addition-theorems, and conse- quently the existence of the eliminant equation is found to be coextensive with that of the elliptic functions. In Chapter I some preliminary notions are given. In particular it is found that the rational and the trigonometric, and later, in Chapter V, that the doubly periodic functions may be expressed in terms of simple elements, and it is seen that all three forms of expression are the same; a treatment is given of infinite products and also of the primary factors of an integral transcendental function; analytic functions are defined. The properties of functions which have algebraic addition-theorems are considered in Chapter II, and it is shown that these properties exist for the wliole region in which the function has a meaning. After establishing the existence of the simply and doubly periodic func- tions in Chapters III and IV and after studying the nature of the periods, we proceed in Chapter V to the actual formation of the doubly periodic functions. It is shown that the doubly periodic functions may be repre- sented as the quotients of two Hermitean "intermediary functions," of which the Jacobi Theta-functions are special cases. The derivation of such functions with their characteristic properties is then treated. Further, by a method also due to Hermite, it is shown that the most general elliptic functions may be expressed in terms of a simple func- tional element, which is in fact the simplest intermediary function. INTRODUCTION. vii After proving the theorem that the most general elliptic function may be expressed algebraically through an elliptic function of the second order (the simplest kind of an elliptic function), a form of eliminant equa- tion is derived in which the derivative appears only to the second power. The functions connected with this equation are treated by means of the Riemann surface, which is given at length in Chapter VI, where also the " one-valued functions of position" are introduced. The integrals defining the circular functions contain radicals under which the variable appears to the second degree; while the variable appears to the third or fourth degree under the radicals in the elliptic integrals. It is therefore natural to consider the elliptic functions as the general- ization of the circular functions, just as the latter functions may be regarded as limiting cases of the former. The methods followed by Legendre, Abel and Jacobi seem the natural and inevitable methods of presenting these functions. History also gives them precedence. Weier- strass built his theory on the foundation already established by these earlier mathematicians, and it is impossible to realize the real signifi- cance of Weierstrass's functions without a prior knowledge of the older theory. Riemann's theory forms an important extension of the purely analytic treatment of Legendre and Jacobi as well as of the Weierstrass- ian theory. The characteristics of Riemann's theory lie on the one hand in the simple application of geometrical representations such as the two- leaved surface and its conformal representation upon the period paral- lelogram, and on the other hand it shows how the formulas are founded synthetically on the basis of the fundamental properties of the functions and integrals; and thus a deeper and a clearer insight into their true nature is gained. Mr. Poincar6 has said, " By the instrument of Riemann we see at a glance the general aspects of things — like a traveler who is examining from the peak of a mountain the topography of the plain which he is going to visit and is finding his bearings. By the instrument of Weier- strass analysis will in due course throw light into every corner and make absolute clearness shine forth." The universal laws of Riemann are particularized in the one direction of the Legendre-Jacobi theory and in the other direction of the Weier- strassian theory, the two theories being interconnected. Accordingly in the present volume the Legendre-Jacobi functions are first developed and often side by side with them the corresponding Weierstrassian functions. Owing to a theorem due to Liouville, we are able to show the real sig- nificance of the one-valued functions of position on the Riemann surface, viz., they are the general elliptic functions. These one-valued functions form a "class of algebraic functions" or "a closed realm of rationality," since the sum, difference, product, or quotient of any two such functions viii THEOEY OF ELLIPTIC FUNCTIONS. is a function of the realm. This realm of rationality is of the first order, corresponding to the connectivity of the associated Riemann surface, the realm of the ordinary rational functions being of the zero order. The former realm is derived from the latter by adjoining an algebraic quan- tity, which quantity defines the Riemann surface. This latter realm, which we call the "elliptic realm," includes as special cases the natural realm of all rational functions, and also the realm of the simply periodic functions. It therefore follows that all one-valued analytic functions which have algebraic addition-theorems form a closed realm; for every element (function) that belongs to this elliptic realm has an algebraic addition-theorem. Thus simultaneously with the development of the elliptic functions, the realm in which they enter is shown to be a closed one, and the reader gradually finds himself studying these fvmctions in their own realm. The elliptic or doubly periodic realm degenerates into a simply periodic realm when any two branch-points coincide, and it degenerates into the realm of rational functions when any two pairs of branch-points are equal. Thus again it is seen that the elliptic realm includes the three types of functions : rational functions, simply periodic functions, and doubly periodic functions. In Chapter VII the eliminant equation is further simplified and it is finally shown what form this equation must have that the upper limit of the resulting integral be a one-valued function of the integral. The problem of inversion is thereby solved in a remarkably simple manner. Thus by means of the Riemann surface, as it is possible in no other way, we may study the integral as a one-valued function of its upper limit and vice versa. In Chapter VIII the most general integral involving the square root of an expression of the third or fourth degree in the variable is made to depend upon three types of integrals. The normal forms of integrals are derived, and in particular Weierstrass's normal form, in a manner which illustrates the meaning of the invariants. The realms of rationality in which the normal forms of Legendre and of Weierstrass are defined are shown to be equivalent. The further contents of this volume are indicated through the headings of the different chapters. To be noted in particular is Chapter XIV, in which it is shown how the Weierstrassian functions are derived directly from those of Jacobi; in Chapter XX are given several different methods of representing any doubly periodic function; while in Chapter XXI we find a method of determining all analytic functions which have algebraic addition-theorems. A table of the most important formulas is found at the end of this volume. Professor Fuchs made the Riemann surfaces fundamental in his treat- ment of the Theory of Functions and the Differential Equations. It was INTEODUCTION. ix my privilege to hear him lecture on these subjects, and the present work, so far as it has to do with the Riemann surfaces, is founded upon the theory of that great mathematician. Although Professor Weierstrass lectured twenty-six times (from 1866 to 1885) in the University of Berlin on the theory of elliptic functions including courses of lectures on the application of these functions, no authoritative account of his work has been published, a quarter of a century having in the meanwhile elapsed. It is therefore difficult to say in that part of the theory which bears his name what is due to him, what to other mathematicians. I have derived considerable help in this respect from the lectures of Professor H. A. Schwarz, the results of which are published in his Formeln und Lehrsdtze zum Gebrauche der elliptischen Functionen. While it has not been my purpose to make the book encyclopedic, I have tried to give the principal authorities which have been of service in its preparation. The pedagogical side is insisted upon, as the work in the form of lectures is intended to be introductory to the theory in question. To Messrs. John Wiley and Sons, Scientific Publishers, and to the Stanhope Press, I am under great obligation for the courteous co-op>eration which has minimized my labor during the progress of printing. HARRIS HANCOCK. 2415 AtTBtTRN Ave., CiNcrNNATi, Ohio, Xov. 1, 1909. CONTENTS CHAPTER I PRELIMINARY NOTIONS ARTICLE PAGE 1. One-valued function. Regular function. Zeros . ... . . 1 2. Singular points. Pole or infinity 2 3. Essential singular points . . 2 4. Remark concerning the zeros and the poles . . . . .... 3 5. The point at infinity .... . . . . .4 6. Convergence of series . . . . . . 4 7. A one-valued function that is regular at all points of the plane is a constant . 5 8. The zeros and the poles of a one-valued function are necessarily isolated 6 Rational Functions 9-10. Methods (1) of decomposing a rational fraction into its partial frac- tions; (2) of representing such a fraction as a quotient of two products of linear factors . . ... 6 Principal Analsrtical Forms of Rational Functions 11. First form: Where the poles and the corresponding principal parts are brought into evidence .... ... 8 12. Second form: Where the zeros and the infinities are brought into evidence 9 Trigonometric Functions 13. Integral transcendental functions . . . . 10 14. Results established by Cauchy ... . 10 15. 16. The fundamental theorem of algebra extended by Weierstrass to these integral transcendents . . . 12 Infinite .Products 17, 18. Condition of convergence 14 19. The infinite products expressed through infinite series . . 16 20, 21. The sine-function ... . . 17 22. The cot-function . . . 19 23. Development in series . 20 The General Trigonometric Functions 24. The general trigonometric function expressed as a rational function of the cot-function . . . . 22 25. Decomposition into partial fractions . . 22 26. Expressed as a quotient of linear factors . 25 xii CONTENTS. Anal3rtic Functions ARTICLE PAGE 27. Domain of convergence. Analytic continuation 26 28. Example of a function which has no definite derivative .... 29 29. The function is one-valued in the plane where the canals have been drawn 29 30. The process may be reversed ... .... 30 31. Algebraic addition-theorems. Definition of an elliptic function ... 31 Examples . . . . . . . . 31 CHAPTER II FUNCTIONS WHICH HAVE ALGEBRAIC ADDITION-THEOREMS Characteristic properties of suck functions in general. The one-valued functions. Rational functions of the unrestricted argument u. Rational functions of the TTIU exponential function e " . 32. Examples of functions having algebraic addition-theorems . . 33 33. The addition-theorem stated ..... 34 34. Moray's eliminant eqvMion . . .... . 34 35. The existence of this equation is universal for the functions considered 36 36. A formula of fundamental importance for the addition-theorems . 37 37. The higher derivatives expressed as rational functions of the function and its first derivative .... . 39 37a, 38. Conditions that a function have a period . 41 39. A form of the general integral of Moray's equatioa . 43 The Discussion Restricted to One-valued Functions 40. All functions which have the property that 4>{v, -\- v) may be rationally ex- pressed through 4>{u), '(u), {v), 'j>'{v) are one-valued .... 44 41-45. All rational functions of the argument u; and all rational functions of the TJ7:i exponential function e "" have algebraic addition-theorems and are such that Hu + v)= FWu), '{u), 4>{v), 4>'{v)], where F denotes a rational function ... 45 46. Example showing that a function {u) may be such that 4>{u + v) is ration- ally expressible through ^(m), 4>'{u)., (v), 4>'{v) without having an alge- braic addition-theorem . . . ... 52 Continuation of the Domain in which the Analytic Function {u) has been Defined, with Proofs that its Characteristic Properties are Retained in the Extended Domain 47. Definition of the function in the neighborhood of the origin ... 54 48-50. The domain of ^(m) may be extended to all finite values of the argu- ment u, without the function 4>{u) ceasing to have the character of an integral or (fractional) rational function ... 55 51. The other characteristic properties of the function are also retained. The addition-theorem, while limited to a ring-formed region, exists for the whole region of convergence established for ^(w) . . . , 59 CONTENTS. xiii CHAPTER III THE EXISTENCE OF PERIODIC FUNCTIONS IN GENERAL Simply Periodic Functions. The Eliminant Equation. ARTICLE PAGE 52-54. When the point at infinity is an essential singularity, the function is periodic .... ... 62 55. Functions defined by their behavior at infinity . 67 The Period-Strips 56. The exponential function takes an arbitrary value once within its period- strip . . ... .... 67 57. The sine-function takes an arbitrary value twice within its period-strip . 69 58. It is sufficient to study a simply periodic function within the initial period- strip .70 59. General form of a simply periodic function . 70 60. Fourier Series ... 71 61-63. Study of the simply periodic functions which are indeterminate for no finite value of the argument; which are indeterminate at infinity; which are one-valued, and which within a period-strip take a prescribed value a finite number of times . 73 The Eliminant Equation 64. The nature of the integrals of this equation . . . 76 65. A further condition that an integral of the equation be simply periodic. Unicursal curves . 77 66. A final condition. . 78 Examples . . . 80 CHAPTER IV DOUBLY PERIODIC FUNCTIONS. THEIR EXISTENCE. THE PERIODS 67. 68. The existence of a second period 82 69. The distance between two period-points is finite 84 70. The quotient of the two periods cannot be real 85 71. Jacobi's proof ....... . . 86 72. 73. Other proofs ...... 87 74. Existence of two primitive periods . . 88 75. The study of a doubly periodic function may be restricted to a period- parallelogram . . 89 76. Congruent points. . , 90 77. All periods may be expressed through a pair of primitive periods 91 78. A theorem due to Jacobi 92 79. Pairs of primitive periods are not unique 80. Equivalent pairs of primitive periods. Transformations of the first degree 81. Preference given to certain pairs of primitive periods 93 95 96 82. Numerical values ■ ^7 xiv CONTENTS. CHAPTER V CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS Hermite's Intermediary Functions. The Eliminant Equation. ARTICLE PAGE 83. An integral transcendental function which is doubly periodic is a constant 99 84. Hermite's doubly periodic functions of the third sort ... . 100 85. Formation of the intermediary functions . . . 102 86. Condition of convergence . . 104 87. 88. The Chi-function. The Theta-functions . 106 89. Historical . . . 108 90. Intermediary functions of the fcth order 109 91. The zeros .... 110 92. Their number within a period-parallelogram 111 93. The zero of the Chi-function 113 The General Doubly Periodic Function Expressed through a Simple Transcendent 94. A doubly periodic function expressed as the quotient of two integral tran- scendental functions .115 95. Expressed through the Chi-function .... . 116 96-98. The Zeta-f unction. The doubly periodic function expressed through the Zeta-function . . 117 99, 100. The sum of the residues of a doubly periodic function is zero 121 101. Liouville's Theorem regarding the infinities ... . . 122 102. Two different methods for the treatment of doubly periodic functions . 123 The Eliminant Equation 103. The existence of the eliminant equation which is associated with every one- valued doubly periodic function . . . 123 104. A doubly periodic function takes any value as often as it becomes infinite of the first order within a period-parallelogram ... 123 105. Algebraic equation connecting two doubly periodic functions of different orders. Algebraic equation connecting a doubly periodic function and its derivative .... . . ... 125 106. The form of the eliminant equation . .... 126 107. The form of the resulting integral. The inverse sine-function. State- ment of the " probtew o/ inrersioTi " . . .... 126 CHAPTER VI THE RIEMANN SURFACE 108. Two-valued functions. Branch-points 128 109. The circle of convergence cannot contain a branch-point ... . 129 110-112. Analytic continuation along two curves that do not contain a branch- point .... . . 130 113. The case where a circuit is around a branch-point 133 114. The case where a circuit is around two branch-points 133 CONTENTS. XV ARTICLE PAGE 115. The case where the point at infinity is a branch-point 134 116. Canals. The Riemann Surface s^=ij(z) . . . . ... 134 The One-valued Functions of Position on the Riemann Surface 117. Every one-valued function of position on the Riemann Surface satisfies a quadratic equation, whose coefficients are rational functions . . . . 137 118. Its form is w = p + qs, where p and q are rational functions of z ... 138 The Zeros of the One-valued Functions of Position 119. The functions p and q may be infinite at a point which is a zero of w ... 139 120. The order of the zero, if at a branch-point . . 140 Integration 121. The path of integration may lie in both leaves .... 142 122. The boundaries of a portion of surface . 143 123. The residues ...... 144 124. The sum of the residues taken over the complete boundaries of a portion of surface . . 145 125. The values of the residues at branch-points 146 126. Application of Cauchy's Theorem . 148 127. The one-valued function of position takes every value in the Riemann Surface an equal number of times . 149 128. Simply connected surfaces ... 149 129. 130. The simple case where there are two branch-points. The modulus of periodicity. The sine-function . 150 Realms of Rationality 131. Definitions. Elements. The elliptic realm . 153 CHAPTER VII THE PROBLEM OF INVERSION 132. The problem stated . 155 133-135. The eliminant equation further restricted ... . 156 136. The elliptic integral of the first kind remains finite at a branch-point and also for the point at infinity . 158 137. The Riemann Surface in which the canals have been drawn 159 138. 139. The moduli of periodicity ... .... .160 140. The intermediary functions on the Riemann Surface . 162 141. The quotient of two such functions is a rational function . 164 142. The moduli of periodicity expressed through integrals . 164 143. The Riemann Surface having three finite branch-points . 165 144-146. The quotient of the two moduli of periodicity is not real . 165 147. The zeros of the intermediary functions . 169 148. The Theta-functions again introduced . . 171 149. The sum of two integrals whose upper limits are points one over the other on the Riemann Surface 172 xvi CONTENTS. ARTICLE PAGE 150, 151. The upper limit expressed as a quotient of Theta-functions . . . 172 152. Resume . .... . . . . 173 153. Remarks of Lejeune Dirichiet .... . . . 174 154. The eliminant equation reduced by another method . . . 175 155. A Theorem of Liouville . .... ... . 175 156. 157. A Theorem of Briot and Bouquet ... 176 158. Classification of one-valued functions that have algebraic addition-theorems . 178 159. The elliptic realm of rationality includes all one-valued functions which have algebraic addition-theorems 179 CHAPTER VIII ELLIPTIC INTEGRALS IN GENERAL The Three Kinds of Integrals. Normal Forms. 160-165. The reduction of the general integral to three typical forms. The parameter . .... . . 180 Legendre's Nonnal Forms 166-167. Legendre's integrals of the first, second and third kinds. The modulus 184 168. The name " elliptic integral" .... 187 169. The forms employed by Weierstrass 187 170. Other methods of deriving Legendre's normal forms 188 171. Discussion of the six anharmonic ratios which are connected with the modulus . . . 190 172. Other methods of deriving the forms employed by Weierstrass . . 191 173-174. A treatment of binary forms 191 175. The discriminant . .... . . 193 176-178. The two fundamental invariants of a binary form of the fourth degree 194 179. The Hessian covariant ... ... 196 180-181. The two fundamental covariants . . 197 182-183. Hermite's fundamental equation connecting the invariants and the covariants . . . .... 198 184. Weierstrass 's notation . ... 200 185. A substitution which changes Weierstrass's normal form into that of Legendre . 200 186. A certain absolute invariant . 201 187. Riemann's normal form 202 188. Further discussion of the elliptic realm of rationality 202 Examples 204 CHAPTER IX THE MODULI OF PERIODICITY FOR THE NORMAL FORMS OF LEGENDRE AND OF WEIERSTRASS 189. Construction of the Riemann Surface which is associated with the integral of Legendre's normal form . . 206 190-192. The moduli of periodicity. Definite values, in particular the branch- points, are taken as upper limits, and the values of the integrals are then expressed through the moduli of periodicity 207 193. The quantities K smd K' 212 CONTENTS. xvii ARTICLE PAGE 194-195. The moduli of periodicity for Weierstrass's normal form. The values of the integrals when branch-points are taken as the upper limits 213 196. The relations between the moduli of periodicity for the normal forms of Legendre and of Weierstrass ... 216 197-198. The conformal representations of the Riemann Surface and the period- parallelogram . . . 216 Examples ... ... 219 CHAPTER X THE Jacob: theta-functions 199-200. The Theta-functions expressed as infinite series in terms of the sine and cosine .... . . 220 201-202. The Theta-functions when multiples of K and iK' are added to the argument . . . 222 203. The zeros ... .224 204. The Theta-functions when the moduli are interchanged . . . 225 Expression of the Theta-Fuaictions in the Form of Infinite Products 205-206. Products of trinomials involving the sines and cosines and a constant quantity ..... . 226 207. Determination of the constant . . 228 The Small Theta-Functions 208. Expressed through infinite series . 229 209. Expressed through infinite products . . 230 210. Jacobi's /undomentai iAeorem for the addition of theta-functions . 231 211. The addition-theorems tabulated . . ..... 234 212. Reason given for not expressing the theta-functions through binomial products . . 237 Examples . . 238 CHAPTER XI THE FUNCTIONS sn n, en u, dn n 213-216. The elliptic functions expressed through quotients of the theta-func- tions. Analytic meaning of these functions . . . 239 217. The zeros of the elliptic functions ... 244 218. The argument increased by quarter and half periods. The periods of these functions . .... . 245 219. The derivatives ....... 246 220. Jacobi's imaginary transformation . ■ 247 221-222. The co-amplitude . 248 223. Linear transformations . ... 248 224. Imaginary argument ... 250 225. Quadratic transformations. Landen's transformations ... 250 226. Development in powers of u ■ 252 xviii CONTENTS. Development of the Elliptic Functions in Simple Series of Sines and Cosines ARTICLE PAGE 227. First method . . .254 228. Formulas employed by Hermite ... . 255 229-231. Second method, followed by Briot and Bouquet . 257 Examples . . . . . . 261 CHAPTER XII DOUBLY PERIODIC FUNCTIONS OF THE SECOND SORT 232. Explanation of the term . 264 233. Definitions . 264 234. Representation of such functions in terms of a fundamental function 265 235. Formation of the fundamental function 267 236. The exceptional case . 268 237. Different procedure for this case . 269 238. A preliminary derivation of the addition-theorems for the elliptic functions 273 239-240. Hermite's determination of the formulas employed by Jacobi relative to rotary motion . . . ... 275 Examples ... . . . . . . 281 CHAPTER XIII ELLIPTIC INTEGRALS OF THE SECOND KIND 241. Formation of an integral that is algebraically infinite at only one point 282 242. The addition of an integral of the first kind to an integral of the second kind . . 284 243. Formation of an expression consisting of two integrals of the second kind which is nowhere infinite . . . . 285 244. Notation of Legendre and of Jacobi . 286 245. A form employed by Hermite. The problem of inversion does vot lead to unique results ... . . . 286 246. The integral is a one-valued function of its argument u 286 247. The analytic expression of the integral. Its relation with the theta- function ... ... ... . 287 248. The moduli of periodicity . 289 249. Legendre's celebrated formula 290 250. Jacobi's zeta-function ... 291 251. The properties of the theta-function derived from those of the zeta-func- tion; an insight into the Weierstrassian functions . . 292 252. The zeta-function expressed in series .... 295 253. Thomae's notation . . . 295 254. The second logarithmic derivatives are rational functions of the upper limit 296 Examples . . . . . . . 296 CHAPTER XIV INTRODUCTION TO WEIERSTRASS'S THEORY 255. The former investigations relative to the Riemann Surface are applicable here ... ... .298 256. The transformation of Weierstrass's normal integral into that of Legendre gives at once the nature and the periods of Weierstrass's function . . 298 CONTENTS. xix ARTICLE PAGE 257. Derivation of the sigma-function from the theta-function . . . 299 258. Definition of Weierstrass's zeta-function. The moduli of periodicity 299 259. These moduli expressed through those of Jacobi; relations among the moduli of periodicity . . 302 260. Other sigma-functions introduced ... . 304 261-262. Sigma-functions expressed through theta-functions and Jacobi's elliptic functions expressed through sigma-functions . . . . 304 263. Jacobi's zeta-function expressed through Weierstrass's zeta-function 307 Examples ... . . 308 CHAPTER XV THE WEIERSTRASSIAN FUNCTIONS ^.^n, ^u, cru 264. The Pe-function 309 265. The existence of a function having the properties required of this function 311 266. Conditions of convergence . . ... .311 267. The infinite series through which the Pe-function is expressed, is absolutely convergent . . . . 313 268. The derivative of the Pe-function . . . . 314 269. The periods . 316 270. Another proof that this function is doubly periodic . 316 271. This function remains unchanged when a transition is made to an equiva- lent pair of primitive periods 317 The Sigma-Function 272. The expression through which the sigma-function is defined, is absolutely convergent; expressed as an infinite product 318 273. Historical. Mention is made in particular of the work of Eisenstein 320 274. The infinite product is absolutely convergent . 321 275-276. Other properties of the sigma-function . . 323 The fn-Function 277. Convergence of the series through which this function is defined . 324 278. The eliminant equation through which the Pe-function is defined . 325 279. The coefficients of the three functions defined above are integral functions of the invariants . ... 325 280. Recursion formula for the coefficients of the Pe-function. The three functions expressed as infinite series in powers of « . . . . 326 281. The Pe-function expressed as the quotient of two integral transcendental functions . . . . 328 282. Another expression of this function 329 283. The Pe-function when one of its periods is infinite . . 332 284-286. The Pe-function expressed through an infinite series of exponential functions ... 332 287-290. The zeta- and sigma-functions expressed through similar series . 336 291. The sigma-function expressed as an infinite product of trigonometric functions; the zeta- and Pe-f unctions expressed as infinite summations of such functions. The invariants . . . ... 341 292. Homogeneity .... • • • 343 293. Degeneracy ... 343 Examples ... .... 345 XX CONTENTS. CHAPTER XVI THE ADDITION-THEOREMS ARTICLE PAGE 294-295. The additioa-theorem for the theta-functions derived directly from the property of these intermediary functions . . . . 346 296. The elliptic functions being quotients of theta-functions have algebraic addition-theorems which may be derived from those of the intermediary functions 349 297. Addition-theorem for the integrals of the second kind . . 350 Addition-Theorems for the Weierstrassian Functions 298. A theorem of fundamental importance in Weierstrass's theory . . . . 351 299. Addition-theorems for the sigma-functions and the addition-theorem of. the Pe-function derived therefrom by differentiation .... 352 300-301. Other forms of the addition-theorem for the Pe-function . . 353 302. The sigma-f unction when the argument is doubled ■ 355 303. Historical. Euler and Lagrange . . . . 356 304-305. Euler's addition-theorem for the sine-function . . . 357 306-307. Euler's addition-theorem for the elhptic functions 360 308. The method of Darboux ... 362 309. Lagrange's direct method of finding the algebraic integral . 365 310. The algebraic integral in Weierstrass's theory follows directly from La- grange's method . . 366 311. Another derivation of the addition-theorem for the Pe-function . . 367 312. Another method of representing the elliptic functions when quarter and half periods are added to the argument . . 367 313. Duplication 368 314. Dimidiation .... . . 368 315-316. Weierstrassian functions when quarter periods are added to the argu- ment . . . . 369 Examples 370 CHAPTER XVII THE SIGMA-FUNCTIONS 317. It is required to determine directly the sigma-function when its character- istic properties are assigned ... 372 318. Introduction of a Fourier Series . . . . 373 319. The sigma-function completely determined ... . 374 320. Introduction of the other sigma-functions; their relation with the theta- functions ... . 377 321. The sigma-functions expressed through infinite products. The moduli of periodicity expressed through infinite series . 378 322. The sigma-function when the argument is doubled 380 323. The sigma-functions when the argument is increased by a period . 380 324. Relation among the sigma-functions ... . . 381 CONTENTS. XXI Difierential Equations which are satisfied by Sigma-Quotients ARTICLE PAGE 325. The differential equation is the same as that given by Legendre 381 326. The Jacobi-functions expressed through products of signaa-functions 382 327. Other relations existing among quotients of sigma-functions . . . 383 328. The square root of the differences of branch-points expressed through quo- tients of sigma-functions . . 384 329. These differences uniquely determined 385 330. The sigma-functions when the argument is increased by a quarter-period 386 331. The quotient of sigma-functions when the argument is increased by a period . . 386 332-333. Additional formulas expressing the Jacobi-functions through sigma- functions . . . 386 334. The sigma-functions for equivalent pairs of primitive periods . 388 Addition-Theorems for the Sigma-Functions 335. The addition-theorems derived and tabulated in the same manner as has already been done for the theta-functions 388 Expansion of the Sigma-Functions in Powers of the Argument 336. Derivation of the differential equation which serves as a recursion-formula for the expansion of the sigma-function . . . 391 Examples 394 CHAPTER XVIII THE THETA- AND SIGMA-FUNCTIONS WHEN SPECIAL VALUES ABE GIVEN TO THE ARGUMENT 337-338. The theta-functions when the argument is zero . 396 339-340. Two fundamental relations due to Jacobi . . . . .... 398 341. The moduli and the moduli of periodicity expressed through theta-functions 400 342. Other interesting formulas for the elliptic functions; expressions for the fourth roots of the moduli .... 401 343. Formulas which arise by equating different expressions through which the theta-functions are represented; the squares of theta-functions with zero arguments . . . 403 344. A formula due to Poisson . . 407 345. The equations connecting the theta- and sigma-functions; relations among the Jacobi and the Weierstrassian constants . . 408 346. The Weierstrassian moduli of periodicity expressed through theta-functions 409 347. The sigma-functions with quarter periods as arguments ... . 410 Examples 411 CHAPTER XIX ELLIPTIC INTEGRALS OF THE THIRD KIND 348. An integral which becomes logarithmically infinite at four points of the Rie- mann Surface .... . 412 349. Formation of an integral which has only two logarithmic infinities. The fundamental integral of the third kind . . 413 CONTENTS. ARTICLE 350. Three fundamental integrals so combined as to make an integral of the first kind . ... 414 351. Construction of the Riemann Surface upon which the fundamental integral is one-valued . . ■ • ..... 415 352. The elementary integral in Weierstrass's normal form . . 416 353. The values of the integrals when the canals are crossed . 417 354-355. The moduli of periodicity . . 417 356. The elementary integral of Weierstrass expressed through sigma-functions. Interchange of argument and parameter 419 357. Legendre's normal integral. The integral of Jacobi . . 420 358. Jacobi's integral expressed through theta-functions . 420 359. Definite values given to the argument . 420 360. Another derivation of the addition-theorem for the zeta-function . 422 361. Integrals with imaginary arguments . . . 422 362. The integral expressed through infinite serias . . . 423 The Omega-Function 363. Definition of the Omega-function. The integral of the third kind expressed through this function . '423 364. The Omega-function with imaginary argument . 424 365. The Jacobi integral expressed through sigma-functions . 425 366. Other forms of integrals of the third kind .... . . 425 Addition-Theorems for the Integrals of the Third Kind 367. The addition-theorem expressed as the logarithm of theta-functions . . 426 368. Other forms of this theorem . . 428 369. A theorem for the addition of the parameters ... . 428 370. The addition-theorem derived directly from the addition-theorems of the theta-functions . . 428 371. The addition-theorem for Weierstrass's integral . . 429 Examples . . 430 CHAPTER XX METHODS OF REPRESENTING ANALYTICALLY DOUBLY PERIODIC FUNCTIONS OF ANY ORDER WHICH HAVE EVERYWHERE IN THE FmiTE PORTION OF THE PLANE THE CHARACTER OF INTEGRAL OR (FRACTIONAL) RATIONAL FUNCTIONS 372. Statement of five kinds of representations of such functions . ... 431 373. In Art. 98 was given the first representation due to Hermite. This was made fundamental throughout this treatise. The other representations all depend upon it . 431 374. The first representation in the Jacobi theory . . . 433 375. The same in Weierstrass's theory . . . . . 434 376. The adaptability of this representation for integration . . . 435 377. Liouville's theorem in the Weierstrassian notation . . . 435 378-379. Representation in the form of a quotient of two products of theta-func- tions or sigma-functions . . . 436 380. A linear relation among the zeros and the infinities ' . 438 381. An application of the above representation 441 CONTENTS. xxiii ARTICLE PAGE 382-384. The fourth manner of representation in the form of a sum of rational functions . 442 385. The function expressed as an infinite product . . . 445 386. Weierstrass's proof of Briot and Bouquet's theorem as stated in Art. 156 . 446 387. The expression of the general elliptic integral 449 Examples .... 450 CHAPTER XXI THE DETERMINATION OF ALL ANALYTIC FUNCTIONS WHICH HAVE ALGEBRAIC ADDITION-THEOREMS 388. A function which has an algebraic addition-theorem may be extended by analytic continuation over an arbitrarily large portion of the plane without ceasing to have the character of an algebraic function . . . 451 389. The variable coefficients that appear in the expression of the addition-theorem are one- valued functions .... . . 453 390. These coefficients have algebraic addition-theorems. The function in ques- tion is the root of an algebraic equation, whose coefficients are rationally expressed through a one-valued analytic fv.nption, which function has an algebraic addition-theorem ... 456 Table of Formulas 458-498 CHAPTER I PRELIMINARY NOTIONS Article 1. One-valued function. — A function of the complex vari- able u = X + iy is said to be one-valued when it has only one value for each value of u; for example, - . sin u, tan u are one-valued functions. u' If we represent the variable u = x + iy by a, point on the plane with coordinates x and y, we also speak of the function as being one-valued in the whole plane, or in any part of the plane for which the function is defined. Regular function. — A one-valued function is regular* at a point a when we may develop this function by Taylor's Theorem within a circle with a as center in a convergent series of the form /(M) =/(a) + ^^/'(a)+ ("-")' /"(«) + • • • + ^-^^^^/'">(a)+ • • • , 1! 2! n! the exponents 1, 2, . . . , n, . . . being positive integers. The power series on the right is denoted by F(m — a). Any such point o is called an ordinary or regular point of the function, and the function is said to behave regularly ^ in the neighborhood of such a point. At these points the function has the character of an integral function. Zeros. — If the function /(it) is regular for all points in the neighbor- hood of a, and if /(a) = 0, the point a is a zero of the function f{u) ; if /' (a) 7^ 0, the point a is a simple zero, or a zero of the first order. If the derivatives /'(a), /"(a), . . . , /'"-^Ho) are all zero, while /<"'(«) ^ 0, the zero w = a is of the nth order. In the latter case the function f{u) may be written f{u) = {u- ayg{u), * Weierstrass, Zur Theorie der eindeutigen analytischen Functionen, Werke, Bd. 2, p. 77; Berl. Abh. 1876, p. 11; Abhandlungen aus der Funktionenlehre, Werke, Bd. 2, p. 135; Zur Funktionentheorie, Ber. Ber. 1880, p. 719; Werke, 2, p. 201. Mittag-Leffler, Sur la representation analytique des fonctions monogi-nes uniformes. Acta Math., Bd. IV, p. 3. t "Ich sage von einer eindeutigen definirten Function einer Veranderlichen u, doss sie tick in der Nahe eines bestimmten Werthes ih der letzteren regular verhalte, wenn sie tick far alie einer gewissen Umgebung der SteUe u„ angehorigen Werthe von u in der Form einer gewohnlichen Potenzreihe von u—u„ darstellen lasst." Weierstrass, Werke, 2, p. 295, 1883. 1 2 THEORY OF ELLIPTIC FUNCTIOKS. where g'(M) is a regular function that is not zero for u = a. The function g{u) may consequently be developed in a convergent series of the form g(u) = g(a) + ^ j'(a) + ^^^^' 9"(a) + ■ ■ ■ . Art. 2. Singular points, ^li the one- valued function /(u) is not regular at a definite point a, we say that this point is a singular point or a singularity of the function. It is an isolated singular point when we may draw around a as center a circle with radius as small as we wish, within which there is no other singularity of the function. Pole or infinity. — A singular point a is a pole or infinity when it is isolated and when the function regular in the vicinity of this point becomes at the point infinite in the same way as, say, the function fiu) (m - a)"' where n is a positive integer and where (u) is a regular function at the point a and 0(o) 7^ 0. The function 4>{u) may be expanded in a con- vergent power series of the form 0(w) = 4>(a) + ^^^—^ )__n_ I g'{u) f{u) u - a g(u) y^^' being a regular function at the point u = a. Similarly it is seen that if m = o is a pole of order m of the function /(m), it is a simple pole of residue —m ioT-'--^- fiu) * Briot and Bouquet (Fonctions ElUptigues, p. 94) employ what seems a more appro- priate name, "point d'indetermination." 4 THEORY OF ELLIPTIC FUNCTIONS. For writing (u — a)"* , /'(m) - m ^ G'{u) we have •'— ^' = h ^, ' < f{u) u — a Giu) where — ^ is a regular function at the point u = a. G(u) Art. 5. The point at infinity. — If we write m = - , a definite point in the V M-plane corresponds to a definite point , in the w-plane, and vice versa. The infinite point in the w-plane corresponds to the origin in the ti-plane. Hence if the function f{u) is regular at the point 2< = oo , the function /(-) must be regular at the point v = 0. It must consequently for small values of v in the vicinity of -y = take the form /[-) = flo + ai'w + o-iv"^ + . . . = P{v), say, where the a's are constants. It follows that for large values of u we must have f{u) =ao+^+^+.-.+^ + ^ + -.. . If the function is regular in the neighborhood of the point oo , the infinite point is a zero of the nth order, when ao = = ai=- • =a„-i; a„ 7^ 0. This function then vanishes at infinity as — (where m = oo ). The point at infinity is a pole or an essential singularity of the function /(m), when n = is a pole or essential singularity of /( i). If m = oo is a pole, we must have for small values of v /(-)= ^ + ^ + • • • + ^+ Co + Cit, + c^v^ + ■ • • , \V/ V v^ v"- where the A's and c's are constants; or, for large values of u, f(u) = Aim + Azu^ + • • • + A„m» + cq + ^ +' -^ + ■ • • . u u^ The part Aiu + AiU^ + ■ • • + AnvT-, which becomes infinite at the pole u= CO, is the principal part relative to this pole and n is the order of the pole. Art. 6. Convergence of series.— We have spoken above of the con- vergence of the series which represents the function f{u) in the neighbor- hood of a point a. We said that the function /(w), one-valued in a defined region, is regular at a point a of this region, when it is developed by Taylor's Theorem in a circle with a as the center. PRELIMINARY NOTIONS. 5 This series is convergent * within the circle having a for center and a radius which extends to the nearest singidar point of the function f{u). We shall presuppose the fundamental tests for absolute convergence. The criterion for uniform convergence as stated by Weierstrass is as follows : The infinite series ui{z)+ 1*2(2)+ 1*3(2)+ • • • , the individual terms of which are functions of 2 defined for a fixed interval, converges uniformly within this interval, provided there exists an absolutely convergent series, Ml + M2 + ■ ■ ■ , where the M's are quantities independent of 2 and are such that within the fixed interval the following inequality is true: I M„(3) |s Mn, where n=[i, fi being a fixed integer. (See Osgood, Lehrbu^h der Funktionentheorie, p. 75.) Art. 7. A one-valued function that is regular at all points of the plane {finite and infinite) is a constant. For the function supposed regular at m = is developable in the series /(m) = ao + OiM + a2U^ + • ■ = •?("), say, which is convergent within a circle which may extend to infinity, since by hypothesis there are no singular points in the plane. Writing u = -, the expansion in the neighborhood of infinity is V /A) = ao + ^ + ^ + --- \v/ V v^ This function being by hypothesis regular in the neighborhood of infinity, 'can contain no negative powers. It follows that ai = = a2 = as = . . . , and consequently fin) = /Q = ao. Another statement of this theorem is the following: A one-valued function that is finite at all points of the plane {including the infinite point) is a constant. For at each one of its poles a one-valued function becomes infinite. It may also be shown that if the variable u tends towards an essential singu- larity in a manner which has been suitably chosen, the modulus of the function increases beyond limit. If then a one-valued function is every- * See Cauchy, Cours d' Analyse de I'Ecole Royale Polytechnique, I*™ Partie. Analyse Alg4brique, Chapitre 9, § 2, Th^or^me I, p. 286. Paris. 1821. Unless stated other- wise, by "convergent" is meant absolutely convergent. (See Osgood, Lehrbuch der Funktionentheorie, pp. 75 et seq.; pp. 285 et seq.) ; and when the variable enters, uni- formly convergent. I^ the latter case by "within the circle of convergence' we understand "within any interval that lies wholly within this circle." 6 THEORY OF ELLIPTIC EUNCTIONS. where finite, it cannot have singular points; it is regular throughout the whole plane and reduces to a constant. Art. 8. The zeros and the poles of a one-valued function, which has no other singularities than poles in the finite portion of the plane, are neces- sarily isolated the one from the other. By this we mean to say that there cannot exist a point a of the plane in whose immediate neighborhood there are an infinite number of poles or an infinite number of zeros. In other words, wherever the point a is situated, one may always draw around o as center a circle with radius sufficiently small that within the circle there are (1) neither zero nor pole; or (2) a zero but no pole; or (3) a pole but no zero. This follows immediately from the preceding developments. For if a point a is taken in the plane, three cases are possible: (1) the function /(m) may be regular at a without vanishing at this point; or (2) the point a is a zero of f(u) ; or (3) the point a is a pole of f(u). In the first case we may draw about a as center a circle with radius sufficiently small that within the circle there is neither zero nor pole; in the second case we may draw a circle sufficiently small that it does not contain a pole and contains the only zero m = a, and similarly in the third case. It follows that if for a one-valued function there exists a point a such that within an area as small as we choose inclosing this point there exists an infinity of poles or an infinity of zeros, this point is an essential singularity. The function is not regular at this point. As examples of what has been said are the rational functions and the trigonometric functions, which shall be first studied as introductory to the general theory of elliptic functions. Rational Functions. Art. 9. Methods are given here, (1) of decomposing a rational fraction into its simple (or partial) fractions; (2) of representing such a fraction as a quotient of two products of linear factors. The same methods will be adopted later in the general theory of elliptic functions, there existing analogous relations for these functions. Consider first as a particular case * the function f{u) = (u - 1) (m - 2) which is regular at all finite points of the plane except the points m = 1 and u = 2. These points are poles of the first order. The principal part of f{u) relative to the pole m = 1 is = ^i(m), say, u - 1 * See Appell et Lacour, Fonctions Elliptiques, p. 7. PRELIMINARY NOTIONS. 7 as is seen by noting that the difference /(w) - ^i(m) is regular at the point w = 1. The residue relative to the pole m = 1 is — 1. Similarly the principal part relative to the pole m =2 is u - 2 with the residue 2. At the point u = co the function is regular, for v) (1 - ?)) (1 - 2 V) is a regular function at the point v = 0. It is further seen that •y = Oorw= oo is a simple zero. The function f{u) has then two simple poles u = 1, w = 2 and two simple zeros m = 0, w = 00 . The function is said to be of order or degree 2. It may also be observed that the equation /(w) = c has two roots, whatever be the constant C. Further, since the functions ^i(m) and j>2{u) are everywhere regular except at the poles w = 1, m = 2, the difference /(m) - l{u) - 2{u) all vanish for w = oo , this constant is zero. We therefore have /(U) = ^l(w) + ^2(W), a formula, which gives immediately the decomposition of the rational function /(w) into its simple fractions. Art. 10. The general case. — A rational function ., ^ ^ apu^ + aiM^-i + ■ ■ ■ + am. ^ Q\(u) ■'^'^' boW + biu^-^ + ■ ■ +bn Qiu) ' where Qi and Q are integral functions (polynomials) of degree m and n, is a function which has no other singularities than poles in the finite portion of the plane or at infinity. At a finite distance it has as poles the roots of Q(u) = 0. The number of these poles at a finite distance, where each is counted with its order of multiplicity, is n. 1°. If m > n, the point at oo is a pole of order m — n. Hence the total number of poles at finite and infinite distances is n + m — n = m. There are also m zeros, viz., the roots of Q\{u) = 0. It is thus seen that tHe function f{u) has ro zeros and m poles. We say that it is of 8 THEORY OF ELLIPTIC FUNCTIONS. order or degree m. The equation/(M) = C has m roots, whatever the value of the constant C. 2°. If n > m, the point oo is a zero of order n — m. The function has n poles and an equal number of zeros. For there are m zeros at finite distances, viz., the roots of Qi{u) =0 and n — m zeros at infinity. The function is of order n and the equation /(m) = C has n roots. 3°. If 771 = n, the point at infinity is neither a pole nor a zero. There are also here as many zeros as infinities, and the function is of order m = n. It follows that a rational function f{u) has always in the whole plane, including infinity, as many zeros as poles. The number of zeros or poles is the order of the function, and the equation /(«) = C, where C is an arbitrary constant, has a number of roots equal to the order of the func- tion /(m). In particular we note that the rational functions have only polar singularities. Principal Analytical Forms of Rational Functions. Art. 1L First form: where the poles and the corresponding principal parts are brought into evidence. Decomposition into simple fractions. Let Oi, a2, . . . , a^be poles of order rii, 712, • ■ • , w^ of the function f{u) and let the principal parts with respect to these poles be 4>,{u) =^ii^ + ^^^ + • • • + ^ -^-^ ^„ , u — ai (u — ai)2 {u — ai)"' Mu) = ^^^ + r^^^ + ■ ■ ■ + , ^"^% , M — O2 (M — 02)^ (m — O2)"' 4>.{u) = ^iii^ + -^^^ + • ■ • + ^"•' u — a^ {u — a„)2 (u — a^)^v Further for the most general case, suppose that the point 00 is also a pole, which is the case in the previous Article when m > n; and let the principal part relative to this pole be 4>{u) = ^low + A2fiu? + • - • + A,oU', where s = 771 — w is the order of the pole. Since each of the principal parts is everywhere regular except at the associated pole, the difference f{u) - (j>i(u) - 4>2iu) - ... - ^(m) is regular everywhere including infinity and consequently is a constant, = A, say. PRELIMINAEY NOTIONS. It follows that f{u) = A + Aiou + Azou^ + • • • + Asqu^ Aii , , ^uji where the index i refers to the indices of the poles ai, 02, . . . , Up. This formula may be written in a somewhat simpler form if we symbolize - by u u — ao, where oq = 00, and let wq = s. We then have f{u) where the summation index i refers to the indices of the poles oi, 0,2, ■ • ■ , a", 0.0- If we put = Vi, we have finally u — ai fin) = A + S(a... - A^./ + f ./' - . . . ± ^_%L_ ./--)). The formula is convenient especially for the integration of a rational function.* Art. 12. Second form: where the zeros and the infinities are brought into evidence. It is sufficient here to decompose the polynomials Qiiu) and Q{u) of the preceding article into their hnear factors, so that f(u) = C (^ ~ gi) (m - C2) ■ ■ ■ (m - Cm) ^ {U — 61) (m - 62) • • • (W - bn) ' where C is a constant. Of course, some of the factors may be equal. We may derive the second form from the first by noting (Art. 4) that 1 + • • • + f'(u) fin) = u 1 - Ci + — u J_ C2 1 1 u — c„ 1 U — bi M — 62 U — bn Integrating and passing from logarithms to numbers, we have the form required. In the next Chapter it will be shown that any rational function /(w) has an algebraic addition-theorem; that is, if m and v are two independent variables, f{u + v) may be expressed algebraically in terms oif(u) andf{v). * Cf. Appell et Lacour, loc. cil., p. 9. 10 THEORY OF ELLIPTIC FUNCTIONS Trigonometric Functions. Art. 13. In the presentation of some of the fundamental properties of the trigonometric functions we shall apply methods which are later used in a similar manner in the theory of the elliptic functions. The polynomial ao + a^u + a2U^ + • • • + OnW = F{u) is a one- valued function with a finite number of terms each having a positive inte- gral exponent. This integral function is of the nth degree. Another class of one-valued functions are those where n has an infinite value. Such functions, when convergent for all finite values of the vari- able, are known as integral transcendental functions. For example, u^ , u^ smu = u--+- is a series which is convergent for all finite values of u and is a regular function at all points at a finite distance from the origin. It becomes zero for the values W = 0, ± TT, ± 2 TT, ± 3 7r, • We know that the decomposition of a polynomial into a product of linear factors is the fundamental problem of algebra. It is natural to seek whether the integral transcendents may not also be decomposed into their prime factors. Euler gave the celebrated formula simvu -(■-f)(-f)M>--('-S)' a formula which is true for every finite value of u. Cauchy was the first to treat the subject in general. Although he did not complete the theory, he recognized that if a is a root of the integral transcendent /(w), it is neces- sary in many cases to join to the product of the infinite number of factors such as 1 a certain exponential factor e^"', where P{u) is a power series in positive powers of u. Weierstrass gave a complete treatment of this subject. Art. 14. We may establish first the results derived by Cauchy. With Hermite {loc. cit., p. 84) suppose that ai, 02, 03, . . . are the roots of the integral transcendental function f{u) which are arranged in the order of increasing moduli. Further suppose that they are all different and none of them is zero. Suppose first that the series r{ mod Qi PKELIMINARY NOTIONS. 11 formed by the inverse of the moduU of the roots, is convergent. The same will (as shown below) also be true of the series ■^ mod (Ot — u) whatever the value of u, excepting the values m = Oi, 02, . . . , which z=oo 1 make the series infinite. It follows then that V will represent an ir{ w - Oi analytic function in the whole plane.* To prove the above statement consider the two infinite series IIu„ and 2vn, of which the first is convergent. The second series will also be convergent if we have Vn < kun (k constant) Tf wp. writfi ?i_ = mod On for all values of n starting with a certain limit. If we write tin = and Vn = ; r , the condition just written is mod (a„ — u) mod On ^ J. mod (On — u) From the inequality mod a„ < mod (On — u) + mod u, we have mod On ^ 1 J. mod u mod {On — u) mod (On — u) which demonstrates the theorem since ; decreases indefinitely mod {On — u) when n increases. It is seen at once that f'{u) y 1 f{u) ^U - On is a regular function for all finite points of the plane. This difference we may represent by G' (m) = • We thus have f{u) ^^ U — On Multiplying by du and integrating, taking zero as the lower limit, we have log-S^ -X log fi - -)= ^(«)- ^(<')= ^i(")' "^y; f(p) ^ \ anj /(O) \ On/ * See Osgood, Lehrhuch der Funktionentheorie, p. 75 and p. 259. 12 THEORY OF ELLIPTIC FUNCTIONS. where the product is to be taken over the finite or infinite number of factors 1 _ Ji T — "^ a I a2 This result is due to Cauchy, Exercises de Mathematiques, IV. Art. 15. We may next consider the general case and, following the methods of Mr. Mittag-LefHer,* establish the important results of Weier- strass t who extended to these integral transcendents the fundamental theorem of algebra. When the series of the preceding article X ; — ^ "^^ mod a„ is not convergent, the sum "V no longer represents an analytic ^-^ u — an function; but by subtracting from each term a part of its development arranged according to decreasing powers of n, Mr. Mittag-Leffler has shown that it is possible to form with these differences an absolutely convergent series. Let a, so that — ^ + PJu) = — u" u — an an"'{u — a„) We may next show that by a suitable choice of oi we may render the series -^Xu — an / -^ a„"(M - a„"(M — an) convergent. In the first place it may happen that V being divergent, the ^-f mod an series formed by raising each term of the divergent series to a certain power is convergent. For example, in the case of the divergent har- monic series V - , we know that V — , where u > 1 , is convergent. Hence we may fix a number w such that the series V is -^ mod On^+i convergent. We may then conclude from this series the convergence of 1 X ; : ; . and consequently of V — ^ mod an^iu - an) h j ^ ^^^^^ an) * See Mittag-Leffler, loc. cit., p. 38; and Comptes rendus, t. 94, pp. 414, 511, 713, 781, 938, 1040, 1105, 1163; t. 95, p. 335. t Weierstrass, Werke, Bd. Ill, p. 100. See also Casorati, Aggiunte a recenti lavori dei Sigi Weierstrass e Mittag-Leffler; Annali di Matematica, serie ii, t. X; Harkness and Morley, Theory of Functions, p. 188; Forsyth, Ttieory of Functions, p. 335. PRELIMINARY NOTIONS. 13 For, if we put mod an^+i mod an^ion — u) we have for the ratio —^ the same value as before, Vn _ mod an Un mod {On — U) We must, however, always know that we are passing to a convergent series when we raise each term of the divergent series to a certain power. For example,* consider the divergent series y . It is seen that Xj ^ log n ^ is also divergent, however great u) be taken, (log n) For writing it is seen that Note that (log 2)- (log 3)- (log nY n - 1 Sn> (log nY (log nY (log nY (log nY and that the first term on the right increases with increasing n, while the second term tends towards zero. The series is therefore divergent. Art. 16. In such cases as the above Weierstrass took for lo a value which changes with n. With Weierstrass write oj = n — I. The given series may be written a„"-i(w-a„) u^^r,li_u\ This series is convergent; for writing -.71 Un = mod ■ \ aj an it is seen that Wn tends towards zero forn = oo . We know (cf . Art. 86) that it is sufficient for this limit to be less than unity for a convergent series. It follows as before that the expression />)-yr-^+p.(«)i f{u) ^[u-an J * This example is due to Mr. Stern and cited by Hermite, loc. cit., p. 86. so that We have at once 14 THEORY OF ELLIPTIC FUNCTIONS. is a function that remains regular for all finite values of u. It must therefore be expressible in a convergent power series in ascending powers of u. Write this series = ^^^ ; and for brevity write du QM = - + - +•••+-' Jo 0„ 2a„2 ojOn.^ \an/ JM = gGw tt\(i- yL\e^"i^) m V I V On) which formula gives an analytic expression, in which the roots are set forth, of the integral transcendental function. The quantities (1 )e "va./ are called primary functions* by \ On/ Weierstrass. Suppose next that f{u) has equal roots, say, of the pth order of multi- plicity. We see immediately that the formula does not undergo any analytic modification, it being sufficient to raise the corresponding primary factor to the pth power. Finally if we admit the case of a function having a zero root of the gth order, we have only to proceed with the quotient ^-^^, the result differing u" from the preceding only by the presence of the factor u«. (See Hermite, loc. cit.) Infinite Products. Art. 17. It may be shown that the infinite product (1 + ai) (1 + az) . . . {l+an) . . . has a definite value, if I ai I + I a2 I + • • ■ + I a„ I + • • • represents a convergent series. f * See Osgood, Ency. der math. Wiss., Band IF, Heft 1, pp. 78 et seq.; Forsyth, Theory of Functions, pp. 92 et seq.; Weierstrass, Werke, II, p. 100; Harkness and Morley, Theory of Functions, p. 190. t Cf . Mittag-Leffler, Acta Math., Vol. IV, pp. 30 et seq.; Dini, Ann. di mat. (2), 2, 1870, p. 35; Harkness and Morley, Theory of Functions, p. 82; and especially Pringsheim, Ueber die Convergem unendlicher Producte, Math. Ann., Bd. 33; Weierstrass, Werke, I, p. 173. PRELIMINARY NOTIONS. 15 For write P„ = (1+ oi) (1 + az) . . . (1 + a„). Then evidently Pn — Pn- 1 = anPn- 1, and P„ = 1 + Oi + 02^1 + a^Pz + • • • + a„P„_i. Hence when n becomes indefinitely large, the series P« will tend towards a definite limit if the series 1 + ai + O2P1 + a3P2 + • ■ • + OnPn-l + a^ + lPn + ' • (1) is convergent, the limit, if there is one, being the sum of this series. Consider first the case where the quantities Oi, 03, . . . are real and positive or zero. The quantities Pi, P^, - . ■ are then at least equal to unity, and consequently, in order that the series (1) be convergent, it is necessary that the series ai + 02 + 03 + • ■ • + a„ + ■ • (2) be convergent. Further, if (2) is convergent, it may be shown as follows that (1) is convergent. The product Pn = (1 + Oi) (1 + 02) ... (1 + an), when developed is 1 + ai + 02 + ■ • • + a„ + O1O2 + ■ • • + aiGz ■ ■ an. Writing An = a-i + a^ + • ■ • + a„ and A = oi + 02 + ■ • ■ + an + a„ + i + • • • , it is seen that A A 2 An Pn 1 + Sl("V + S2<"V2 + . . . + Smf")/-™ if leaving m fixed we let n increase indefinitely, it is seen that P{r) > 1 + Sir + S2r2 + . . + s„r'". Since the sum Sm{r) of the m first terms (m indefinitely large) of the series 1 + Sir + S2r2 + • ■ • (5) is less than a finite quantity P{r), we conclude that this sum tends toward a limit S{r) which is less than or equal to P{r). * See Briot et Bouquet, Fonctions Elliptigues, pp. 301 et seq.; Osgood, Lehrbuch der Funktionentheorie, pp. 460 et seq.; Tannery et Molk, Fonctions Elliptigues, t. I, pp. 28 et seq.; Picard, Traiti d' Analyse, I, 2, p. 136; Bromwich, Theory of Infinite Series, pp. 101 et seq. PRELIMINARY NOTIONS. 17 On the other hand, each of the terms of the product Pn(r) = 1 + Si(»'r + S2<"V2 + . . . + Sn^'^h'' being less than the corresponding term of the sum Snir) = 1 + Sir + S2r-2 + • • + SnV, the sum tends towards a limit Om- The series 1 + (Tiu + azu^ + asu^ + . . . = P(u), say, (6) is convergent, since the moduU of its terms are less than the correspond- ing terms of (5). The sum Sn{u) of the n first terms of this series contains all the terms of Pn(u). Further, the terms of the difference Sn{u) — Pn{u) have for their moduli the corresponding terms of the difference Sn{r) — Pn(r) and con- sequently tend towards zero, when n increases indefinitely. We conclude that Sn{u) tends towards a limit P(u). Thus the function defined by the product (4) is developable in a uni- formly convergent series (6) arranged according to increasing powers of u. Aht. 20. The sine-function. — As an example of Art. 16, we note that the function f(u) = ^i5_!E^ has for its roots all the positive and negative TTU integers ± 1, ± 2, ± 3, • • • . The series V — = — is here divergent, but the series ^^ IS ' mod o„ ^ (mod a„)2 convergent. We may consequently put w = 1 in Weierstrass's formula. The primary factors are therefore ('-r- 18 THEORY OF ELLIPTIC FUNCTIONS. Noting that /(o) = 1, and admitting* that G{u) = (see Vivanti- Gutzmer, Eindeutige anaiytische Funktionen, p. 163), we have the formula Sm TZU _ TT TTU n \ 1 1--K n= ±\, ± 2, ± 3, Uniting the integers that are equal and of opposite sign we have Euler's formula: sin TZU KU (-f)('-7)---(-S--- The periodic property of the sine-function may be deduced from this definition. For write F{u) = Au{u — 1) (m — 2) . . . {u — n) midtiplied by {u + 1> (m + 2) . . . (m + n), where 4 is a constant. Changing u into u + 1, we have F(u + 1) = A(w + 1) M (m — 1) . . . (u — n + I) midtiplied by {u + 2) (m + 3) . . . (w + n + 1). It follows that Fiu +1)= F(u) ^ + ^ - 1 ; u — n or, when n = oo , F{u +1)=- F{u). From this we may derive at once the relation sin (u + 7z) = — sin u, or sin (m + 2 w) = sin u. Art. 21. We may write m ( \ mn/ ) where the product extends over all integers m = ± 1, ± 2, ± 3, . . . , the accent over the product-sign denoting that m does not take the value zero. u Owing to the factor e"", the above product is convergent whatever be the order of the factors. For any one of the factors (l - ~\e^ msiy be written \ mnl gBiir "K mnl ^^ LzVmi/ 3\mir/ J * If we expand the sine-function on the left by Maclaurin's Theorem, and equate like powers of u on either side of the equation, it follows that e^(") = 1. sm u PRELIMINARY NOTIONS. 19 and passing to the product of such terms we note that the series 2 ^ \mn) ' 3 ^ [mn) ' ' ' ' are absolutely convergent. Since m takes all integral values from — oo to +oo excepting zero, we may change the sign in the above product and have sm Next changing m to — m and comparing the resulting product with the one previously derived, we see that sin ( — m) = — sin u. The point m = oo is an essential singularity of sin u. For if we put w = - we see that within an area as small as we wish about v = 0, the function sin - admits an infinity of zeros v = — . m being any indefinitely large V rmz integer. It follows from what we saw in Art. 3 that v = Oorw= <» ia an essential singularity. Art. 22. The funption cot u. — This function may be derived from the sine-function from the formula cot M = — log sin u. du It follows from the above formula * for sin u that cotu=l+(^-+l)+(-V + f)+--- |/ 1 A|/ 1 J_U... \u-\-n n) \u-\-2tz 2n) From this expression we have at once cot ( — w) = — cot u. We also note that the points 0, ± tt, ± 2 ;r, . . . are simple poles and that the residue with respect to each of these poles is unity. With respect to any of these poles, say u = n, the difference cot u u — n is a regular function in the neighborhood of m = ;r. * Eisenstein (Crelle's Journ., Bd. 35, p. 191) makes use of this formula for sin u together with the expression for cot u and establishes a complete theory for the trigo- nometric functions. 20 THEORY OF ELLIPTIC FUNCTIONS. The point u = oo is an essential singularity. In a more condensed form we may write cot M = - + 'S! { 1 ) ' u ^ \u — mn rrm/ where the summation extends over all integers from — oo to + oo excepting zero. The function d . 1 , V 1 sin^ u du u^ m {u — mn)^ ion which has as double poles the poir The principal part relative to the pole u = mn is is an even function which has as double poles the points 0, ± ;r, ± 2 ;r, ■ 1 (u — mn)^ From the preceding formulas the periodicity of the circular functions is easily established. The expression of -r^ — is seen to remain unchanged when n is added sin^ u to u. For the cotangent consider the difference cot (u + n) — cot u. We find that the expression \u + n uj \u u — n/ \u — iz u — 2n/ VM + 27r u + n) Vw + Stt u + 2-k} is zero. Further, from the relation cot (m + tt) = cot u we may derive the periodicity of the sine-function. For multiplying both sides of this expression by du and integrating, we have log sin {u + n) = log sin u + log C, or sin iu + n) = C sin u. In this formula put m = — - , and we have C = — 1. Akt. 23. Development in series. — If we note that 1 _ J_ _ M m2 u — mn mn mP-n"^ m^n^ it is seen from the expansion of the cotangent that cot. = l-«,^-s,Mf-.3!£f_ ... , M n'' n* n^ PEELIMINARY NOTIONS. 21 where sj = V'X, sa = V'-L, S3 = V'^, etc. The sums V'A, 2^ ~3> etc., are evidently zero, since the positive terms are destroyed by the corresponding negative terms. To determine the values Si, S2, . . . , multiply the above formula by du and integrate. We thus have log sin M = log A + log w — sin u = All e ^ "' * "* 2 7C^ 4 ;r* 6 7r« Since - Further, since = 1, when u approaches zero, it follows that A = 1. sin u = u \- we have by equating like powers of u, after the exponential function on the right has been developed in series, sr-j, s. 32-5' S3 = 2n<^ 33 . 5 . 7 ' (see Bertrand's Calcul Differentiel, p. 421). Noting that = T'^ = 2 V— ^ we have * ^-i -^2 + ■ ■ 7:2 _ 1 6 6 2 7:'2 2! -^. -i-^ + ■ ■ n* _ 1 90 30 2^7:4 " J 4! -ie -h + ■ ■ 7r6 1 945 42 257:6 6! -^3 -is + ■ ■ n» 1 9450 30 2^7:8 8! ^+210 + 1 310 + • • Trio 5 93555 66 29 Trio 10! The numbers -, — -, j_ 1 5 • • • are the so- called J 6 30 42 30 66 bers (cf. Staudt, Crelle's Joum., Bd. 21, p. 372). * See Biermann, Theorie der analytischen Functionen, p. 326; Jordan, Traits d' Analyse, t. I, p. 360. 22 THEORY 0¥ ELLIPTIC EUNCTIONS. The General Trigonometric Functions. Art. 24. We know that sin 2 w = ^ and 1 + cot'' u „ cot^ M — 1 cos 2 u = — • cot"' u + 1 Further, since any rational function of a trigonometric function may be expressed rationally in terms of the sine and cosine, we may consider as the general case any rational function of sin u and cos u which in turn is a rational function of cot - . These functions remain unchanged when 2 we add to the argument u any positive or negative multiple of 2 tz. We say that 2 tt is a primitive period of these functions. Writmg cot ^ = '^i we have here to consider any rational function of t. Such a function is consequently a one-valued function of r and has only polar singularities.' As in the case of the rational functions we shall find two forms for the representation of the trigonometric functions, the one corresponding to the decomposition of rational functions into partial fractions and the other corresponding to the expression of a rational function as a quotient of linear factors. Art. 25. First form. — Write ,, s F(sm u, cos u) G (sin u, cos u) where the numerator and denominator are integral functions of sin u and cos u. Further, since sin u = and cos u = 1 2t 2 it is seen that f(^. ^ ..in ^o(e^'")"'+ Ai(e2'")'"-i + ■ + ^^_ie2'"+ A^ ■'^ Bo(e2'«)" + Bi(e2i-)n-l + . . . + Bn-ie^^" + Bn ' where v is zero or is an integer and where the A's and B's are constants or zero. Through division we may express f{u) in the form * f{u) = P(e^") + Q(e«), where P(e'") is composed of integral (positive or negative) powers of e'"- But in Q(e'") the degree of the numerator is not greater than that of the denominator and this denominator does not contain e^" as a factor. Hence Q(e'") = ^{u), say, remains finite when u = and also when m = — oo. * Cf. Hermite, "Cours," loc. cit., p. 121; and also Hermite, Cours d' Analyse de VEcole Polytechnigue, p. 321. PRELIMINARY NOTIONS. 23 We shall next study the function 4>(m). Consider the integral / ^{u)du, where the integration is taken over the contour of the rectangle ABCD in which OM = xo, MN = 2 n, AN = NB = a. If we denote by {AB) the value of this in- tegral taken over the Une AB, we have by Cauchy's Theorem (see Art. 96) {AB) + {BC) + {CD) + {DA) = 2zil., where 2 denotes the sum of the residues of 4>(w) corresponding to the poles that are situated on the interior of this rectangle, p. , Since xq is an arbitrary length, the sides of the rectangle may always be so taken that they are free from the infinities of ^{u). For any point along the line DC we may write u = xo+ i-c, where t is a real quantity that varies from —a to -I- a. We may there- fore write X+o (xo + ix)d-: and similarly a C B M N D / V {AB) = i (^"(^{xo + 2n + i \j —a ix)dx. These two integrals are equal since (m) = (m -I- 2 ;r) . It follows that {AB) -\- {CD) = 0, and consequently {DA) ^ {BC) =2i7rS; or J'*2 jr /'S jr 4>(xo- lo -l-T )dr - I *(xo + ia + T:)dx = 2w2. ^0 (1) Next let the constant a become very large and let the corresponding values of ] = Q[e+°+'(^'>+'>] and ^(xo + ia +t) = Q[e'(^+'-''+')] = Q[e -"+••(-»+')] be respectively G and R. Formula (1) becomes then G -E = i^ or S = i{E - G), an expression which gives the sum of the residues of 4>(m) for all the poles that are situated between the parallels AB and CD when indefinitely produced. 24 THEORY OF ELLIPTIC FUNCTIONS. We apply this result to the product cot(^)4>(u). ./t-u\ .e*-">+l .e^'' + e''" cot ( = I — ^ = I ■ Note that p.nt ( — , „ „ , 2 j eHt-u) _ I git _ giu and that this quantity is equal to — i for m = <» and to +i for m = — oo . Hence the sum of the residues of cotj )*^(m) that are situated between the two parallel lines above, is equal to — G — H. We may next compute these residues and equate the sum of the residues computed to — G — H. Let the poles of 4>(w) be ai of order wi, a2 of order n2, ttv of order n„. We know that the residue with respect to a pole ai is, if we put h = u — Oi, the coefficient of - in the development according to ascend- h ing increasing powers of h of the expression cot( ^-"j -^ Yia, + h). By Taylor's Theorem ^.t — ai — h ^, t — ai h d ,t—ai , cot = cot cot *- + • • • 2 2 1\ dt 2 (ni - 1)! dr'-i°° 2 • • • • Further, the expansion of (ai + h) in the neighborhood of aj is of the form (ai +fe) = ^ + 4i + - • •+ 4^^=^ +^0 + positive powers of h. If we put 4 = C„; Ai = ^i, ^2=^; ■ • . ; A„._i= -^ , it is 1! ^! (wi — 1) seen that the coefficient of - in the above quotient is h PRELIMINARY NOTIONS. 25 The sum of the residues which correspond to the poles of ^(u) is therefore represented by Further, with respect to the pole w = f, if we write u = t + h in the quotient t — u cos ■ t — u sm it is seen that the coeflBcient of -, when h is very small, is — 2 (0. We thus have -G-H^ ^^\Cu cot ^^- - C2i |-cot ^^^ + . . . ■_ij_ ^ dt 2i ±^-d^i^°*4^']-2*« or a formula which is similar to the decomposition of a rational function into its simple fractions (see Art. 11). Art. 26. Second form. — If the function f(u) becomes zero on the points Ci, C2 ■ . ■ , Cm and infinite on the points bi, 62, • • • , b„, it follows at once from the expression of f(u) above that Cg2iM_ g2t6A Cg2iu_ g2 ib2'\ _ _ _ (g2iu_ g2t6„'\ ^ ^g^i sin (m - ci) sin (w - ca) ■ ■ . sin (m - c^) ^ sin (m — 61) sin (m — 62) • • ■ sin (u — 6„) where /i is an integer and C and A are constants. We shall see later (Arts. 373, 380) that there are analogous representa- tions of the general elliptic function. Remark. — The functions which we have just considered admit the period 2;r, so that f(u + 2;:) =/(w). 26 THEORY OF ELLIPTIC FUNCTIONS. If we chana-e the variable by writing m = — , so that/(M) =/(--) = w \io / fi{u), it is seen that /i(M + 2 a;) =/i(m), and consequently 2 w is the period of the new function ; and further all .u Kl — rational functions of e" are now rational functions of e ". In the next Chapter we shall show that any trigonometric function /(m) has an algebraic addition-theorem; or, in other words, /(m + v) may be expressed algebraically through /(m) and/('y). Analytic Functions. Art. 27. We have already referred to certain expressions as being analytic. The general notion of an analytic function may be had as follows.* Suppose that 'the function /(m) has a finite num- ber of singular points pi, p2, . . . , pn in the finite portion of the u-plane.t From each of these points we suppose a line drawn toward infinity, the only restriction being that no two of the lines intersect or approach each other asymptotically. J These lines we may consider replaced by canals which can never be crossed. The canals we suppose infinitesimally broad, so that all the points of the w-plane excepting pi, P2, ■ ■ . , pn are either on or outside of the banks of the canals, the points p being the sources from which the canals flow. We suppose that the function f{u) may be expanded in convergent power series in positive integral powers of the variable at all points except * Weierstrass, Abhand. aus der Functionenlehre, pp. 1 et seq.; Werke, 2, p. 135. See also Vivanti-Gutzmer, loc. cit., pp. 334 et seq.; Goursat, Cours d'analyse, t. 2; Forsyth, Theory of Functions, pp. 54 et seq.; Harkness and Morley, Theory of Functions, p. 105. Osgood {Funktionenlheorw, p. 189) defines a function as analytic in a fixed realm when it has a continuous derivative at any point within this realm. It is then regular at all points within this realm. t We have supposed the function defined for the whole plane; it may, however, be restricted to any portion of this plane. X Mr. Mittag-Leffler's "star-theory" suggests that the plane be cut so as to have a starlike appearance before the initial Mittag-Leffler star is formed. See references and remarks at the end of this Chapter. PRELIMINARY NOTIONS. 27 Pi) P2, • • • , Pn- Let a be any such point and let P{u — a) denote the power series by which the function f{u) may be represented in the neigh- borhood of a. The domain of the absolute convergence of this series is a circle having a as center and with a radius that extends to the nearest of the points p (see Osgood, loc. cit., p. 285). There may be a point c in the u-plane which lies without this domain and at which the function has a definite value. The function /(m) may also at c be expressed in the form of a power series which has its own domain of convergence. The question is: What connection is there between the two power series? Suppose next that the points a and c are connected by any line which does not cross a canal. Take any point aj on this line which lies within the circle of convergence about a. The value of the function /(m) at the point ai is therefore given by P(ai — a), and also the derivatives of /(m) at the point ai are had from the derivatives of this power series after we have written oi for u. It is thus seen that the values of /(m) and of its derivatives at ai involve both a and aj. Next draw the circle of convergence about aj where the arbitrary point tti has been so chosen that the circle about a and the circle about a^ inter- sect in such a way that there are points common to both circles and also points that belong to either circle but not to both. For all points u in the domain of ai the function /(m) may be represented by a power series, say Pi{u — a^). We may show as follows that the coefficients of this power series involve both a and ai: For the domain about a we have the series (I) fiu) =/(a) + ^/'(a)+ ('^-^)V "(a) + ■ ■ ■ ^ P{u -a); and for points common to the domains of both a and ai we have Pi(w - ai) = Piu - a) = PicLi - a -{- u - a{) = P{a, -a)+ "^^ P'{a, - a) + ■ ■ ■ In the domain about ai we have (II) /(^*)=/(ai)+^^/'(«i)+^^^^f^ /"(«!)+■ • -^iC^-^i). where in this domain /W(ai) = Pt'-Mai -a), /< = 1, 2 . . . , which quantities are known from (I). Since the coefficients of Pi(m - aj) involve both a and ai, the power series Pi(m — Oi) is sometimes written Pi(m - ai, a). 28 THEORY OF ELLIPTIC FUNCTIONS. At a point u situated within the domains common to both a and oi the two series P{u — a) and Pi{u — Ui) give the same value for the function /(u). Hence the second series gives nothing new for such points. But for a point u situated within the domain of ai but without the domain of a, the series Pi{u — ai, a) gives a value oi f{u) which cannot be had from P(u — a). The new series gives an additional representation of the function. It is called a continuation * of the series which represents the function in the initial domain of a. Next take a point 02 situated within the domain of a-i and upon the line joining a and c. This point a2 is to be so chosen that its domain coin- cides in part with the domain of ai, the other portion of the domain of a 2 lying without that of a^. The values of /(m) and its derivatives at 02 are offered by the power series Pi{u — ai, a) and its derivatives when for u we have written 02. It is seen that for all points common to the domains a I and 02 P2{u - aa) = Pi{u - ai)= Pi{a2 - Oj + m - 02) = Pi(a2-ai)+^i-^Pi>2-a,)+ • In the domain about 02 (III) /(M) = /(a2) +'^jf^f'{a2)+ (H.^^/"(a)+. . .= P2{u-a2), where in this domain /W(«2) = PiW(o2 - ai), fi= 1,2, . . . , which quantities are known from (II). It is thus seen that the coefficients of the new power series P2{u — 02) which represents /(m) in the neighborhood of a2 involve the quantities a and fli, and it may consequently be written P2{u — a^ = P2(u — 02, a, Oi)- At those points u in the domain of 02 which do not lie within either of the two earlier circles the series P2(w — 02, a, aj) gives values of /(m) which cannot be derived from either P{u — a) or Pi{u — ai). Thus the new series is a continuation of the older ones. Proceeding in this way we may reach all the points of the M-plane where the function behaves regularly. In an indefinitely small neighborhood of tho^e points p which are essential singularities of the function f{u), the * Weierstrass, Werke, Bd. I, p. 84, 1842, employed the word Fortsetzung; M^ray, who also did much towards the foundation of the theory of functions by means of integral power series, used the expression cheminement, a series of circles (see M^ray, Lemons nouvelles sur I'analyse infiniUsimale et ses applications giomHriques. Paris, 1894-98). PRELIMmARY NOTIONS. 29 function can take any arbitrary value (Art. 3) ; consequently the function may be continued up to this neighborhood but not to the points them- selves; while it may be continued up to those p's which are polar singu- larities (cf. Stolz, Allgemeine Arithmetik, Bd. II., p. 100). The combined aggregate of all the domains is called the region of con- tinuity of the function. With each domain of the region of continuity con- structed so as to include some portion not included in an earlier domain, a. series is associated which is a continuation of the earlier series and gives at certain points values of the function that are not deducible from the earlier series. Such a continuation is called an element * of the function. It is seen from above that any later element may be derived from the earlier elements by a definite process of calculation. The aggregate of all the distinct elements is called an analytic function, or more correctly a monogenic analytic function, the word monogenic meaning that the function has a definite derivative. As only functions occur in the present treatise that have definite derivatives, the word monogenic will be omitted as superfluous. Art. 28. We may note that there are functions which although finite and continuous have no definite derivatives. Weierstrass {Crelle's Journ. , Bd. 79, p. 29; Werke, Bd. II., p. 71) shows this by means of the function t f{u) = Ha" cos 6"u., which, although always finite and continuous, never has a definite deriva- tive, if h is an odd integer and (1st) oh > I + %7z or (2d) o62 > 1 + 3 n^, where in the first case ah > I and in the second case ah must be S 1. Art. 29. If c is any point in the region of continuity but not neces- sarily in the circle of convergence of the initial element about a, it is evi- dent that a value of the function at c may be obtained through the con- tinuations of the initial element. In the formation of each new domain (and therefore of each new element) a certain amount of arbitrary choice is possible; and as a rule there may be different sets of domains (for example in the figure of p. 26 along another path ahi 62 • • • c), which domains taken together in a set lead to c from the initial point a. So long as we do not cross a canal and consequently do not encircle any of the singular points J), the same value of the function at c is had, whatever be the method of continuation from the initial point a. The function is one-valued in the plane where the canals have been drawn. * Weierstrass, Werke 2, p. 208. t See also Jordan, Traite d' Analyse, t. 3, p. 577; Dini, Fondamenti per la teorica deUe funzioni di variabili reali, § 126; Wiener, Crelle, Bd. 90, p. 221; Picard, Traite d' Analyse, t. 2, p. 70; Forsyth, Theory of Functions, p. 138; Hadamard's Thesis, Journ. de Math., 1872; Darboux, Mimoire sur V approximation, etc., Liouv. Journ., 1877; Osgood, Lehrhuch der Funktionentheorie, p. 89; Pringsheim, Ency. der Math. Wiss., Bd. II,' Heft 1, pp. 36 et seq. 30 TPIEORY OF ELLIPTIC FUNCTIONS. In Chapter VI it will be seen that if the crossing of a canal is allowed we may have different values of the function at c; in fact, the function has at c just as many values * as there are different elements P{u — c) which lead back to the same initial element at a. Art. 30. The whole process given above is reversible when the function is one-valued. We can pass from any point to an earlier point by the use if necessary of intermediate points. We thus return to the point a with a certain functional element, which has an associated domain. From this the original series P{u — a) can be deduced. As this result is quite general, any one of the continuations of a one-valued analytic function repre- sented by a power series can be derived from any other; and conse- quently the expression of such a function is potentially given by any one element. This subject is treated more fully in Chapter VI. To effect the above representation of an analytic function it is often necessary to calculate a number of analytic continuations, for each of which we must find the radius of the circle of convergence. Thus (cf. also Mr. Mittag-Leffler,t one of the greatest exponents of Weierstrass's Theory of Functions) it is seen that the manner given above of repre- senting a function by means of its analytic continuations is an extremely complicated one. It seems that' Weierstrass scarcely regarded the ana- lytic continuation other than as a mode of definition of the analytic func- tion. As a definition it has great advantages. But the theory of Cauchy (cf. again Mittag-LefHer) , which is founded upon quite different principles, has in most other respects greater advan- tages. The representation of a function by means of the integral 2 ni Js z — u the integration being taken over a closed contour's situated within the region for which f{u) is defined, is fundamental in the derivation of Taylor's Theorem for a function of the complex argument. Mr. Mittag-LefHer t gives an extension of Taylor's Theorem in his " star-theory " by means of which he treats the " prolongation of a branch of an analytic function " in a very comprehensive manner. General methods of representing an analytic function in the form of * Vivanti (see Vivanti-Gutzmer, loc. cit., p. 109) gives a method by which a many- valiied function may be considered as a combination of one-valued functions. See also Weierstrass, Abel'sche Transcendenten, Werke, 4, p. 44. In the sequel we shall by means of canals so arrange our plane or surface on which the function is represented, that the function may be always regarded as one-valued. t Sur la representation analytique, etc., Acta Math., Bd. 23, p. 45. t Mittag-LefHer, Sur la representation, etc., Seconde note. Acta Math., Bd. 24, p. 157; Troisi'eme note. Acta Math., Bd. 24, p. 205; Quatrieme note, Acta Math., Bd. 26, p. 353; Cinquicme note. Acta Math,, Bd. 29, p. 101. PKELIMINAEY NOTIONS. 31 an arithmetical expression are given by Hilbert, Runge, and Painleve (see Vivanti-Gutzmer, loc. cit., pp. 349 et seq. ; Osgood, Encyklopddie der Math. Wiss., Bd. II2, Heft 1, pp. 80 et seq.). Art. 31. Algebraic addition-theorems. — We have seen that the rational functions are characterized by the properties of being one-valued and of having no other singularities than poles. These functions possess algebraic addition-theorems. We have also seen that the general trigonometric functions (rational functions of sin u and cos u or of cot u/2) have only polar singularities in the finite portion of the plane. These functions have periods which are integral multiples of one primitive period 2 k. These properties, however, do not characterize the trigonometric functions; for they belong also to the function esinu -which is not a trigonometric function. To character- ize the trigonometric functions, it is necessary to add the further con- dition that they have algebraic addition-theorems, as is shown in the next Chapter. We shall call an elliptic function * a one-valued analytic function which has only polar singularities in the finite portion of the plane and which has periods composed of integral (positive or negative) multiples of two primitive periods, say 2 w and 2 cj' ; for example, f{u + 2(v) = fiu), f{u -t- 2 a;') = /(") and f{u + 2 mat + 2nw') —f{u), where m and n are integers. A further condition is that these functions have algebraic addition- theorems. Weierstrass characterized as an elliptic function any one-val- ued analytic function as defined above which has only polar singularities in the finite portion of the plane and which possesses an algebraic addi- tion-theorem, the trigonometric functions being limiting cases where one of the primitive periods becomes infinite, as are also the rational func- tions which have both primith^e periods infinite. EXAMPLES 1. Prove that ™ ^ \ m + hl ) where m takes all integral values, negative, zero, and positive. 2. Show that m= +00 „= -00 ( V m- a) ) sin TO •un cot iro ♦ To be more explicit, such a function is an elliptic function in a restricted sense. The more general elliptic functions include also the many- valued functions (see Chapter XXI). 32 THEORY OF ELLIPTIC FUNCTIONS. 3. Show that m=+oo m= — « 4. Show that = -t-oo , > "^ \ > = ;i[cot n{u + a)— cot Jro]. •^ (u + a — m a — m) - u log [Gauss.] 5. Show that 3 cos TZX — /r ~ J {x + m) sin r^x (3,x)=V— J^ -.^n" '—' (x + ■ (4; and that (2 ._y 1 ^,/_ 2 1 ^ 1 V ■^ (x + m)^ \ 3 sin^ nx sin^ Ttx/' ■^(x + m)^" Lsin^ !rx sin^ m sin^" ttxJ g I 1^-^)_V 1 _ ^;„+ir b, COSTO ^ fe^COSTO ^ I feg cos TTl l where the coefficients a^, a^, . . ; 6i, bj; • a-re connected with the BernouHi numbers in a simple manner and may be found by successive differentiation. Eisenstein, Crelle, Bd. 35, p. 198; Euler, Introductio in analysin infinitorum. 6. Prove that 3(4,x) = {2,xy +2{l,x)(3,x); 3(2, 0) = si CHAPTER II FUNCTIONS WHICH HAVE ALGEBRAIC ADDITION-THEOREMS Characteristic properties of such functions in general. The one-valued functions. Rational functions of the unrestricted argument u. Rational functions of the niu exponential function e " Article 32. The simplest case of a function which has an algebraic addition-theorem is the exponential function {u) = e". It follows at once that 6" + "= e" • e", or (j>{u + v)= 4>{u){u), (j){v) and (ji{u + v) is expressed through an algebraic equation, and consequently the addi- tion-theorem is called an algebraic addition-theorem. The theorem is true for all values of u and v, real or complex. The exponential function e" is perhaps best studied by deriving its properties fiom. its addition-theorem. The sine function has the algebraic addition-theorem sin {u + v) = sin u cos v + cos u sin v = sin u Vl — sin^ w + sin v Vl — sin^ u. The root signs may be done away with by squaring. We also have tan u + tan t; . tan (w + v) = . etc. 1 — tan u tan v We note in the above algebraic addition-theorems that the coefficients connecting 0(m), 0W, and ^(zt + v) are constants, that is, quantities independent of u and v. With Weierstrass * the problem of the theory of elliptic functions is to * Cf. Schwarz, Formeln und Lehrsdtze zum Gebrauche der elliptischen Functionen, pp. 1 et seq. The Berlin lectures of Prof. Schwarz have been of service in the prepa- ration of this Chapter. 33 34 THEORY OF ELLIPTIC FUNCTIONS. determine all functions of the complex argument for which there exists an algebraic addition-theorem. Every function for which there exists an algebraic addition-theorem is an elliptic function or a limiting case of one, those limiting cases being the rational functions, the trigonometric and the exponential functions. Art. 33. We may represent a function of the complex argument by <^(m) = ?; and further we shall write 4){V) = 1), 4)(u + v) = C- We may assume either that the function is defined for all real, imaginary, and complex values of the argument, or that this function is defined for a definite region, which, however, must lie in the neighbor- hood of the origin. Further it is assumed that has an algebraic addition- theorem. We therefore have, if G represents an integral function w;th constant coefficients, G(f, 71, = 0. We may now derive other properties of such a function from the property that there exists an algebraic addition-theorem. Art. 34. If we differentiate the function G with respect to u, then, since $ is independent of tj, we have — - - — I 7^ = 0, and similarly af du dl^du dGdji dGd^ _Q drj dv dC, dv Write u + v = hand note that ^ = # . i = ^ . du dh dv We consequently have by subtraction aGdf _ dGdji^Q 9f du dT] dv There are two cases possible: First. The quantity (^ may appear in the coefficients — > — ; or Second. The quantity ^ does not appear in these coefficients. Consider the first case. We have the two equations Gi?, V, = 0, flf du dnj dv ALGEBRAIC ADDITION-THEOREMS. 35 The first of these equations may be written G(f, 7), = aoC™ + air-i + • • + o^- iC + a„ = 0, where the a's are integral functions of f and -q; the second equation may be written where the A's are integral functions of f, ij, — and ^ . If (T is elimi- du dv nated from these two expressions, we have In the second case where ^ does not appear in the coefficients— and -- , we have at once an equation connecting f, — , i) and -^ • oTj du dv This case is, however, the very exceptional one. We have by the above considerations put into evidence a new property of the function ^, viz.: If the function (j) has an algebraic addition-theorem, there is always an equation of the form where H represents an integral function of its arguments with constant coeffi- cients. The equation is true for all values of u and v which lie within the ascribed region. •This equation being true for all such values of v, we may give to « a special value, and have consequently between f and — an equation of du the form 4i) = °' where / denotes an integral function of its arguments. This equation we shall call the eliminant equation.* We may write it in the form /[ t^) we give v a definite value, and if the value resulting of the coefficient is different from zero, then in the above devel- opment we have an equation connecting $ and — . du But if this value of v causes /lU, -^ J to be zero, we try another value and continue until we find a value of v that causes this coefficient to be different from zero, if this be possible. If, however, the function /jU, Jj is zero for every value of v, we have an equation of the form Amv), 4>'{v)] = 0, where /i is an integral function of its arguments. This equation, how- ever, expresses the same thing as the equation f[6{u), cj)'(n)] = 0, ALGEBKAIC ADDITION-THEOREMS. 37 only in the first case the argument is v and not u, which of course makes no difference. If any of the coefficients in the development of the function H con- tained r) alone, /2(rj) being such a coefficient, then since /2 is an integral function of finite degree it can vanish only for a finite number of values of Tj, and we have only to give nj a value such that/2(j;) ^ 0. The theorem is therefore true without exception for every analytic function for which there exists an algebraic addition-theorem with con- stant coefficients; and conversely, as will be shown in Chapters VI and VII, if a one-valued analytic function {u) and its first derivative 4>'{u) there exists an algebraic equation whose coefficients are independent of the argument u, the function has an algebraic addition-theorem. This eliminant equation (see also Forsyth, Theory of Functions, p. 309) must be added as a latent test to ascertain whether or not an algebraic equation connecting $, tj, C. is one necessarily implying the existence of an algebraic addition-theorem. We must not suppose that every algebraic equation G(e, ^, C) = necessarily exacts the existence of an algebraic addition-theorem; neither does the relation 4,{u + i;) = ¥\<^{u), <^'{u), (t>{v), ^'{u)}, where F denotes a rational function of its arguments, always indicate the existence of such a theorem. (See Art. 46.) Art. 36. If we solve the equation /I with respect to — , we have du m-^ du where "^(0 is an algebraic function of f. This equation may be written # =du, n d$ where uq and a denote constants. It is thus seen that in the case of every analytic function $ = {u), for which there exists an algebraic addition-theorem with constant coefficients, the quantity u may be expressed through the integral of an algebraic func- tion of$. 38 THEORY OF ELLIPTIC FUNCTIONS. We may so choose the initial value a that mq = 0, thus having In a similar manner V = I - — - and u + V = I -7-— • On the other hand we have , _ r^_dt , n dt " JaMt) JcMt)' We thus have the equation n dt r^_dt^ _ p dt JaMt) Ja„^it) JaMt)' a formula which is of fundamental importance. To illustrate the significance of the above formula consider the follow- ing examples: 1. Let f = 4>{u) = e"; (f>'{u) = e" We therefore have as the eliminant equation du and also 'f'iS) = f- Since ^ = 1 when ao = 0, we may write ndt u = f — ■ Ji t On the other hand, ^ = ^(w +«) = e"+"= e" • 6"= ^(m) ^(■u) = f • ij. It follows that pd< ^ p^i _ P" i<^ t/l i t/i i «yi t or log !■ + log jj = log f -J). 2. Let f = ^{u) = sin m; <^'(m) = cos u = \/l — sin^ u = Vl — f^. It follows that •>lr{$) = Vl — f2, and consequently since w = for f = 0, Jo V] Vl -<2 Further, since C =^(W + V) =fVl - T]^ + IjVl - ^, we have Jo \/l - <2 Jo V'l - ^2 Jo Vl - t^' or sin - 1 f + sin - 1 )j = sin - 1 [f Vl — ij2 + ,j v'l — f 2]_ ALGEBRAIC ADDITION-THEOREMS. 39 3. If $ = tan u =(f>{u), we have dt n dt p dt p-^", Jo l + fi Jo 1 + <2 Jo 1 tan-i e + tan-i tj = tan-if" ^ "^ " LI - f >? Art. 37. We have seen that for every function $ = 4>{u) for which there exists an algebraic addition-theorem, there exists without excep- tion a differential equation of the form M{u),4>'{u)] = 0, or/(e,£)=o, where / denotes an integral function of its arguments and where u does not appear explicitly in the equation. If f = ^(w) is known for a definite value of u, then from the above equation we may determine — , there being one or more values according du to the degree of the equation in — ■ du We may now prove the following theorem: // the function 6 = (/>(tt) has an algebraic addition-theorem, the values of all the higher derivatives of 4> (u) with respect to u may be expressed as rational functions with constant coefficients of the function itself and its first derivative; so that if the values of the function and its first derivative are known, the higher derivatives are uniquely determined. There are exceptions to the theorem which are noted in the following proof: If we write — = ^' , the equation above becomes du fi$, $') = 0, or, say, oo?'" + aif'"-i + ■ ■ ■ +an-i$' + an = 0, where n is a positive integer and the a's are integral functions of $. We may assume that /(f , $') is an irreducible function, that is, it cannot be resolved into two integral functions of f , f ' ; for if this were the case, one of the factors put equal to zero might be regarded as the integral equation connecting f and $'. We form the derivative ■' ^ ' , which is an integral function in f , f '. The degree of this derivative in $' is one less than the degree of /(*, ?') inf. Further, the equation ■' ^'" '^ ' = is not satisfied for all pairs of values f, f which satisfy the equation /(f, $') = 0. For if this were the case. 40 THEORY OF ELLIPTIC FUNCTIONS. the two equations would have a greatest common divisor, this divisor appearing as a factor of both functions. But by hypothesis f{?, $') is irreducible. The two equations /(f , f) = 0, are satisfied by only a finite number of pairs of common values $, $'. For their discriminant with respect to f is an integral function in the a's; and as this discriminant put equal to zero is the condition of a root common to both equations, we have an integral equation in the a's, that is, in f. There are consequently only a finite number of values of $ which satisfy this condition. These common roots constitute the exceptional case mentioned at the beginning of the article and are excluded from the further investi- gation. They may be called the singular roots. We next consider a value m = Uq of the argument, for which (f>{uo) = fo, '(uo) = $o', where ^o, fo' satisfy the equation /(f, f) = but not the equation ^■■^^'^' ^'^ = 0. ^ ar By differentiation we have g* + |*' = 0. We further assume that the point in question is such that the function has for it a definite derivative. We may write df = $'du, df = ^"du. It then follows that or e" = ar From this it is seen that $" = (wo) = f = ^(.ui), Further let (j>{u) be an analytic function with an algebraic addition- theorem, and in the neighborhood of Uq and Wi let the function 4>{u) be regular. Finally, it is assumed that >0; that is, Co' does not belong to the singular roots of /(f, f) = 0. We assert that 4'{u) under these conditions is a periodic function and that Ui — uq is a period of the argument* Since the function <^(m) is regular in the neighborhood of uq, it may be developed by Taylor's Theorem in the form iu) = {Uo) + ^^^ cf>'{uo) + (^-^^oP ^»(^^) + . . . . In a similar manner we also have {u) = 'iuo)='iui)=^o'- The derivative " (mq) ni^'Y be expressed as a rational function of (f> (mq) , "{ui) has the same form in (l>{ui), It follows that (f)" (uq) = (j)"{ui), and in a similar manner "'iuo) = "'iui), <^(">(Mo)=<^«(i^i), Let Uo + V be a point that lies within the region of convergence of the first of the above series and let wi + v be a point situated within the region of convergence of the second. * Cf . Biennann, Theorie der analytisclien Funktionen, p. 392. 42 THEORY OF ELLIPTIC FUNCTIONS. Instead of u write uq + v and mi + -y in the two series respectively. They become 2 {uo + v)= 4>{uo) + v4>'{uo) + '"-" (uo) +• • • ' {ui) + v(j>'iui) + |-^"(mi) + • • • ■ Consequently, owing to the relations above, 4>{uo + v)=(j){Ui + v). Next write ui — uq = 2ct) or ui = uq + 2(i), and we have 4>{uo + v) = 4>{uq + V + 2(d). The quantity v may be regarded as an arbitrary complex quantity, and must satisfy the condition that uq + v belongs to the region for which {u) is periodic, if it has an algebraic addition-theorem and if there are two points, uq and Wi, that are not the singular roots of/[^(w), 0'(m)]= 0, for which 4>{uQ) = {u{) and (f>'{uo)= 4>'{ui). Art. 38. If we have only the one condition that ^(mq) = 0(wi), we cannot without further data draw the same conclusions about periodicity. If the equation connecting 'iu) is of the first degree in 4>'{u), as is the case of the exponential function, then the second condition, viz., 4>'{uo) = (j>'{ui) follows at once. In general this is not the case. We may, however, effect a conclusion if the assumptions are somewhat changed: Suppose that n is the degree of the equation /[^(m), ^'(w)] = with respect to 4''{u). To every value of ^(w) there belong at most n values of 4>'{u). . Suppose next that n + 1 points uq, ui, . . . , Un may be found, at which CO = {un) ; and suppose also that cf){u) is regular in the neighborhood of each of these points, and further suppose that $o is not a singular root of /(f, f) = 0. Write 4>'{uo)= a>o, 4>'{Ui)=U}i, ^'{Un)= (On- These n + 1 values of 4>'{u) belong to one value of fo = ^("o) = <^(wi) = • • • = '{u) belonging to one ALGEBRAIC ADDITION-THEOEEMS. 43 value of {u), it follows that two of the above values of '{u) must be equal, and consequently where a and /? are to be found among the integers 0, 1, 2, . . . , n. But by hypothesis we also had It follows from the theorem of the preceding article that (j)(u) is periodic,. Ux — U0 being a period of i^(m). We have then the following theorem: * // it can he shown that a function having an algebraic addition-theorem takes the same value on an arbitrarily large number of positions in the neigh- borhood of which the function is regular, the function is periodic. Art. 39. We have seen that in the equation connecting $ and ^^ , viz., du /' M)=«' the quantity u does not explicitly appear. Suppose that $ = {u) is a particular solution of this differential equa- tion. As this differential equation is of the first order, the general solution must contain one arbitrary ccJnstant. We may introduce this constant by writing the arbitrary constant v being added to the argument. It makes no difference whether we differentiate with regard to u or with regard to u -\- V since u does not enter the equation explicitly. We consequently have ^ = ^^ . = '{u + v), du d{u + v) from which it is seen that the differential equation is satisfied by ^(u + v). We may therefore write f[{u-{-v), {v), ^(w + i.)] = 0. As 4>(v) is a constant, we may determine (p{u + v) as an algebraic function of ^(w) from this equation. It is thus shown that the general integral of the differential equation f[^{u), 4>'(u)] = * See Daniels, loc. cit., p. 256. 44: THEORY OF ELLIPTIC FUNCTIONS. is an algebraic function of the particular solution ^(w). We note that this theorem is not true for every differential equation in which the argu- ment does not enter explicitly, but only for those functions for which there exists an algebraic addition-theorem. If one succeeds in integrating the differential equation in two ways, the one being by the addition of a constant to the argument of the function and the second in any other way, the addition-theorem is at once deduced by equating the two integrals. (See Chapter XVI.) The Discussion Restricted to One-valued Functions. Art. 40. We proceed next with the consideration of the two equations of Art. 34: dG d$_ _ dG dji ^ Q (1) 6f du dr] dv G{$, 71, = 0. (2) The first of these equations may be written in the form AoC*^ + ^iC*^-^ + ■ ■ -I- Afc-1 c + Afc = 0, where the A's are integral functions of $, tj, $', -q' , while the second equation has the form aoC + aiC""^ + ■ • + ^".-iC + flm = 0, the a's being integral functions of f, tj. By the application of Euler's method for finding the Greatest Common Divisor of these functions, it is seen that this divisor is an integral function of the A's, and a's and i:^, say go:, ?, V, f, v')- (3) This function equated to zero is the simplest equation in virtue of which equations (1) and (2) are true. If g is to be a one-valued function of its arguments and if $, tj, $', rj' have each a definite value for a definite value of u, then ^ also must have a definite value, so that the equation (3) must be of the first degree in ^. Hence (^ must have the form \ du dv/ where F is a rational function of its arguments. We shall leave for a later discussion (Chapter XXI) the determination of all analytic functions which have algebraic addition-theorems. At present we shall only seek among such functions those which have the property that (^ = 4>{u + v) may be expressed rationally in terms of {u), (j)'{u), cji(v), {u + v) may be expressed rationally through 4>M, 4^'{u), 4'{v), 4>'M- Much emphasis is put upon this theorem, which is proved in Art. 158. Thus while the general problem has been restricted, we have in fact only limited the discussion in that one-valued analytic functions are treated. It may be remarked here that the rationality of <^(m + v) in terms of 4>(u), '{u), '{u), cPiv), '{y). These functions are (cf. Art. 293) limiting cases of elliptic functions; those under heading I are not periodic and those under II are simply periodic. Finally, we have III. The elliptic functions, which are doubly periodic. These functions have the properties just mentioned under I and II. We shall see in Art. 78 that there do not exist one-valued functions which have more than two periods. Hence every function for which there exists an algebraic addition-theorem is an elliptic function or a limiting case of one. Art. 42. Let 4>{u) be a rational function of finite degree and let f = (1>{U), T) = {v), l; = (f)iu + V). By means of these three equations we may eliminate u and v and then have an equation of the form (A) G(f, v, = 0, where G denotes an integral function of its arguments. Writing (1) e={u), (2) £ = K'' I) ^ '' which is an ordinary differential equation in which the variable u does not appear explicitly. The equation (A) and the latent test (B) are sufficient to show that every rational function has an algebraic addition-theorem. We shall next show that in the case of the rational functions the argu- ment u may be expressed rationally in terms of $ and - — du We assume first that the two equations ^=cb{u) and ^=4>'{u) du have only one common root, which may be a multiple root. By the method of Art. 40 we derive an equation which is either of the first degree in u, in which case we may solve with respect to u and thus have u ration- ally expressed through f and — ; or it is of a higher degree in u, of the du form, say aoM'" + aiW"-! + a2W™-2 ^ . . . j^ ^^ = 0, where the a's are functions of f and — • du Since this equation must represent the multiple root, it must be of the form ao(u — Mo)'" = 0. This expression developed by the Binomial Theorem becomes aoM'" — maou'^'^uo + • • • . It follows from the theory of indeterminate coefficients that ai = —maouo or uq = — — ^• mao Since ai and Oq are integral functions of ^ and — , it is seen that Wq du may be expressed rationally through these quantities. We may there- fore write R ^ £)■ where R denotes a rational function. We thus see that for the case where the equations $ = {u) and ^ = ^'(m) du ALGEBRAIC ADDITION-THEOREMS. 47 have only one common root, we have *(. + „ = , [«(,,£) + «(,,!)]. Further, since (j) and R both denote rational functions, it is seen that ( au dv) where F denotes a rational function. Art. 43. We shall next show that the two equations au cannot have more than one common root. For assume that they have the common roots wi and U2. It follows that iU2), (1) <}>'{Ul)=f-=cf>'(U2). (2) au Smce these two expressions exist for continuous values of $ and ^> du we may regard Wi and W2 as two variable quantities. Taking the differential of (1) it follows that (p'{Ui)dUi = (j>'{U2)dU2. If we exclude as singular all values of u for which ^'(Ml) = = («! + C). This expression is true for an arbitrarily large number of values of ui, and since the degree of 4>{u) is finite we must have the identical relation <^(ui) s {ui + C). Further, for Wi we may write any arbitrary value in the identity, say Ml + C, and we thus have 4,{ui + C) = 4>{ui + 2 C) = 4>{ui). 48 THEORY OF ELLIPTIC FUNCTIONS. Hence the roots of the identity are Ui, Ui + C, Ui + 2 C, • • • . If then C 7^ 0, the equation has an infinite number of solutions. This, however, is not true, since the equation is of finite degree. If follows that the constant C = and conse- quently the two equations can have only one common root. We have thus shown that every rational function of the argument u has an algebraic addition-theorem and has the property that '{u), (piv), (l>'{v). Art. 44. We shall next show that the theorem of the last article is also true for all functions that are composed rationally of the exponen- tial function e " . Let /i be a real or complex quantity different from and oo and write t = e''", and fit) = {u), (1) where f denotes a rational function. Further, let s = e"" and f{s) = ^W. (2) It follows that g^^u-^v) == e''" • e'" = < • s, and fit • s) = 4>iu + v). (3) From the three equations (1), (2) and (3) we may eliminate f and ij, and have (A) G{iu),4>iv),iu + v)]=^Q, where G denotes an integral function. We have under consideration a group of one-valued analytic functions which have everywhere the character of an integral or fractional func- tion and which are simply periodic, the period of the argument being 2 ni n = 2 iu)=fit), f- = 4>'{u)=f'it)fit. du If t is eliminated from these equations, we have the eliminant equation where / denotes an integral function. ALGEBRAIC ADDITION-THEOREMS. 49 It follows from equations (A) and (B) that the function 4>{u) has an algebraic addition-theorem. Art. 45. It may be shown as in the case of the rational functions that when the equations f = -f (0 and ^ = ■^'it)/it du have one common root * in t, then we may express t in the form where jB denotes a rational function. It also follows that {u + v) = F[<^{u), cj>'{u), 4>{v), 4>'{v)], where F is a rational function. Suppose next that the two equations have more than one common root. Suppose that ii and ^2 are two roots that are common to both equa- tions, so that f = t(lr'{ti)dh = -^'{t2)dt2, which divided by the expression (2) becomes dti _ dt2 ii ~ <2 ' or log p). Writing q — p = m, sl positive integer, we have C" = 1. (2) It is thus shown that C is an mth root of unity, and as m is the smallest integer that satisfies this equation it is a primitive mth root of unity. It is easy to see that the quantities C,C^,C^, . . ., C'"-\C"' are all different. For if C = C'ihj mm), then is C^-' = 1 (where i — j = m' ! = 2, so that Cai — g2 = 0, or 02 = Cai. In a similar manner, since the left-hand member of the equation van- ishes for 02, one of the factors on the right-hand side must vanish for t = a2, say Cg2 — a„ = 0, where v is to be found among the integers 1, 3, 4, . . . , ^, say v = 3. We thus have Ca2 — 03 = 0, or as = Ca2 = C^Ui. Continuing this process we derive the relations ai = oi, g2 = Cai, a^ = C^a^, . . . , g^ = C^-^ai. Further, since C, C^, . . . , C""~^ are all different, it is seen that oi, 02, • • • , flm are all different. The quantities gi, Cai, C^ai, .... C™-%i form a group of roots of the equation, and after Cote's Theorem {t - ai) {t - Cai) {t - C^a{) ... {t - C^-igi) = f" - gi"*- This factor t™ — gi"* may consequently be separated from the two sides of the equation (I). If further there remain linear factors in the numer- ator of equation (I), we repeat the above process until there are no such factors. The same is also done with the denominator. When all such factors have been divided out from either side of the equation (I), there remains t±>- = (CO*", so that C'' = 1. It follows at once that fi must be a multiple of m and consequently '>lr(t)= Ait^) "■ (^^ ~ '^ ' / (^'" ~ ^p™) ■ • • 52 THEORY OF ELLIPTIC EUNCTIONS. We have thus shown that if the two equations du have more than one root in common, there exists an integer to, such that ■^{t) may be expressed as a rational function of t"^ Writing t = e"", it follows that i™ = e'"''" and 4>{u)= ylr{t)= ■»|^i(«'")= f'i(e'"'"'). In the further discussion we may use '^i{t™) in the place of ■^(t). It may happen that the two equations f = ■fi{e"'i'^) and — = i|ri (e'"'-") m/ie""'" du have more than one common root. By repeating the above process we may diminish the degree of V^i and replace the function ■f'i{e'"'"^) by the equivalent function ■^2{e'^^' ''") , where m' is an integer, etc. Since the original function yjr was of finite degree, a finite number of divisors must reduce the degree to unity. It therefore follows that in the process of diminishing the degrees of the functions -vlf, "v/^i, ^1^2, • • • , we must come to a function, say f = •iu), 'iu), {v), '{v)l where F denotes a rational function. Example. — Apply the above theory to the examples sin u, cos u, tan u. Write sin u = = , where t = e'". 2i 2i t Aet. 46. It may be shown by an example that a function i^(m) may have the property that 4>{u + v) is rationally expressible through ^(w), 4>'iu), {v), '{v) without having an algebraic addition-theorem. Take the function ^(m) = ^e"" + Be'"', (1) where A, B, a, b are constants and a j^b. It follows that ?!>'(«) = aAe°» + 6Be''". (2) ALGEBEATC ADDITION-THEOREMS. 63 From (1) and (2) we have 4e«« = ^^(m) -'iu) _ b ~ a b — a We further have (f>{u + v) = Ae^^e"" + Be'^'e'"' _ 1 b(j){u)~'iu) _ b(j)iv)-^'iv) _|_ 1_ ~act>(u)+(l>'{u) _ -a(f>iv)+4>'{v) ^ A b—a b—a B b—a b—a from which it is seen that {u + v) may be expressed rationally in terms of iu), '{u), {v), '{v). We shall now show that ^(m) has not an algebraic addition-theorem. We so choose a and b that the ratio - is an irrational or complex quan- tity. *" In Art. 35 we saw that without exception the differential equation where / denoted an integral algebraic function, existed for all functions which had algebraic addition-theorems. If therefore we can prove that such an equation does not exist for 0(u), we may infer that <^{u) does not have an algebraic addition-theorem. Suppose for the function 4>{u) there exists an equation of the form fmu), <^'(w)] = o, where / denotes an integral function. Since (f>{u) and <^'(m) may be expressed through e°" and e''" where only constant terms occur in the coefficients, we may write the above equa- tion in the form /i[e°'', ef-"], where /i like/ denotes an integral function of finite degree. This equation must be satisfied for all values of u for which the function ^(m) is defined. We give to u successively the values , 2ni , ini Wo, Mo H , "o H ' • • • • a a The quantity e»" has the same value, viz., e""" for all these values of u. But corresponding to one value of e""", the equation above being of finite degree can furnish only a finite number of different values of e''". On 54 THEOEY OF ELLIPTIC FUNCTIONS. the other hand there correspond to the one value e""" an infinite number of values e''" of the form . buQ+ —2ni 6i{v), 4>'{v)\, F denoting a rational function, does not necessarily imply the existence of an algebraic addition-theorem. Continuation op the Domain in which the Analytic Function {u) HAS BEEN Defined, with Proofs that its Characteristic Prop- erties ARE Retained in the Extended Domain. Art. 47. In the previous discussion we have supposed that ^(w) was defined for a certain region which contained the origin. This region we may call the initial domain of the function cf>{u). We further assume that (j){u) has an algebraic addition-theorem and is such that ^(m -I- v) may be rationally expressed through 4>{u), 4>'{u), '{'o) within this initial domain. These properties are expressed through the two equations (I) G[(t>{u), 4>{v), <^(M + „)] = 0, (II) 4>{u + v)=F[,p{u), cj>'{u), 4>{v), 4>'{v) }, where G denotes an algebraic function and F a rational function. We also assume that u and v are taken so that u + v lies within the initial domain.* We shall now prove the following theo- rem: If the function 4>{u) has the properties above mentioned, it has the character of an integral or a (fractional) rational function in the neighborhood of the origin. In the equation (II) we write u + V in the place of u, — 1) in the place of v, u in the place oi u + v. We thus have -;6(m) = F{4>{u + v),(f>'iu + v),{- v), '{- v)}. (1) Fig. 3. * Cf . Weierstrass, Abel'schen Functionen, Werke 4, pp. 450 et seq. ALGEBRAIC ADDITION-THEOREMS. 55 Such values are chosen for v that for these values the functions (}){v) and {— v) belong to the mitial domain. We develop ^(m + v) by Taylor's Theorem in the form {u + v) = '{u ^v)= ) + v4>"{v) + ^.^'"W + • • • . These series may therefore be substituted in formula (1). We thus have <^(w) expressed as a rational function of u, which as the quotient of two integral functions takes the form - /^N ^ Pio(w) ^ gp + axu + a^u^ + ■ ■ ■ P2o(w) 60 + hxu + 62^2 + • • •' where the two series are convergent so long as | u | is less than a certain quantity, say p. If 60 7^ 0, (^(w) has the character of an integral function in the neigh- borhood of the origin m = 0; if 60 = = ^1 = • • = 6it and at the same time tto = = «! = ■ • = a*, then 4>{u) has the character of an integral function at the origin; but if one of the a's just written is different from zero, then <^{v) becomes infinite for M = but of a finite integral degree. It then has the character of a ratipnal function at the origin, and its expansion by Laurent's Theorem* has a finite number of terms with negative integral exponents. Art. 48. We may next prove the following theorem: The domain of (j){u) may be extended to all finite values of the argument u without the func- tion 4>{u) ceasing to have the character of an integral or {fractional) rational function. Fundamental in the proof of this theorem is the expression of ^(m) as the quotient of two power series where the two series are convergent so long as | m | does not exceed a definite limit p. If we draw the circle with radius p about the point w = 0, then within * In this connection see a proof of Laurent's Theorem by Professor Mittag-Leffler, Acta Math., Bd. IV, pp. 80 et seq., where the theorem is proved by the elements of the Theory of Functions without recourse to definite integrals. 56 THEOEY OF ELLIPTIC FUNCTIOlSrS. this circle the function (piu) is completely defined. In order to extend or continue this region, we may use the equation {u + v)=F{iu),'{u),i>{v),'iv)}. We shall at first assume that we may write u = v without the function F taking the form 0/0. We then have * for m = v, 4>{2u)=F{{u),4>'{u),4>{u),4>'{u)]. The right-hand side of this equation is true for all values of u that lie within the circle with radius p. It follows then that through this expres- sion the function ^ on the left-hand side is defined so long as its argument lies within the circle with radius 2 p. If then we write u in the place of 2 m in this equation, we have {u), as the quotient of two power series, = ^'"^ ' ■ Further, P2,o{u) 4>'{u) may also be expressed as the quotient of two power series. These values substituted in F give {u) defined as the quotient of two new power series, say P2.liu) Since 5 u has been written for u in the two new power series, they are convergent so long as | m | < 2 p. We cannot apply the above method, if for u = v the function <;6(m) takes the form 0/0. Nevertheless we may proceed as follows and extend the region of convergence at pleasure. In the equation {u + v) = F{{u),'{u),(l>{v),cf,'(v)}, we write instead of u, 1 + a and instead of v, 1 +a where a is a real quantity such that J < a < 1. We have in this manner The function F being a rational function, we may express ^(w) as the quotient of two power series in which the numerator and denominator are ♦ See Daniels, Amer. Joum. Math., Vol. VI, p. 255. ALGEBRAIC ADDITION-THEOREMS. 67 analytic functions of u and a. The denominator cannot vanish for all values of a. We shall therefore so choose a that the denominator is differ- ent from zero. We may then express 0(u) as the quotient of two power series in the form 'P2,l(w) where the series are convergent for values of u such that I M I < (1 + a)p. Since a = \ the series is convergent if | m | < i p. This process, as weU as the one employed in the previous article, may be repeated as often as we wish, so that we have eventually P2.nW where the power series Pi,n{u) and P2.n{u) are convergent so long as Hence ^(m) may be defined for an arbitrarily large portion of the plane as the quotient of two power series which may be expanded in ascending powers of u.* Art. 49. As an example of the above theory, consider the function • = P{u), {u) ire = tan u = sm u Pi .o(m) ^ I u3 ^ 2 w5 I COSM ^2,o(w) 3 15 Pi.oiu) = " ~ ^ + ^ ~ ■ * ■ ' P,,oiu) = l-^ + '^ . At the points 0, n, 2n, 3n, . . . , the function tan u is zero, and is ■ r. ■. , 7t 3 n 5 7t miinite at -> — > — > ■ • • . For the point m = oo , the function tan u is not defined, this point being an essential singularity of the function. The function is convergent for all points within a circle described about the point u = 0, whose radius extends up to the infinity ^ of tan u, so that we may take P = -^ • * Weierstrass, Werke IV, p. 6, savs that from the fact that {u) has an algebraic addition-theorem we may show that it is a uniquely defined function having the char- acter of an integral or rational (fractional) function and that starting with this we may derive a complete theory of the elliptic functions. ? THEORY OF ELLIPTIC FUNCTIONS. Using the formula , , , s tan u + tan v tan [u + V) = 1 — tan u tan v we may extend the definition of tan u to an arbitrarily large region. For writine v = u, then is , o 2 tan u tan 2u = tan^ u Further, if we put ^ m in the place of u, we have tan u = 2 tan ^ M _ 2P{iu) ^ Pi,i(u) 1 - tan2 i M 1 - P2(J m) P2, i(u) ' where Pi, i(w) and P2, i(m) are convergent so long as ^|m|<^7: or |M|<7r. We see that here the new circle of convergence passes through the points + 7: and — n and that the old region of convergence has been extended by a ring-formed region. By another repetition of the same process we have P2.i{u) \P2,1{U)J tan.- ^-■'(") ^g^^ The radius of convergence of the two series on the right-hand side is now 2 7:, so that the tangent function is defined for all points within the circle whose radius is 2 n. By continuing this process we are able to define tan u for all finite values of the argument u without its ceasing to have the character of an integral or (fractional) rational function. Art. 50. Returning to the general case we shall see whether the function which has been thus defined for all points of the plane is the same as the function (j){u) with which we started and which was defined for the interior of the circle with the radius p. We shall show that such is the case and that the new function is the analytic continuation of the one with which we began.* We shall first show that the two functions are identical within the circle with radius p. It is seen that the expression of (f>{u) as the quotient of two convergent power series is characteristic of this sort of function. We limit u to the interior of the circle with radius J p within which v is also restricted to remain. The points u, v, u + v evidently lie within the domain for which ^(m) was defined, and the property expressed through the formula (u), ^'{u), cf.{v), 4>'{v) ] is true for this domain. * Weierstrass {Definition der Ahel'schen Functionen, Werke 4, pp. 441 et seq.) empha- Bues this fact. ALGEBRAIC ADDITION-THEOREMS. 59 On the right-hand side we again write instead of u 1 +a and instead of v, 1 +a with the limitation that the absolute values of these quantities be less than ^ p. Writing first (i){u) = — ''°^ and then making the formal computation P2,o(m) as above, we have c6(w) = ^'^ { ■ These two quotients are identical * •P2, i(w) within the circle with radius ^ p, so that . P2.0{U) P2.liu)' or Pi,o(m)P2,i(w) =P2,o(w)Pi,i(m). (1) If we multiply these two power series on either side of the equation, we will have the equality of two new power series, which is true for all values of u, such that | w | < i p. Now Pi,o and Pi, 1 are convergent by hypothe- sis within the circle of radius p, while P2,o and P2,i are convergent within the circle of radius | p . Within the circle with radius J p the coefficients of u on either side of the equation are equal. But as these coefficients are constants we conclude that the two series on the right and left of equation (1) must be the same within the extended realm, the circle with radius p. It follows that the representations of {u),{v),{u + v)} =0, (II) cl>{u + t)) = F{cj>(u), 4>'{u), {v), '{v)}, are also retained for the extended region, f First take \u\ < h p and \v\ < i pso that \u + V \ < p and therefore lies within the initial domain. * Weierstrass, loc. cit., p. 455. t This theorem has the same significance for the properties of the elHptic functions as the fact that the functions themselves may be analytically continued as emphasized in Chapter I. 60 THEORY OF ELLIPTIC FUNCTIONS. In the equation G = write ^l^sinl for Mu), il^°M. for Mv) and ?Lo(m±«) P2,o(w) jP2,oW P2.o{u + v) for (j){v + v). Multiply the expression thus obtained by the least com- mon multiple of the denominators and we have an integral power series in u and V on the left equated to zero. This power series is convergent so long as | tt | < i p and \ v \ < ^ p. If this power series is arranged in ascending powers of u, the coefBcients are functions of v which may also be arranged in ascending powers of v. Since the right-hand side is zero, the coefficients of u are all zero and consequently the power series in V are identically zero. Making use of equation (II) we derive the second development for ^(m), viz., -P2,i(m) This value and the corresponding values of (« + v) are now sub- stituted in (I). We thus make another integral power series in u and v on the left equal to zero on the right as in the previous case. These two power series must be the same so long as \u\ < ^ p and \v\ < ^ p. But as here the coefficients of u are all identically zero, this must also be true in the extended region. By repeating this process we have the theorem: The addition-theorem, while limited to a ring-formed region, exists for the whole region of convergence established for the function (l>{u). If the point w = oo is an essential singularity, the function ^(w) will have this point as a limiting position, that is, the function may be con- tinued analytically as near as we wish to this point, but at the point oo the function need have the character of neither an integral nor a (frac- tional) rational function. CHAPTER III THE EXISTENCE OF PERIODIC FUNCTIONS IN GENERAL Simply Periodic Functions. The Eliminant Equation. Article 52. In the previous Chapter we have studied the characteristic properties of one-valued analytic functions which have algebraic addition- theorems. These properties were considered in the finite portion of the plane. The function may behave regularly at infinity or this point may be either a polar or an essential singularity of the function. In the latter case the function is quite indeterminate (Art. 3) in the neighborhood of infinity. When the point at infinity is an essential singularity, we shall show that the function is periodic. To prove this we have only to show that the function may take certain values at an arbitrarily large number of points (cf. Art. 38) of the M-plane. Suppose that m is the number of points at which <^(m) = $o> say, where fo is a definite constant, and denote these points by a i, 02, . . . , am- Let Oj, be any one of these points, and with a radius r^ draw a circle C ^ about o^ as center. Take r^ so small that within and on the periphery of C^ none of the other points ai, a2, . . . , a^^-i, a^+i, . . . , an lies, and also within and on the periphery of this circle suppose that (u ) is everywhere regular. Next let u take a circuit around C in the w-plane; then in the plane in which (f>(u) is geometrically represented (u) _ d\cl>(u)-$o\ ^ Hu) - fo (f>{u) - fo and expressing ^(u) — fo in the form ^(w) - fo = re*', it is seen that d^{u) _ d\re''\ ^e''dr rie^ + 4>(.u) — fo re" re*' re = '^+ide. r 61 62 THEORY OF ELLIPTIC FUNCTIONS. If next we integrate around S^ in the i^(w) -plane, we have r d4>{u) ^ r dr^ r .^g_ The first integral on the right is log r, which is here zero, since the curve returns to its initial point, making the upper and lower limits identical. We thus have f Jf^- f ""'■ JSp 4>{u) — fo Js^ On the other hand, r d4>{u) r 4>'{u)du Js„ 4>{u)— fo Jc^ 4>iu) - fo where the integration of the first integral is taken with respect to the elements d(j>{u) and consequently over iS^ in the ^(w)-plane, while the integration over the second integral is with respect to du and there- fore over the circle C,, in the w-plane. The function <^(w) when developed in powers of w — a^, is of the form or, since i;6(a^) = ^o, ^{u)-$o='^{u~ a,) + ^ (« _ a,)2 + . . . On the right-hand side a number of the coefficients may vanish. Let Ak= ^ '' be the first of the coefficients that is different from zero and fc ! let k = /e„ say, be the order of the zero of the function 4>{u) — $o at the point a,,. We therefore have i)iu) -$0 = Af,{u - a^)*" -I- • • • ' and consequently kjjdu ic^u — a^ all the remaining terms having vanished. f Since r J^£^ = 2 nik, JCiiU — a^ "111 it follows that 2 k^Tci = f idd, or k^ = -i— / JSu 2 Tii Js^ dd. PEEIODIC FUNCTIONS IN GENERAL. 63 In other words, the order of zero of the function (j^iu) — fo at the point u = a^; that is, fc^ is equal to the number of circuits which the curve in the ^(u) -plane makes around fo corresponding to the circle C^ made around the point a^ by the variable u in the w-plane. The integer fc^ is at least unity. Suppose in the place of a^ another point Oq is written, and about this point let a circle Co be described with a radius so small that within and on the circumference of the circle none of the points Ui, a2, . . . , an lies, nor any of the infinities of the function. We know then that the integral Jc, 4>'(u)du ■ ^ ^ IS zero, Co0(w) - fo where the path of integration is taken over the circle Cq. We have accordingly proved * that the integral -f '■mJs d4>{u) s 4>{u) - fo' where 4>{u) is a regular function for all points on and within the interior of the contour S, indicates the number of times that the function (pin) takes the value fo within S, provided each point a^, say, at which (f>{u) takes the value $0, is counted as often as the order k^ of the zero of 4>{u) — fo it the point a^. Art. 53. Next in the place of fo take another value f i, which also lies within S^, so that the corresponding value of u lies within C^ . Then the number of circuits of the curve about fo is the same as the number of circuits about fo, since all the circuits encircle both points. It follows that ^'{u)du 'm Jsi,4>{u) — f 1 " Jc^ 2m JSi,'p{u) — fi Jci,iu) - fo 1 gJ^—]-G2(-^—] gJ—^—]=p{u), \u — ail \u - 02/ \M — Om/ 4>{u) - fo where P{u) is a power series with positive integral exponents. The function P(w) cannot reduce to a constant, for then 4>{u) would be a rational function and the point w = 00 would not be an essential singu- larity. It follows that the absolute value of the above difference exceeds any limit if we take values of u sufficiently distant from the origin. We may therefore by taking u sufficiently large make ^{u) — fo as small as we wish. If further the point f 1 is taken very near the point $0, the value ci is certainly taken by the function <^(w) as u is made to increase. Hence the function ^(m) takes the value f 1 at least m times in the finite portion of the plane and another time towards infinity. Since by hypothesis 4>{u) is indeterminate for m = 00 , it appears that <^(w) — f is zero for some value of u such that m < 00. Call this value am+i. By repeating the above process it may be shown that we may find such values of the function 4>(u) which may be taken arbitrarily often by that function. Art. 54. We may derive the above results in a somewhat more explicit manner by means of our eliminant equation /i (^■l)=°- We have excluded as being singular all values of the function f = ^(w) which satisfy the equation dfi?, n _ * See Weierstrass, Zur Theorie der eindeutigen analytischen Functionen, Werke, Bd. II, pp. 77 et seq.; Weierstrass, Zur Functionenlehre, pp. 1 et seq.; Hermite, Sur quelques points de la thiorie des fonctions, Crelle, Bd. 91, and " Cours," U>c. cit., p. 98; Mittag- Leffler, Sur la representation analytique, etc., Ada Math., Bd. IV, p. 8. PERIODIC FUNCTIONS IN GENERAL. 66 In the present discussion we shall also exclude the roots of the equation /(f, 0) = 0. In other words, the function f = ^(m) is not allowed to take those values of u which make f ' = (f>' (m) = 0. If we denote by fo any finite value that (j){u) can take, then all the points at which (f>{u) can take this value fo are simple roots of the equation {u) — f o = 0; for this difference can only become infinitesimally small of the first order since iu) = f + ^ (« - Wo) + ^ (w - uo)^ +■ ■ ■ , and by hypothesis f o' t^ 0. It follows that the quantities oi, a2, . . . , «„ above are simple roots of the equation <^(u) — f o = 0, and consequently '■J 2m J 4>{u) — fo if the integration is taken over a closed curve in the ^(m) -plane that corresponds to a circle made by u about any of the points ai, 02, . . . , am- We also saw that the above integral indicates the number of circuits made by the function (f>{u) about fo in the (^(M)-plane. As this integral equals unity, we see that there is one circuit made in the positive direc- tion about f corresponding to the circle made in the w-plane about any one of the points a. All values f 1 which belong to the surface included by the circuit about fo are therefore taken once by the function 0(zt) if u takes all values within the corresponding circle C^ about a„. We describe about $0 as center a circle C with so small a radius that it lies totally within the above circuit S^ about f o- We shall show that every value 61 within this circle is taken once and only once by the function {u) when u takes all possible 'values within the circle C^. We saw that the integral d(f>{u) — f 2-KiJc 'c 4>{u) - $1' where ^(w) is regular on and within the contour C, is equal to the number of points at which the value f 1 is taken within C, provided each point is counted as often as the order of the zero of {u) 2m. J {u) -fo' where 4>{u) takes the value f q at the points u = a^, (I2, . . . , flm- By hypothesis 4>{u) — f is zero of the first order on each of these points. By Laurent's Theorem we may develop in the neighborhood of each of these points; and, if the first term of the development in the neighborhood of a^ is denoted by ^ — , it is seen that u —a^ 1 "^ "^^^giu), 4>{u) — f ^ u where g{u) has the character of an integral function for all finite values of the argument. Since g{u) cannot be a constant, as otherwise 4)(u) would be a rational and not a transcendental function, it is seen by taking values of u sufficiently removed from the origin that (piu) — fo may be made arbitrarily small. Suppose that f 1 is a value of ^(m) which lies within the interior of the circle above. It is clear that for values of u sufficiently distant from the origin the function 4>{u) is equal to f 1. We have also shown that besides this value of u the function f/)(M) takes the value f 1 at m other points and consequently {u) takes the value f 1 at m + 1 points. By continuing this process it may be shown that there are an indefinite number of values ivhich do not belong to the singidar values of the function 4>{u), and which maij be taken hij (j>{u) an arbitrarily large number of times.* It follows from what we saw in Art. 38 that {u + v) = F[cf>{u), '{v)l we have seen that the region of u may be extended by anah'tic continu- ation to the whole plane without the function 4>{u) ceasing to have the character of an integral or (fractional) rational function for all values of the argument. If <^(w) has at infinity the character of an integral or (fractional) rational function, then <^(m) is a rational function of u; but if the point at infinity is an essential singularity, then 4>{u) is a ■periodic function. It may happen that all the periods may be expressed as positive or negative integral multiples of the same quantity. In this case the function is simply periodic and the quantity in question is the primitive period of the argu- ment of the function. If all the periods of a function can be expressed through integral multiples of several quantities, the function is said to be multiply periodic. The functions with two. primitive periods are called doubly periodic, the two periods constituting a primitive pair of periods. The Period-Strips. Art. 56. Consider the simple case of the exponential function e^- We shall first show that e'' + 2« = e" for all values of u. Writing u = X + iy, it is seen that gu _ gx + iy = e^(cos y -\- i sin y) = e^ cos y + ie^ sin //. If now we increase u by 2 Tzi, then y is in- creased by 2 71, and consequently gu+2ni = gx cog (y _^ 2 t:) + ie^ sin {y + 2 7r) = e^ cos y + ie-' sin y = e^ + '^ = e"- It follows that if we wish to examine the function e", then clearly we need not study r '--^ this function in the whole M-plane but only ^S^iy,m■^ , vn,(l within a strip which lies above the Z-axis pjg 4 and has the breadth 2 tt. For we see at once that to every point Uq which lies without this period-strip* there corresponds a point Ui within the strip and in such a way that the func- tion e" has the same value at uq as at u^. For example in the figure * Cf. Koenigsberger, Elliptische Funclionen, p. 210. The lines including a period- strip need not be straight, if only the difference between corresponding points is a period. 68 THEOEY OF ELLIPTIC FUNCTIONS. Suppose that p = a + i/? is an arbitrary complex quantity, and con- sider the equation e" = p = a + i/3. Let us first see whether this equation can always be solved with respect to u; and in case it is always possible to solve it, let us see how many values of u there are within the period-strip which satisfy it. We have e" = e^ cos y + ie^ siny ■= p = a + i/?, and consequently e"^ cos y = a, e^ sin !/ = /?■ It follows that (,2x^ (^2 + ^2^ or gx_ Vq,2 + ^2. Since a; is a real quantity, the positive sign is to be taken with the root. This equation determines x uniquely, since we have at once X = log ■s/a^ + ^2. To determine y, we have tan y = ^ ■ a Suppose that yo is a value of y situated between and tz which satisfies this equation (we know that there is always one such value and indeed only one). It follows also that tan (yo + t:) = tan yo- It appears then as ii yo + 7t satisfies the conditions required of y. This, however, is not the case, since we have cos (yo + t:) = - cos yo, sin (yo + n) = - sin yo, and consequently the equations e^ cos y — a, e^ sin y = P are not satisfied by the value j/o -I- n. Hence within the period-strip the equation e" = a -!- -i/? is satisfied by only one value of m — a; + iy, and this value of u is u = log Va^ + /?2 -\- iijQ^ On the outside of the period-strip, however, the equation is satisfied by an indefinite number of values of u. These values are had if we increase or diminish by integral multiples of 2 m that value of u which satisfies the equation within the period-strip, that is, if we keep x unchanged and increase or diminish the value yo by 2 n. PERIODIC FUNCTIONS EST GENERAL. 69 Art. 57. We shall next study two other simple functions, cos u and sin u. These functions may be defined through the equations cosw = i(e*" + e -'■"), sinu = — (e'" - e-'"). It follows at once that cos (m + 2 ;r) = cos u, sin (m + 2 tt) = sin u. Both functions have the period 2 tt. We may therefore limit the study of these functions to a period-strip with breadth 2 ^ measured along the lateral axis. It is evident that to every point Uq lying without this period-strip there is a corresponding point Ui within the strip at which cos u and sin u have the same values as at wq. For example in the figure cos Mo = cos (Mi -|- 6 Tt) = cos Wi, sin Wo = sin (wi + 6 tt) = sin ui. B* %;iW^ PI Uo Fig. 5. Suppose next that p is an arbitrary complex quantity, and let us see whether for the equation cos u = p there is always a solution. If there is one, there is an indefinite number. For if ui satisfies the equation, then from the above it is also satisfied by the values mi -)- 2 tt, Mi + 4 tt, • • • . We shall show that there are always two values of u within the period- strip which satisfy the equation. ■For writing cosu= i(e'" + e-'") = p, we have e''" + e-»"" = 2 p. Writing e'" = t, this equation becomes t^ -2pt + l =0. (1) From this it follows that t = p±\/p2 _ 1. We thus have two values of t = e'"- Let the corresponding values of u be Ml and M2, so that therefore ti = e'"', «2 = e'"=. It follows that we have for iui and iu2 values of the form iui = fji + ki 2ni, iU2 = 1J2 + ^2 2 Tvi, where fci and ki are positive or negative integers. 70 THEORY OF ELLIPTIC EUNCTIONS. Dividing by i we have at once Ml = — ijji + fci 27r, U2 = — iTj2 + ^2 2 71. Hence clearly there are two solutions of the equation cos u = p within the period-strip, and these solutions are different from each other. From the quadratic equation (1) it follows that We therefore have and consequently or ii •^2 = 1, ore'"'e'^''2 = 1. i{ui + U2)= (mod. 2 7ri), u-i + U2 = (mod. 2 tt). It follows that the two values of u which satisfy the equation cos u = 'p within the period-strip are such that their sum is equal to 2 tt. We may derive similar results for the function sin u. It is thus seen that the two functions cos u and sin u take any arbitrary value within the period-strip twice, while the function e" takes such a value only once within its period-strip. Art. 58. The period a of a simply periodic function f{u) is in general a complex quantity. We have fiu + a) - and if we write u that /(«) = /(0), fiu), = 0, it follows that is, the function f{u) has at the origin the same value as it has at the point a in the it- plane; and also at the points ...,3a, 2 a, a, — a, — 2 a, . . it has the same value as at the origin. We draw through the origin an arbitrary straight line OL, and through the points a, 2 a, 3 a, ..., — a, — 2 a, ... we draw lines parallel to OL. The entire w-plane is thus distributed into an indefinite number of strips. That strip which is made by OL and the straight line through + a par- allel to OL we call the initial strip. Fig. 6. PERIODIC rUNCTIONS IN GENERAL. 71 Let M be a point in any strip. There is always a point v! in the initial strip at which /(w) has the same value as at u. For if through the point u we draw a line parallel to the line that goes through the points 0, a, 2a,..., and on this line measure off distances a until we come within the initial strip and call u' the end-point of the last distance measured off, then u and u' differ only by integral multiples of a, so that the function /(w) has the same value at both points. In the above figure, for example, w = w' + 2 a, so that/(w) = f{u' + 2 o) = /(w')- Hence every value that the function can take in the w-plane is had also in each single strip. We therefore need investigate every simply periodic function only within a single period-strip. This we have done above for the simple cases of e", sin u, cos u. Akt. 59. If a represents any complex quantity, we saw in Art. 26 that a simply periodic function with a as a period may be readily formed. 2 Tci ti Such a function was e " . Consider next the series An) = X c.e - " k" -00 where the constants Ck may always be so chosen that the series is conver- gent.* It is clear that the function just written has the period a; and, since the constants Ct may be determined in different ways, it is clear that an arbitrarily large number of such functions may be formed, all of which have the period a. Such a function is ft=+ot> ,2in k 1 i:=-oo w = r — 7-7- = (h(u), say, k u where the du's are also constants. All such functions have the property that there is no essential singu- larity in the finite part of the plane and they are indeterminate for no finite value of u. For the point w = oo the exponential function is indeterminate (Art. 21), and for all other values of u it is seen that the function (l){u) is one-valued. Art. 60. Suppose that f{u) is a one-valued simply periodic function with period o = 2 w, and which has only polar singularities in the finite portion of the plane. * Cf. Briot et Bouquet, Fonctions EUiptiques, p. 161. 72 THEORY OF ELLIPTIC FUNCTIONS. If we put «u e " = t = re*«, it is seen that io) log r , 0)6 u = — ^ — I n n Hence in the ^-plane, when 6 varies from to 2 ;r, the variable t describes a circle about the origin with radius r, while in the M-pIane the variable u describes the straight line AA', where A = — icu log r and A' = — ia> log r+2oj. Further, when d varies from 2 n to 4:71, u varies from A' to A", where again A' A" = 2 (o, etc. Next if we give to t the value se**, it is seen that when t describes a circle about the origin in the <-plane with radius s, u describes the straight line BB' , where B = — ico log s and B' = — iio log s + 2 w. Pig_ 7. It follows that in the w-plane the rectangle AA'BB' corresponds to the ring included between the two circles with radii r and s in the i-plane, and corresponding to the initial period- strip in the w-plane is the entire <-plane. Further, any period-strip is, as we may say, conformally represented on the i-plane. There being an in- definite number of these strips, it is evident that to any value of t in the i-plane corresponds an infinite number of values in the w-plane differing by integral multiples of 2 oj. Suppose that the rectangle AA'BB' is taken so as not to include any of the singularities of /(m). Then if F{t) = f{u), it is seen that F{t) is regular at all points at which f{u) is regular and consequently may be expanded by Laurent's Theorem in a series of the form n = +00 71 = —00 which is convergent for all values of t situated within the ring-formed surface in the i-plane that corresponds to the rectangle AA'BB'. It also follows that for all points within this rectangle, f{u) may be expressed in a convergent series of the form n=+QO nidu f{u)= 2) cne~^; n= —00 or expressed in a Fourier's Series, n = oo /(m) = V (an cos n-u + bnskin-u], where an = Cn + C-n, bn = i{Cn — C-„). PERIODIC FUNCTIONS IN GENERAL. 73 Prof. Osgood, loc. dt., pp. 406 et seq., gives more explicitly the limits within which such series are convergent.* Art. 61. We next propose to study all those simply periodic functions which first are indeterminate for no finite value of u, which therefore in the finite portion of the plane have no essential singularity, while they are inde- terminate for u — infinity; which secondly are one-valued; and which thirdly vnthin a period-strip take a prescribed value a finite number of times. Suppose that (m) behaves within the period-strip in a similar manner as do the rational functions in the whole plane. For if m = $(w) is -a rational function of u, then (w) is one- valued and for every given value of w there is only a finite number of values of u. In Art. 63 it is shown that at the end-points of the period-strip the function has definite values. It is easy to see that the function ^(w) which we are considering must be indeterminate at infinity in the direction of the line through 0, a, 2 o, . . . (see Fig. 6). For let Mq be a point within the initial period-strip. Draw through Uq a line parallel to the line through 0, a, 2 a, ■ ■ ■ . On this line, starting from uq, we measure off distances a an indefinitely large number of times. We thus come finally to infinity and the function takes at the end of the last distance that has been laid off the value 4>{uo). Next if we start with another point Wi and proceed to infinity in the same way as before, the function will take for the infinitely distant point the value 4>{ui). Hence at infinity there appear all possible values which the function ^(m) can take, and the function is thus said to be indeterminate at infinity (cf. Art. 3). Aet. 62. Let w = ^(u) be a simply periodic function with the period a which satisfies the three postulates made above. Further, write 2 m t = e" , so that t and w have the same period a and may consequently both be considered within the same period-strip of the w-plane. Next suppose a given value is ascribed to t. Within this period-strip there is (Art. 56) one definite value of u which belongs to the prescribed value of t. If we write this value of u in the function 4>iu), then w = {u) has a definite value. It is thus shown that to every value of t there belongs a definite value of w. If next we consider not only one period-strip but the whole M-plane, then there belongs to the given value of t an infinite number of values of u, namely in each period-strip one value. And if u is one of these values then all the other values have the form u + ka, where k is a positive or negative integer. If we write all these values in 4>{u), then w = (f){u) takes always the same value, since 0(m + ak) = (j>{u). Hence * See also Henri Lebesgue, Lemons sur les series trigonomUriques. 74 THEORY OF ELLIPTIC FUNCTIONS. also when we consider the whole M-plane, for every definite value of t there is one definite value of w. Thus we have shown that w is a one- valued function of t. For a definite value of w there are after the third of the above postulates only a finite number of values of the argument u in each period-strip. Let those values of u belonging to the strip in ques- tion be Ml, U2, . . . , Urn, and let the corresponding values of t be 2ni 2ni 2 jri Ui U2 Uwi «i = e " , <2=e " , . . . , «„ = e " . There are no other values of t which belong to the given value of w; for if we extend our consideration to the whole w-plane, that is, if with the given value of w we also associate those values of u which differ from Ml, U2, . . . , Um by integral powers of a, we still have for t always one of the values ti, t2, . . . , t^. We have previously seen that to each value of t there belongs only one value of w. We now see that to every value of w there belong m values of t and therefore that t is an w-valued function of w. It follows that w and t are connected by an algebraic equation which is of the first degree in w and the mth degree in t, say, F{w, t) = 0. Solving this equation we have w = ijrit), where ijr denotes an algebraic function of t. On the other hand we saw that w was a one-valued function of t, and since one-valued algebraic functions are the rational functions, it follows 2tn u that w is a rational function oi t = e" . We have then the important theorem: Every simply periodic function <^{u) which is indeterminate for no value of u, and has an essential singularity * only at infinity, which is one-valued and within a period-strip can take an ascribed value only a finite number of 2 m w times %s a rational function of t = e '^ , where a is the period of {u). All such functions may therefore be written in the form k= m 2 in Sk — u Cke " w = ^ ^- = cj^iu) k = n , 27n k=0 where the Cj and d^ are constants. * A treatment of simply periodic functions which have essential singularities else- where than at infinity is given by Guichard, Theorie des points si.nguliers essentiels [These, Gauthier-Villars, Paris. 1883J. PERIODIC FUNCTIONS IN GENERAL. 75 There are no other simply periodic functions which have the required properties. Art. 63. We may make m and n equal in the above expression without affecting its generality. For suppose n < m. We have then to put all the d's in the denominator equal to zero from d„+i to dm- If n > m, we make the corresponding change in the numerator. It follows that all simply periodic functions belonging to the category defined above may be expressed in the form Cke " Xct<* ^ fe-O . 2jri k~m ^(u) =W = f^ -— = ^ = ^}r{t), k — u «: = 4 = where i/^ is a rational function of t. Hence the points f = ± oo , < = are not essential singularities of ■<^{t) and consequently also {u) has definite values for u = ± 00 . In other words, the end-points of the period-strips of the function (w) are not essential singularities. We may write the above equation in the form (C„ - dmw)V^ + (C„-i - dm-i w)t^-^ + ■ ■ ■ + {Co- daw) = 0, 2jn where m represents the number of values which < = e « " can take for a given value of w, or, in other words, the number of points in each period- strip at which w = 4>{u) takes a definitely prescribed value. We call m the degree or order of the simply periodic function w = (u) = "^{t), where y\r is & rational function. « Further, since e " = f, we have du a where ^l^i is also a rational function. By eliminating t from the two expressions we have the eliminant equa- tion (Art. 34) /I (-£)-«' where / denotes an integral algebraic function. In Art. 41 we said that if there existed an eliminant equation for a one-valued function w = ^{v), then <;6(w) had an algebraic addition- theorem and belonged to one of the categories of functions I. Rational function of u, or u II. Rational function of 6 ° (simply periodic), or III. Doubly periodic function. In his Cows d' Analyse a I'Ecole Polytechnique, in 1873, Herraite observed that if the equation admits a one-valued integral (that is, if w is a one- valued function of u), we may express w and -— rationally in terms of an auxiliary variable t, du if the integral w is a rational function of u, or if it is a simply periodic function of u; and that w and — - may be expressed through formulas du PERIODIC FUNCTIONS IN GENERAL. 77 which include no other irrationaUties than the square root of a polynomial of the fourth degree, if ii> is a doubly periodic function.* Art. 65. The following question arises: What further conditions must be satisfied in order that an integral of the equation f I w, — ]= 0, belong to the categori/ of functions defined in Art. 61? Such a function must, as we have already seen, be expressible as a rational function of t, say ^/^(O, and its derivative is also a rational function '^lit). If we put -— = •«, the above equation is du f(w, v) = 0. We may regard this integral algebraic equation as the equation of a curve. Strictly speaking, however, this can only be done if w and v are real quantities ; still we may speak of a curve, for the sake of a graphical representation, even if as here w and v are complex quantities. From what was shown above, if we write for w a certain rational function -^{t) and for V a rational function ■)= {v- 2Y + It follows that a/ _ 2 10 (v - 1) =0, ■^= 2(t)- 2) + v?=fd, dv and consequently w) = 0, f = 2 is the double point. PERIODIC FUNCTIONS IN GENERAL. 81 Hence, if we write w = (v — 2)t, it follows from the equation of the curve that 1 + (v- l)t^ = 0, or I, = 1 _ _ = ^j(i) and w = - t = !/»(<). The curve is therefore unicursal; and further to every value of v there belong two values of t, but of these only one can satisfy the equation for w when to v a fixed value is given. It is also seen that and consequently we have the fourth case. It follows that t = — {u + y) and w = h u + ■)■, u+ y and being a rational function of u does not belong to our category of functions. 3. Show that the integrals of the differential equation are simply periodic functions. Note that the equation fiw, v) = (v- 2)2 (v + 1) = (3 u)^ + 2f is satisfied by i; = 3(1 + e) (1+3 t^), w =3(t + f). 2 + 4 w' - 27 w" = 4. Show that the integrals of (dwV _ /dw\ \du/ \du/ are rational functions of u. [Briot et Bouquet.] 5. Show that the integrals of \du/ \du/ are simply periodic functions of u. [Briot et Bouquet.] CHAPTER IV DOUBLY PERIODIC FUNCTIONS. THEIR EXISTENCE. THE PERIODS Article 67. Returning to the exponential function e"", we know that -^ = 2 oi, say, is its period. P- The constant ji is taken real or complex and dififerent from zero or infinity. wn Write < = e"" = e " , and consider the function (j){u) = V^(0, where here ■^ is not necessarily a rational function. Draw the period-strip as in the figure and let u be any point within or on the boundaries of this strip. Let I M I be r and | 2 a» | be s, so that 2 0) se*« s = -[cos ({u) = (u + 2(d) and that 4>{u) has the character of an integral or (fractional) rational function for all points within the period- strip except the two points ± oo. We shall show (cf. Art. 62) that if ^(m) is a one-valued function of u, it is also a one-valued function of t. Let ux be a point within the period- strip. We therefore have in the neighborhood of mi {u) = ^/^(i) may take the value fo an arbitrarily large number of times. The theorem is then proved. (2) The function (j){u) may take the value co a finite number of times, say m, within the period-strip. Let the corresponding values of t be ii, t2, • ■ . , tm- In the neighborhood of any one of these points develop , ^ ^ by Laurent's Theorem. Then as in Art. 53 it is seen that the absolute value of this expression surpasses every limit for values of t as we approach one or the other (or 84 ' THEORY OF ELLIPTIC FUNCTIONS. possibly both) of the points t = or t = oo . There are then values $i, say, in the neighborhood of f o which are taken by {u) = -^{t) at least m + 1 times. By continuing this process it is shown as in Art. 38 that ^(w) must have another period 2 w' and consequently (j>{u + 2w) = (li{u), (i>{u + 2 w') = («)• Art. 69. It follows at once from the development of {u) in the neigh- borhood of Ml in the form (Art. 53) 4>{u) = G + P{U - Ml), that there are no points in the immediate vicinity of ui at which (j>{u) has the same value * (Art. 8) as it has at mi. We may therefore draw with Ml as center a circle with radius p which is so small (but of finite length) that within the circle the function ^(m) does not take the same value twice. Further, since ^(m + 2ai) = ^(m), it is evident that \2iu\ > p, where p is a finite quantity. The point in the M-plane which represents 2 w we call a period-point. Since 2 a>' is also a period-point, it is evident that (t>{u + 2w - 2 to') = {u), and as above \2o} - 2 a)' \ > p. Fig. 9. It is thus shown that the distance between two period-points is always a finite quantity. It is also evident that if we bound any arbitrary but finite portion of surface (S) in the M-plane, there are only a finite number of period-points within this surface. li A is a period-point and if B and D are the next period-points to A, then C, the other vertex of the parallelogram, is also a period-point. From what we have just seen this parallelogram has a finite area. If then there were an infinite number of period- points within (S), there would be within this area (iS) an infinite number of parallelo- grams with finite area, which is impossible. Fig. lO. * Cf. Burkhardt, Analyt. Funkt., p. 124; Forsyth, he. cit., p. 59; Osgood, Lehr- ittch der Funktionentheorie, p. 398. DOUBLY PERIODIC FUNCTIONS. 85 4>= e or 4> = We therefore have a b' -± r s Art. 70. We consider the following question: If 2 w = a and 2oj' = b are periods of the function F{u) and in the sense that they are not inte- gral multiples of one and the same primitive period, is it possible for the point b to lie on the line joining the origin and the point a? The quantities a and b may be written in the form ,■„ 5 = se^*; and consequently, if b lies upon the straight line + -K. FiK. 11. that is, the ratio - is a real quantity. The above question may consequently b be expressed as follows: Can the quotient of two periods a and b be a real quantity? Suppose this were the case and that the point b lies upon the line Oa. The quantity a is either a primitive period or it is not a primitive period. If it is not, it may be written in the form a = ma, where a is a primitive period and m an integer. We also know that | a | > p, where ,0 is a finite quantity. We measure off upon the line Oa in the direction of the point b distances a and have the points a, 2 a, ... , ka, {k + l)a, ■ • ■ . If 6 coin- cided with one of these points, for example ka, we would have b = ka, a = ma, which is contrary to our hypothesis. It follows that b must lie between two of the dis- tances measured off, say between ka and {k + l)a. Since both b and ka are periods, the distance b — ka is also a period. We therefore have \b — ka\ < I a I . (fc-H)a Fig. 12. Writing b — ka = a', we measure off this new period along the line Oa and make for a the same conclusions as we did above for b. We find that a — la' is a new period, where I is an integer. This period is such that \a -la'\ < a'. By continuing this process we come finally to periods whose absolute values are smaller than any assignable finite quantity p, which is a con- tradiction of what was proved in Art. 69. 86 THEORY OF ELLIPTIC FUNCTIONS. We have thus shown the following: // the quotient - is real, there exists a b friwitive period of which a and b are integral multiples. If a and b are two different periods, as defined at the beginning of this article, then the ratio - cannot be real, and b cannot lie upon the line Oa. b Art. 71. The above theorem is due to Jacobi (Werke, Bd. II, pp. 25, 26), who proved it as follows: Suppose first that the ratio - is rational and write - = 22^ where p2 and pi are integers that are relatively prime, o pi It follows that , a — = — = a, say, P2 Pi and consequently b = p2a and a = pia. To show that a is a period we determine two integers qi, q2, such that Pi5i + P2q2= 1- We know that there are an infinite number of solutions of this equation. Multiplying by a we have Piaqi + P2aq2 = a, or qia + 52^ = «• Thus a is composed of integral multiples of the periods a and b and is consequently a period. Consequently (Art. 70) a and b cannot be con- sidered as two different periods. Suppose next that the ratio - is real but irrational. In the theory of a continued fractions we know that if Un Un+i are consecutive convergents, then -^ ^=^^^ = = — - 1 where £ < 1. b ■ ■ r Hence if we expand - in a continued fraction and if '^ is the nth conver- gent, then is , " o Tn s » J, £« - '^t^ ^ T'2' O"" ^nb- rna = —- a dn Sn^ dn Since dn may be made indefinitely large, it follows that I 8nb — Yna I < p, where p is as small as we choose. Further, since dn and /-„ are integers, the left-hand side is a period. This contradicts what was given in Art. 69. It is thus seen that the ratio - a must be a complex quantity * (including the case of a pure imaginary). ♦ See Pringsheim, Math. Ann., Bd. 27, pp. 151-157; Falk, Acta Math., Bd. 7, pp. 197-200; W. W. Johnson, Am. Journ., Vol. 6, pp. 246-253; Fuchs, Crelle, Bd. 83, pp. 13 et seq.; M6ray, Ann. de VEcole Norn. Sup. (3), t. 1, pp. 177-184. DOUBLY PERIODIC FUNCTIONS. 87 Art. 72. We may, however, prove that if the ratio of any two periods is real it is also rational. For let 2 (02, 2 wi be any two periods whose ratio is real. The ratio — ^ may always be taken positive; for if it were 2 ^1 negative we might substitute the period — 2 W2 in the place of + 2 0^2. We lay off the periods 2a>i, 4wi, 6wi, . . . ; 2a>2, 4^2, 60)2, •• • upon the same straight line (cf. Art. 70). It is evident that 9,, _o^,, .o,, where wi is a positive or negative integer, and 2 W3 < 2 (0\. Similarly we ''^"te 4 W2 = 2 m^i + 2 (Oi, m2 being an integer, and 2 (04 < 2 a>i . It follows that Or, o^ ., — o,, Zi02 — zniiioi = za>3, 4w2 — 2m20ii = 2(1)4, 6 ai2 — 2 wisoi 1 =2^5, and consequently the quantities 2 0J3, 2 0)4, 2 wg, . . . are all periods. There are two cases possible: (1) These quantities are all different; or (2) they are not all different. Suppose that 2 C03, 2 0^4, . . . are all different, and consider the n quantities 2 (03,2 0)4, . . . , 2 aj„ + 2, to which we also add 2 wi, in all n + 1 quantities. Divide the distance between and 2 wi into n equal parts; then, since each of the quantities 2 ^3, 2(D4, . . . , 2 Wn + 2 is less than 2 wi, two of these quantities must lie within one of the n equal intervals. Let these two quantities be 2 a** and 2 lui. It is clear that 2 014 — 2 wj is also a period and less than ^• n Since n is an arbitrarily large integer, it is seen that we have here periods that are arbitrarily small, contrary to what was proved in Art. 69. It follows then that two of the above quantities must be equal (which includes now also the second case). We then have for example 2u)q+2 = 2a)p+2, so that 2 qeriods a and h there are present periods. Their number must be finite (Art. 69). Among all these periods let /? be the one whose perpendicular distance on Oa is the shortest. It is then evident that the period-parallelogram constructed on Oa and op is free from periods. Of course we have assumed that Oa is not an inte- gral multiple of another period. It is evident that j' is a period since «+/?=;'; and it is also evident that there can be no period -points within or on the boundaries of afiy. If for example k were a period-point on the side /?;-, then through X we could draw the parallel to the side 0/3 which cuts the line ^ -r^ tI Oa in 11. We would then have a period-point at [x, which con- tradicts the fact that no period- point lies on Oa. In the same way it may be shown that no period-point lies on ay. Suppose next that a period-point v lies within the triangle /J^a (Fig. 15); then by completing the parallelogram /?va// it is seen that fi is also a period- point and lies within the triangle O^a, which contradicts what we saw above. * Picard, Traiie d' Analyse, t. 2, p. 220, gives an interesting proof of this theorem; see also other proofs in Hennite's " Coun" (4™e ^d.), p. 217, and Goursat, Cours d' Analyse, t. 2, No. 314. Fig. 14. DOUBLY PERIODIC FUNCTIONS. 89 We thus see that within the entire parallelogram OPya, the sides included, there are situated no period-points except at the vertices. It is also evident that if the whole w-plane be filled with the congruent parallelograms, as indicated in Fig. 16, there is nowhere a period-point except at the ver- tices. If for example there were a period-point u in any of the parallelo- Fig. 15 Fig. 16. grams, there exists in the initial parallelogram O^ya a point u' which differs from u only by integral multiples of a period, and contrary to hypothesis there would be a period-point within the initial parallelogram. It is also evident that the vertices of all the parallelograms are period-points since they are of the form ka + Z/3, where k and I are integers. It follows that a one-valued analytic function cannot have three inde- pendent periods a,b,c; for, as we have just seen, these three quantities are expressible in the form a = ka + 1/3, b = k'a + Z'/3, c = k"a + l"j3, where the k's and Z's are integers. We have thus shown that a one-valued analytic function, which {in the neighborhood of at least one point) is developable in an ascending integral power series, cannot have more than two independent periods. We shall see later that the pairs of primitive periods may be chosen in an infinite number of different ways (see Art. 80). Art. 75. It is evident from the foregoing that it is only necessary to consider the values of a doubly periodic function {u) has everywhere the nature of an integral or a (fractional) rational function.. We shall agree that the second period lies to the left if we look from the origin toward 2 oi. (See Fig. 17). 90 THEORY OF ELLIPTIC FUNCTIONS. We may write 2 m' = T = (T + I/O, where by hypothesis \ p\ 5^ 0, since the ratio - — is not real. All points J'^' 201+2(1)' ^itijin the interior and on the sides of this period-parallelogram may be expressed in the form M = 2 to + 2 t'u)', o 2u) where S i = 1, ^ <' = L p. yj The totality of all such values of u may be considered as the analytic definition of a period-parallelogram. The vertices (except the origin) are excluded from the consideration. Further, let w = 2 moj + 2 m'u)' where m and m' are real quantities. It follows that W , ,0)' = m + TO — ; 2 (1) (o and since — is a complex quantity, - — is also complex, = a" + ip' , say. TO = ff' — '— <7- P P Since p is different from zero, the denominator does not vanish, and consequently m and to' are determinate quantities. It is thus seen that every complex quantity w may be uniquely written in the form w = 2 mw + 2 m'co', where m and to' are real quantities. Art. 76. Two points w and w' are called congruent if w - w' = 2kio + 2ho', * where k and I are integers. The fact that w is congruent to w' may be written w = w' (modd. 2 w, 2 w'); or, if no confusion can arise, w = w'. DOUBLY PERIODIC FUNCTIONS. 91 It is also clear that, when w and w' are congruent, then w — u;' is a period of the argument of the function. If we write w — 2 met) + 2 m'u)' , w' = 2nu) +2 n'u)', and if w = w' (modd. 2 oj, 2 (o'), it is evident that the quantities m and n, as also the quantities m' and n', differ only by integers, that is, m — n = integer as is also m' — n'. Art. 77. Suppose that the period-parallelogram formed on the two sides . 2 w and . . 2 w' is free from period-points. We may show analytically that all the period-points in the w-plane are composed through addition and subtraction of 2 w and 2 w'. For let \2oj\ =1. Then, since = a + ip, 2ix} it is seen that |2w'| =l'= I V(02 + <72. Further, since 2o) + 2 cj' = 2id{\ -\- a + ip), it follows that the length of one diagonal of the parallelogram is \2(o + 2oj'\ =1 \/(l + a)2 + p2, while the length of the other diagonal is \2oj' - 2a)\ = Z n/CI - (7)2 + p2. Represent by L the longest of the four sides \2(d\,\2oj'\,\2co + 2 a)' \, |2w'.-2w|. Next divide the two sides . . 2 w and . . 2 w' respectively into n equal parts, so that the period-parallelogram wUl be divided into n^ small parallelograms. The distance between any two points situated within one of the smaller parallelograms is not greater than — ■ If there are periods that cannot be expressed through integral multiples of 2w and 2 a)' and if 2 wi is such a period, we shall construct the con- gruent point which lies within the initial period-parallelogram. We may write 2^1 = 2 fiia) + 2fii'a)', where =/ii < 1 and = fii'i, 0)2, 013 are integral multiple combinations. We may assume that there is no common divisor other than unity of mi, m2, WI3. Let d be the common divisor of TO2 and W3. Of course, d = 1 when W2 and ms are relatively prime. Then, since — ^oii = ^0^2 p W3 and the right-hand side is an a d d integral combination of periods, it follows that — a^i is a period. Since m ■ ■ ■ . ^ —1 is a fraction in its lowest terms, when expressed as a continued frac- tion it may be written twi^ £ _ j. 1 d q dq' where ^ is the last convergent before the proper value. It follows that q qmi , 1 »— a>i - poji = ±-, is a period. DOUBLY PERIODIC PUNCTIOlSrS. 93 a a so that OTiOj + 7112' C02 + Ws'ws = 0. Change—^ into a continued fraction, taking— to be the last convergent Ms' s before the proper value, so that m^ _ ^ ^ I 1_. Then rw2 + sws being an integral combination of periods, is a period = u)' , say. On the other hand, ± W2 = cD2ism2' — rma') = — rai2W3' — s{mia> + ms'ioz) = — misu) — mz {ju)2 + S'to) = — m\Six> — mz'ii)'] also ± ^3 = W3(sTO2' — '■wis'), = sm2'a>3 + r(mia) + 11112102), = miroj + m-iuy' ; and ui\ = dot. Hence two periods oj, lo' exist of which io\, 102, 0J3 are integral multiple combinations.* We may conclude from the foregoing that All one-valued analytic functions are either (1) Not periodic, or (2) Sim-phi periodic, or (3) Doubli/ periodic. Tfiply or multiply periodic one-valued functions do not exist. Art. 79. We may next prove the following theorem: It is possible in an infinite number of ways to form pairs of primitive periods of a doubly periodic function. Let 2 CO, 2 ai' he a, pair of primitive periods, and suppo.se that 2w' 2 0) where o is positive, that is. We wish to form another pair of primitive periods 2 oi, 2 w' such that ■(I) > »• * Cf. Forsyth, Theory of Functions, p. 202; see also Hermite in Lacroix's Calculus, Vol. II, p. 370. 94 THEORY OF ELLIPTIC FUNCTIONS. It is evident that we must have 2 oi = 2 pw + 2 qat' , 2 S' = 2 p'w + 2 q'oj', where p, q, p', q' are integers. Further, p and q must be relatively prime, for otherwise 2 u> would be the integral multiple of a period. The integers p' and q' must also be relatively prime. It follows that „ 2 q'u) — 2 qu)' P9 - 9? Since 2 oi and 2 53' are to be a pair of primitive periods, the period 2 w must be expressible integrally through them. It follows that ^ and 2 — n' - qp' pq' - qp must be integers. We further have 2aj' = —2 — , and consequently pq - qp f and 2 are integers. pq' - qp' pq' - qp' If we put pq' — qp' = A, it is seen that the four quantities above are integers, if A = ±1. For suppose that A is different from ± L It would then follow, since -2- and -2- are to be integers, that q and p have a common divisor other than unity, which is contrary to the hypothesis. The next question is: Are both values A = +1 and A = — 1 admissiblef We required that Rf^Wo and r(^)>0. We have P' + 9- 2 53' _ 2 p'w + 2 q'(x)' ^ ^ ^ oj 2 uji i(2 po) + 2 qoj') i(p + ,^) w Since — = <7 + ip, it follows that 2w' _ f + q'{ a+ ip) _ - (jo' + q' a)qp + (p + qaWp . ^-r y 2 wi t[p + q{a + ip)] (p + g(7)2 + g2^2 L J, and consequently pq' - qp' I2_^\^ \2 m'i) (p + qaY + q^p^ DOUBLY PERIODIC FUNCTIONS. 95 As p is positive by hypothesis, we must have pg' — qp' -positive in order to fulfill the condition It follows then that \2 u)%/ A = pq' - qp' = + 1. Art. 80. Using the condition just written, we may form an arbitrary number of equivalent pairs of primitive periods as soon as one such pair is known.* The transition from one pair of periods to another is known as a. trans- formation, and the quantity A = pg' — qp' is called the degree of the transformation. We have here to consider transformations of the first degree. The quantity A gives the measure of the surface-area of the second period-parallelogram, if that of the first is denoted by unity. Hence all primitive period-parallelograms have the same area, for if 2w = X + iy and 2 5' = a:' -|- iy', the area of the corresponding parallelogram is ± ixy' - yx'). If further, 2 w = f + IT) and 2 w' = $' -I- ii)', the area of the corresponding p)eriod-parallelogram is ± (fj?' - ?'v)- It follows that, if 2^ = 2pw + 2qm' and 2 5' = 2 p'cj + 2 q'a)', ,x = p? + q$', _, {x' = p'$ + q'^', y = pi) + qr]'; and y' = p'v + 3'V; X, y x',y' = ± P, 9 P',q' then and consequently But here pq' — qp' = 1. Hence a primitive period-parallelogram is not unique. The linear substitution 2 5* = 2 pw + 2 qu)', 2u)' = 2p'a) + 2 q'w' is denoted by P, 9 "I y, 9'J' ♦ Cf. Briot et Bouquet, Fonctions EUiptiques, pp. 234, 235, and pp. 268 et seq. 96 THEOEY OF ELLIPTIC FUNCTIONS. One of the substitutions which satisfies the condition A = pg' - qv' = 1 IS In this case we have lU- 2u) =2(1)', 2u)' = -2(i>. A second substitution which satisfies the same condition is C;°} or 2 a* = 2 w, 2Z)' = 2uj ^2o)'. It may be shown that every linear substitution with integral elements and determinant A = 1 may be formed by a finite number of repetitions of these two substitutions. Art. 81. The question arises* whether among the infinite number of equivalent pairs of periods there are those to which preference should be given. There are one, two, and sometimes three pairs of primitive periods which may be chosen in preference to the others. One of the periods in these selected pairs of periods has the smallest absolute value among all the periods. It is clear that such a period exists; indeed there are two such periods differing only in sign. Taking this smallest period as a radius we describe a circle about the origin. Within this circle no period can be situated, but upon the periphery there lie at least two periods (180 degrees from each other). It is also seen that the surfaces of the two circles drawn about these period-points and having the same radii as the first circle must be free of periods. Hence besides the period- points P and P' none can be situated on any part of the periphery of the first circle except the shaded arcs P1P2 and P3P4. On these arcs there may be two periods differing by 180 degrees p- ^g and possibly four periods. In the last ease the period-points must lie at the four points of intersection of the circles, viz., Pi, P2, P3 and P4, so that there may lie upon the first circle two, four, or at most six period-points; and consequently the period of smallest absolute value is either 2-ply, 4-ply, or 6-ply determined. * Cf. Burkhardt, Elliptische Funktionen, p. 194. DOUBLY PERIODIC FUNCTIONS. 97 Denote any one of these six periods by 2 (o, which we use as one of the selected pair of primitive periods. We shall impose a further condition upon the other period of this selected pair. The second period 2a)' must lie to the left of . . 2w. We also know that | 2 w' | > | 2 oi (. We cut a strip out of the plane as indicated in the figure. The second period-point may always be made to lie within this strip; for if it were situated without the strip, by the addition of 2wkt>, where to is a positive or negative integer, it can be caused to lie within the strip, but it does not fall within that part of the strip which belongs to the two circles. Hence the triangle . . 2a} . . 2w' has only acute angles, the right angle being a limituig case. _ 2a/_ 2u) We write + ^/?, Fig. 19. where a^ + ji^ Owing to the substitution we may so choose 2 aj' that It follows that p> i Vs. If further we write h = q it is clear that Sa < 1. a /act which we shall find to be very important in the development of the Theta-f unctions (Chapter X). Fig. 20. Art. 82. We have interpreted the equa- tion 1 = pq' — qp' = 1 as denoting that the parallelograms formed on pairs of primitive periods have the same area. Let 2 S, 2 S' be a pair of primitive periods. The quan- tities 2 w and 2 w' determine a triangle, and all such period-triangles have the same area. Let I 2 oi I = Z. 2S' Then if =~ = a + i^, the area of the 2 ai triangle is ^ and that of the period-parallelogram is pP. This quan- tity being constant for all equivalent primitive pairs of periods, we have const. /J P 98 THEORY OF ELLIPTIC FUNCTIONS. From this it is seen that /? is a maximum when Z is a minimum. If then /? is to have its greatest value, we must choose the first period 2 S so that it has the smallest possible value. If the ratio of the periods is a pure imaginary, then a = and /3 S 1 . In this case \h\ = |e^"| S e-l^" < ^V- EXAMPLE If oi,, oj^ and 0^3 are periods of ^(m) and if 29^3 = 17ft>i + 11 W2, show that oj' = 3 o), + 2(jL>2 — 5 Wj are a pair of primitive periods of (u). [Forsyth.] CHAPTER V CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS Hermite's Intermediary Functions. The Eliminant Equation. Article 83. Having established the existence of the doubly periodic functions, we shall next show how to construct such functions and natur- ally the simplest ones possible. The expression i = + 00 , 2 «■ Xk u k= -00 is a simply periodic function which can be developed in positive ascending powers of M — mq, and which is not indeterminate or infinite for any finite value of u, provided the constants Ak have been suitably chosen. A function which is developable in a convergent power series in ascending positive integral powers and in the finite portion of the plane nowhere becomes infinite or indeterminate is an integral transcendental function (see Chapter I). Such a function is {u) above. The question is asked : Is there an integral transcendental function which has besides the period a another period bf Liouville [Crelle's Journ., Bd. 88, p. 277] answered this question by prov- ing the following theorem: An integral transcendental function which is doiibly periodic is a constant. We need only study the function within the first or initial period paral- lelogram, i.e., the one which has the origin as a vertex and which lies to the right of this vertex. For every point u of the plane is congruent to a point u' in the first parallelogram, that is, u = u' + ka + lb, where k and I are integers. The function has therefore the same value at ■u and at u'. An integral transcendental function becomes infinite for no finite value of the argument. Consequently the function remains finite in the first period-parallelogram and therefore the absolute value of the function in this parallelogram is smaller than a certain finite quantity M. Further, since the function at points without the first period-parallelogram always takes such values as it has icithin this parallelogram, it remains in the 99 100 THEORY OF ELLIPTIC FUNCTIONS. whole plane less in absolute value than M. But an integral transcendental function , . , , 9,^1, g(u) = ao + aiu + azu^ + a-'w* + • ■ ■ which remains finite for arbitrarily large values of m is a constant, since g{u) can remain finite only if ai = = a2 = ^^3 = " " ' ■ The following is a more direct proof of Liouville's Theorem. If i-+0O 2jri 4>{w)= 2^ A^e " ", the condition that (w + 6) = 4)(m) and consequently h _ ;t^6 Since - is an irrational quantity, e " 5^ 1, and therefore As = (A; = ± 1, ± 2, . . . ). It follows that $(m) = Ao\ and consequently there is no integral transcendental function which is doubly periodic. Art. 84. We shall now seek to form a doubly periodic function which has the character of a rational function and which may therefore be written in the form cE.(m) where (u) and '^[''(m) are integral transcendental functions. We may write jt=+oo 2n k-+x 2 m ^{u)= X A^e " ", ^(w)= 2 B,e « ", k" -00 A:=-oo where .4 4 and B^ are constants, so chosen that the two series are convergent. Since (m) and '^{u) both have the period a, their quotient 0(m) has the period a. We therefore have to bring it about that the quotient J^^' has also the period h. ^^' We must so determine the functions («) and '^(w) that 4>(u + 6)= T{u) 4>(w), where T{u) is a function of m. If we succeed in this, then or 9S(m) has also the period h. CONSTRUCTION OF DOUBLY PEEIODIC FUNCTIONS. 101 It will be advantageous to make our choice so that ^{u + b) has the same zero as 4>(it), and consequently (w) does not vanish or become infinite for any finite value of u. This will be effected if we write Tiu)= gCW, where G(u) is an integral function in u. We have then to seek a function 4>(u) and a function '^{u) so that (M + o)= <|)(w), ^{u + a)= ^(li), $(m + b)= eG(«>(M), ^{u + b)= e<^(«)-«/r(M). We shall next bring about a further limitation in that we determine (w) and ^(u) so that G{u) is an integral function of the first degree in u. We will then have 4>(m + a)= (m), ^(m + a) = ■*■((*), $(m + 6)= e^'^+^'^Cw), '^(u + b)= e'^'+'^'i^iu), where A and /j. are constants which are at our disposal. We shall see that there is an infinite number of such functions. Hermite * called them " dovbly periodic functions of the third sort (espece)." If (u + a)= y(w) and 4>(m + 6)= y'<3>(w), where v and v' are con- stants, one or both being different from unity, then 4>(m) is a doubly peri- o'dic function of the second sort; and if y = 1 = v' we have the doubly periodic functions of the first sort, which are properly the doubly periodic functions. Note that the word sort (espece) used here in no manner connects a doubly periodic function of the first sort, say, with an elliptic integral of the first kind (espece) , a term which will be employed later. * Hermite (Lettre a Jacobi; Hermite's (Euvres, 1. 1, p. 18) first considered these func- tions. Briot and Bouquet, Fonctions Elliptiques, p. 236, called them "intermediary functions." They are sometimes called quasi- or pseudo-periodic. See also Hermite, " Cours" (4™® 6d.), pp. 227-234; Hermite, Note sur la theorie des fonctions in Lacroix, Calcul (6™* 6d.), t. 2, p. 384, which is reprinted in Hermite's (Euvres, t. 2, p. 125; Hermite, Note sur la theorie des fonctions elliptiques, Camb. and Dubl. Math. Journ., Yol. Ill (1848); Hermite, OEjvres, p. 75 of Vol. I; Crelle, Bd. 100; Comptes Rendus (1861), t. 53, pp. 214-228, and Comptes Rendus (1862), t. 55, pp. 11-18, 85-91; Biehler, These, 1879; Painlev^, Ann. de la Faculte des Sciences de Toulouse, 1888; Appell, Ann. de I'Ecole Normale, 3d Series, Vols. I, II, III and V; Picard, Comptes Rendus, 21 Mars, 1881. The Berlin lectures of the late Prof. L. Fuchs have also been of service in the preparation of this Chapter. 102 THEORY OF ELLIPTIC FUNCTIONS. Art. 85. From the formula k= +00 ;^2ira" k= —00 it follows at once that k— (u + 6) $(u+6)= 2) ^t*^ " fc= -00 .6 If with Hermite we write Q = e ", it follows that «:-+oo 2 ;ii 4>(M + 6)= 2) ^*Q"e " "• (1) A:= —00 On the other hand we had (m + 6)= e^"+''(M). If on the right-hand side we write for $(m) its value and put X = g, we have k- +00 2 iti ,, . ^iu + 6) =6" ;^ Ake" . (2) In this formula k is an integer and we shall choose the quantities so that g is also an integer. 2ni u If further we write t = e" and equate like powers of t in formulas (1) and (2), we have for the determination of the A's the formula AmQ^"'= e>^Am-o. If we take the logarithms of both sides of this equation, we have fi + log Am-g = log ^„ + 2 TO log Q + n 2 ni, (i) where on the right n 2 ni has been added, since the logarithm is an infinitely multiple-valued function. We shall further write b b ni- V II = ni-v, so that e" = e " = Q", a or fi = V log Q. Since the constant (i is perfectly arbitrary, v is also arbitrary. It follows directly from (i) that logA -log.4„. ^^^_ ^ +!L^. (ii) logQ logQ We note that m, n, and g are integers, and we seek the most general solution of this equation. If for brevity we put ~—~ = c^, the equation (ii) becomes logQ o , 2 TTl ..... Cm-g -Cm=2m— v + n • (m) logQ CONSTRUCTION OP DOUBLY PERIODIC FUNCTIONS. 103 To determine first a particular solution of this equation, write Cj. = afc2 + ^k, where the constants a and /? are to be determined. Since Cm-g = a{m — gY + /?(to — g) and we have from equation (iii) — 2 amg + ao^ — Bg = 2m,— v + n , logQ Since this equation must be satisfied for every value of m, the coefficients of like powers of m on either side of it must be equal. We thus have - 2ag =2, ag^ - ^g ^-v + n -^ ; and consequently 2 m - g + V - n - a=-l ^ = l^g^. g g We may give to the arbitrary constant v a value and we shall write V = g. It follows at once that 1 o — n2 m a = . a = — - — — -—• g g^ogQ These values written in the formula c/c = ak^ + ^k will give the particular solution of the equation Cm-g -Cm = 2m-V + H- • (Ul) logQ We may write the general solution in the form Cm = am? + ^m + Cm, where Cm is a function of m. Writing for Cm its value, we have logj4^ _ ^^2 + j3m + Cm, or logQ /) _ ^am^ log Q + (.0m + C„) log Q Writing for a its value from above and putting /3m + Cm = Dm, we have - — 10 - — Am =e " ° gO-iogO = Q e^"'°Be 104 THEORY OF ELLIPTIC FUNCTIONS. Finally, putting Dm log Q = log Bm, we have _ w? where Bm is a new function of m. Here, indeed, we have not deter- mined Am, since Bm is not determined; but we have found a suitable form for Am- Returning to the original equation AmQ'^'^= cAm-g, it follows that _ m_2 _ (m-gV Q^'^Q 'Bm=Q'Q ' Bm-g, or Bm-g = Bm, where ra and g are integers. The integer g being arbitrary we shall write g = — k, where fc is a positive integer. We thus have Bm+k — Bm- It follows at once that Bk = Bo, Bk+i = Bi, Bk+2 = B2, ■B2A-1 = Bk-i, B2k = Bk = Bq. We thus see that the constants Bq, Bi, B2, - - • , B^-i repeat themselves but are otherwise quite arbitrary. It has thus been shown that the function 7n=+oo wi2„2)n' ^{u) = ^ BmQ^e « m= — w satisfies the functional equations 4>(ii + a) = (m), (m +b) = e " (m). This function (m) is the most general integral transcendental function which satisfies these two equations. It contains the k arbitrary constants ■Bo, Bi, B2, - - . , Bk-i- Art. 86. It remains to be proved that the series through which the function $(m) has been expressed is convergent. Instead of the con- vergence of the series itself, we may consider the convergence of the series of moduli of the single terms, that is, of the series XI I I m^ 2ffi iBmlQr e~'' CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 105 In this series the coefficients | Bq |; | ^i | , • • ■ , | Bk-i \ repeat them- selves. We collect all those terms which contain | Bq | and likewise all those which contain | Bi |, etc., and take | Bq |, | Si |, • • • on the out- side of the summation signs. We thus distribute the above series into k new series of which each is multiplied by one of the quantities | S |. If each of these series is convergent, then the product of each one of them by the corresponding | B | is convergent and therefore also the sum of the products, that is, the above series of moduli, is convergent. If this series of moduli is convergent, it follows also a fortiori that the series which represents (u) is convergent. It therefore remains to prove the convergence of the k single series. To do this we may make use of the following well-known criterion of con- vergence: Suppose we have given a series composed solely of positive terms Vi + V2 + ■ ■ ■ + Vm + This series is convergent if the mth root of the mth term, that is, V^m, tends towards a definite value which is less than unity, with increasing values of m. For if "n/v™ < p < 1, then is Vm < p"" < 1, and v^+i < p""-^^ < 1, etc., so that TiVm is less than a geometrical series in which p < 1. The general term in the above series is j m- I I 2jn \m and the mth root of this quantity is The second of the above factors has for all finite values of u a definite value which is independent of m. For the other factor we may write I ml .b — TTt - \Q * = |e " If we put - = a + i/? (where /? 7^ 0, since -is not real), we have a 1 ■ b ■ r, m - = ma — np a .b I m- < I and Ie "1=1 It follows that ptrin = « ""^ IqI"' =e-'"~K If /? is a positive quantity, the quantity e '' becomes arbitrarily small for increasing values of m, which proves the convergence of each of the above series. 106 THEORY OF ELLIPTIC FUNCTIONS. The condition that /3 be positive need not be regarded as a limitation. For if p is negative, we form the quotient a _ 1 ^ a — ip ^ a , — /? ^• 6 " a + i/? ~ a2 + ^2 a2 + ^2 „2 +^2 ' where the coefficient of i on the right is positive. We may therefore write - = a' + ip', where /?' is positive. If then the coefficient of z in - is nega- tive, we interchange h and a in the whole investigation and thus form a function $(m) of such characteristics that *(m + 6) = 4>(m), -~(.2u+a) ^{u + a) = e * (M). The function ^{u) is defined by the series m=+oo ?n^ 2 jrt m= — 00 .a where Qo = e • Art. 87. If k = \, we have (Art. 85) (w + a) = 4)(m), - — (2« +6) (m), which equations are satisfied by the series 771= +00 771^ 2 Jn _— . -- 771 U m= —CO where B„+i = B„. In this case, since the B's are all equal, we may write 771= +00 2 TTl (w) = Bo X Q-'e"^". m= —00 This is Hermite's function X(m), when we make Bq = 1. It is the simplest intermediary function and is called the Chi-function. For k — 2, we have 4>(m + a) = $(m), -^(2«+6) ^iCm + 6) = e " $(u) m= +00 m^ 2ja and (w)= X BrnQ^e"" -^ " m=-oo 5m+2 = Bm- CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 107 In this case (m) contains the two arbitrary constants Bq and Bi. We may therefore write $(m) = Bo^oiu) + Bi$i(m), where 4>o(u) = ^ Q^^'e " , (m = 2 //) /!= —00 *i(w) = X Q ^" , (m = 2;.+ l). u= — 00 The constants Bq and Bi being arbitrary, we choose Bq = 1, and Si = 0, and thus have a particular solution $o(w) of the functional equations; writing Bq = and Bi = 1 we have another particular solution $i(m). The functions $o(w) and $i(m) are the remarkable functions first intro- duced into analysis by Jacobi and known as the Jacobi Theta-f unctions.* Jacobi employed a somewhat different notation, which we will have, if we write Q2 = e «= q. It follows then that u =■ -»- 00 4 iri Iiu Jl— —00 (i=+oo / 2 ft+i y 2_jr « 12, ^IV 2«(2.+ l) Jacobi further wrote instead of a the quantity 4 K, and instead of h the quantity 2iK', and consequently q = e The above functions become *o(w) = @i(w) = X ^''^ ^' /!= —00 -00 ♦ In his memorial address Lejeune-Dirichlet eulogized Jacobi as follows (see Jacobi, Ges. Werke, I, p. 14) : " Bedenkt man, dass die neue Function jetzt das ?anze Gebiet der elliptischen Transcendenten beherrscht, dass Jacobi aus ihren Eigenschaften wichtige Theoreme der hohreren Arithmetik abgeleitet hat, und dass sie eine wesent- liche RoUe in vielen Anwendungen spielt, von welchen hier nur die vermittelst dieser Transcendenten gegebene Darstellung der Rotationsbewegung erwahnt werden mag, so wird man dieser Function die nachste Stelle nach den langst in die Wissenschaft aufgenommenen Elementartranscendenten einraumen miissen." 108 THEORY OF ELLIPTIC FUNCTIONS. Art. 88. If we put o(w) it IS seen that (i>{u + a) = -— 4 f = ■. ) ' = 9W, 4>o(w + a) {u) is a doubly periodic function hav- ing the periods a and b. This function ^(m) cannot be a constant, for if 4>o(m) then i(tt)= Co(w), which is not true since o(M) is developable in the u even powers of e " while ^i(u) is developable in the odd powers. The functions (m) which have been considered do not become infinite or indeterminate for any finite value of u; they have the character of integral functions and may be developed in power series which proceed in positive integral powers. They are integral transcendental functions (Chapter I). Art. 89. Historical. — Abel (CEuvres, Sylow and Lie edition, T. I, p. 263 and p. 518, 1827-1830) showed that the elliptic functions considered as the inverse of the elliptic integrals could be expressed as the quotient of infinite products. These infinite products Jacobi [Gesam. Werke, Bd. I, p. 198, 1829] introduced into analysis under the name of Theta-f unctions, and by expanding them in infinite series (see Chapter X) he discovered many new properties other than those which had been previously employed in mathe- matical physics by French mathematicians, notably by Foisson and Fourier {Sur la Thcorie de la Chaleur). Jacobi [Fund. Nova, p. 45; Werke, Bd. I, p. 497] founded the whole theory of the elliptic functions upon these new transcendents, which made the elliptic functions remarkably simple, as well as their application, for example, to rotary motion, the swing of the pendulum, and innumerable problems of physics and mechanics; also through these Theta-f unctions the realms of geometry were essentially widened and many abstract properties of the theory of numbers were revealed in a new light. In the present treatise these Theta-functions are to be regarded as the fundamental elements. CONSTEUCTION OF DOUBLY PERIODIC FUNCTIONS. 109 Art. 90. The intermediary functions of the kth order or degree. — It is clear that we may write the function (w) of Art. 85 in the form (w)= Bo^o(w)+ 5ii(w)+ • • • + Bk-i^k-i{u), where (.1 = 0, 1, . . . , A; - 1). Such functions, for reasons given in Art. 92, are said to be of the fcth degree or order. We shall next prove that there are k {and not more than k) independent intermediary functions of the kth order. Suppose that we have k + 1 such functions which satisfy the functional equations 4>(w + a)= 4>(w), ink (2u + b) («) . (m + h)= e - These functions are therefore of the form (a = 1, 2, . . . , A; + 1). We have at once, if we take p( = l) as the coefficient of ■^„(w), = - p^a(.u)+Bo^"^^o(u)+ B/"^4>i(m)+ • • ■ +B,<_>,_i(m) (a = 1, 2 k + 1). In these k + 1 equations we may consider p, $o, "^i- • ■ • , ^k-i as unknown quantities; then, since the equations are homogenous, either their determinant must be zero, or all the unknown quantities are zero. The latter cannot be the case, since p = 1. We must therefore have ^l(M), Bo ^2(m), Bo (1) (2) R '1' Ok-1 R (2) Dk-1 r(A+1) = 0. ^kMu), Bo<*^'', . If this determinant is expanded with reference to the terms of the first column, we have Ci^i(m)+ C2^2(m)+ • • ■ + Cfc + ,^l'i+l(M)= 0, where the Cs are the constant minors (sub-determinants). We thus see that there exists a linear homogeneous equation with con- stant coefficients among any fc -t- 1 intermediary functions of the fcth degree. 110 THEOEY OF ELLIPTIC FUNCTIONS. Art. 91. The zeros. — In the initial period-parallelogram there is a congruent point u' corresponding to any point u in the w-plane, such that u = u' + ka + (lb, where }. and fi are integers. We have (w)= («' + fih + Xa)= «!>( U' + [lb), and further, *(w + b) = -- e (M + 26) = = e - — (2«+36) + h), - — (.2u+5b) <|)(M + 3 6)=e " (w + 2 6), -— [2m+(2p-1)!>)^ {u + fib)=e " ^{u + (fi- 1) b). When these equations are multiplied together, we have -—[2iiu+b(l+3 + S+ ■ •+2,1-1)] <^{u + fib)= e " ^(m), or ^(u + fib)= e " $(m). It follows that -— (2,iU'+«2i«) (m) = (w' + /i6 + -ia) = e " (u'). Since the exponential factor is different from zero, it follows that (m) can only vanish when (w') equals zero. We may therefore limit our- selves to the discussion of («) within the initial period-parallelo- gram. Since an integral transcendental function can have only a finite number of zeros* (Art. 8) within a finite surface-area, it follows that there are only a finite number of zeros of <3'(w) within the period-parallelogram. This parallelogram may be constructed in different ways. If from any point Q in the w-plane we measure off both in length and direction the quantities a and b and draw parallels through the end-points, we have a period- parallelogram of the function with the periods a and b. If starting with this parallelogram we cover the plane with similar parallelograms, it is seen that the plane is differently divided from what it was in the former distribution of parallelograms, where the first initial parallelogram had the origin as one of the vertices. * Cf . Forsyth, Theory of Functions, p. 62. CONSTRUCTION OF DOUBLY PEEIODIC FUNCTIONS. Ill b' g/ r.' / d/ p/ / / Fig. 21. It will be convenient for the following investigation if the initial period- parallelogram is so situated that there are no zeros of the function upon its sides. To effect this let QA'C'B' be any period-parallelogram. As there can be only a finite number b r: of zeros of (w) within this parallelo- gram, it is evident that upon the line QB' there is a point D such that there is no zero of the function on the line DE which is drawn parallel to QA' = a. Similarly there will be a point F on the line QA' such that there is no zero of the function on the line FG drawn parallel to QB' = b. The lines DE and FG intersect in a point P, say. We take P as the vertex of a new parallelogram PACB. We shall see that there are no zeros of the func- tion 4> (m) on the sides of this parallelogram. On the side PE there is by construction no zero. Also upon EA there can be none owing to the relation (u + a) = (w), so that (m) takes the same values upon EA as upon DP. Upon PG likewise by construction there is no zero of the function (w) and upon GB there is also none, since (^(u + b) =e " (m). Hence upon the sides PA and PB there are no zeros of the function. It follows also on account of the two functional equations just written that there are no zeros on the sides AC and BC. Art. 92. We may now apply the following well-known theorem of Cauchy:* If a function (w) within a definite region, boundaries included, is everywhere one-valued, finite and continuous, and if N denotes the number of zeros within this region, then is N IniJ 4>(m) du. where the integration is to be taken over the boundaries of this region and in the direc- tion such that the region is always to the left. This theorem is applicable to our func- tion (w) which is infinite for no finite Fig. 22. value of u. The region in question is the period-parallelogram PACB. We therefore have, if we write ■>^{u) = ^"^ 2mN = f ■f{u)du+ f ■f{u)du+ j ■^{u)du+ j ■ylr{u)du. JpA J AC JCB JbP * Cf. Forsyth, loc. cit., p. 63; Osgood, loc. eit., p. 282. Professor Osgood demands that the curve be analytic (regular) for all points within the boundaries and continuous for all points of the boundaries. See the theorem at the end of Art. 52. 112 THEORY 0¥ ELLIPTIC FUNCTIONS. AVe may transform these integrals of the complex variable into integrals of a real variable i. Let u take the value p at P; then, since PA = a, we may write all the values which u can take on this portion of line PA in the form u = p + at, where S t = I. It follows that J'>jr(u)du= a I y(r{p + at)dt. PA Jo Further, the variable u has at A the value p + a, and since AC = b, we have Jyjr{u)du= b i ■^{p+ a + bt)dL AC Jo Similarly u has at B the value p + b, and therefore all values of u on CB have the form p + b + at, and consequently / ir{u)du = a / ■\lf(p+ b + at)dt = - a / ^{p+ b + at)dt. J AC J\ Jq Finally we have in the same manner / f'{u)du=bl fip + bt)dt= - I ^{p + bt)dt. J BP J I Jo It is thus seen that 2 -iN = \dt. a j yjrip + at) - yjr{p + b+ at)iit + b j ■f{p+ a + bt)- -f (p + 60 c Further, since 4>(m + o) = 4>(w) and (m + 6) = e " <^ follows at once through logarithmic differentiation that fiu+ a) =ir{u) and -«/r(M + 6) =^|>-{u) - ^^. a These values substituted in the above integrals give 2KiN = a r—-dt, Jo « or N ^ k. We thus see* that the number of zeros of the function (u) which lie within the period-parallelogram is equal to the integer k which appears in the second functional equation which («) satisfies. * Cf . Hermite, " Cours " [4th ed.], p. 224. CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 113 In algebra we say an integral rational function which vanishes for k values of u in the w-plane is of the fcth degree. In a corresponding manner we say of our function (w), it is of the fcth degree or order, because it vanishes at k points within the period-parallelogram. Art. 93. For fc = 1, we had in Art. 87 $(m + a) = (m), --(2«+6) $(w + 6) = e " *(m). After the theorem just proved we know that there is one and only one zero of the function (w) which satisfies these two functional equations in the period-parallelogram. We shall seek this zero in the initial period- parallelogram. We had 771= +x 2jn *(w) = X Q'"'e'"^" = X(w). m= —CO Writing m = — (n + 1) in this formula, it becomes 71 = + CO' 2 jri _ -u(n+l) n= — 00 e " e ° n= —00 1= -00 1=4-00 _ 71= -CO If we give to u the value ^ "^ in the above formula, it becomes - [n6-(n+l)a] n-+oo n-+« n=-oo n=-oo If we also write " "^ ^ in the original expression for X(m), it becomes m=+co m=+co X( " "^ ^ ) = X Q'"'Q'"e"'''" = + X Q"''"""(- 1)" \ ^ J m= -00 m= -co Comparing the two expressions thus obtained for X P j, it is seen that they differ from each other only in sign, and consequently it necessarily follows that x(«-±i) = o. 114 THEORY OF ELLIPTIC FUNCTIONS. Since the zero of the intermediary function $(w) of the first order, i.e., of X(w), is the intersection of the diagonals of the initial period-parallelogram, it follows that X(m) =0 at all the intersections of the diagonals of the parallelograms which are congruent to this initial parallelogram. Remark. — The question might be raised as to whether there were zeros of X(m) on the boundaries of the initial period-parallelogram. We saw in Art. 91 that it was always possible so to place the period-parallel- ogram that its boundaries were free from zeros. If, however, we con- sider as we do here a definite period-parallelogram, viz., the one where the origin is the vertex and which lies to the right of the origin, we do not a priori know that there is no zero of X(m) upon its boundaries. Suppose that the period-parallelogram which has m = p as one of its vertices is so drawn that there are no zeros upon its boundaries. There is one zero within the period-parallelogram, since 4>(m) is of the first degree. The value of u at this point may be expressed in the form p + Xa + vb, a Fig. 23. where X and v are proper fractions. If now we cover the w-plane with congruent parallelograms, there does not lie a zero of X(m) on any of the boundaries of these paral- lelograms, and within each parallelogram there is always one and only one zero. Since all the zeros are congruent one to the other and since from above '^ is one of them, we must have 2 p + ka + vh == \- ga -\- lb, where g and I are integers. Every zero of X(m) may be expressed in this form, and therefore also the zero which we suppose may lie upon one of the boundaries of the initial period-parallelogram, ^ say at L, where L = h ^-'da, -d being a proper fraction. We would then have 'L^r^ + ga + lb = b + ■^a, Fig. 24. and consequently - = . ^ ^ b 1 -2i} + 2g But the right-hand side of this expression is a rational number, which is contrary to what has been proved in Art. 71. When L lies upon any other side of the parallelogram, we may derive a similar result and thus by a reductio ad dbsurdum show that there does not lie a zero of X(u) upon the boundary of the initial period-parallelogram. CONSTEUCTION OF DOUBLY PERIODIC FUNCTIONS. 116 The General Doubly Periodic Function Expressed through a Simple Transcendent. Art. 94. We shall next consider a doubly periodic function F{u) which has nowhere in the finite portion of the plane an essential singularity. Such a function has only a finite number of zeros and a finite number of infinities within a finite area. We may limit our study, as shown above, to the initial period-parallelogram. We shall assume that within this parallelogram the function F{u) has the infinities Ui, U2, . . . , Un] and we shall further assume that these infinities are of the first order, so that in the neighborhood of any one of them, ui say, F{u) has the form F{u) = h Co + Ci(m — Wi) + C2(m — Ul)^ + • • , M — Ml where d and the c's are constants. We shall see that every such function may be expressed through the general intermediary functions 4>(m). We shall form such a function where the integer k is taken equal to n + 1 and which therefore satisfies the two functional equations (M + o) = 4)(m), -(n + l)-(2u + 6) (J)(m + 6) = e " (m). There being n + 1 arbitrary constants in this function, we may write it in the form ^{u) = Bo*o(m) + 5x$i(m) + ■ ■ • + Bn^n(u). The constants Bq, Bi, . . . , Bn may be so determined that the function 4»(m) becomes zero of the first order on the points Wi, U2, . . . , Un- For write $(Wi) = Bo^oiUl) + 5ii(Wi) + + Bn^niUi) = 0, *(M2) = JBo*o(W2) + Bl^lM + • • • + Bn^n{U2) = 0, *K) = -Bo*o(Wn) + Bi*i(it„) + • • + jB„4>„(m„) = 0. In these equations we may consider the 5's as the unknown quantities. We have then n equations with n + 1 unknown quantities, from which we may determine the ratios of the B's so that 4>(w) becomes zero of the first order at all the points ux, u^, . . . , Un- By hypothesis F{u) be- comes infinite of the first order on all these points. Form the product /(w) = (u) F{u), -(n + l)-(2M+6) or f(u+b) =e " f{u). From this it is seen that f{u) is also one of the intermediary functions which satisfies the same functional equations as does i>{u). Further, since (-u) becomes zero of the first order at the same points at which F(u) is infinite of the first order, the product f(u) = <3i>(w) F{u} is nowhere infinite in the finite portion of the plane. A one-valued analytic function which does not have an essential singularity in the finite portion of the plane and in this portion of plane is nowhere infinite, is an integral tran- scendental function; and, as there are only n + 1 such functions that are linearly independent (cf. Art. 90), it follows that where the C's are constant. It is also seen that f/^A F{u) = =^AH1 . We consequently have the theorem: Any arbitrary doubly periodic function which has only infinities of the first order may be expressed as the quotient of two integral transcendental functions, both of which satisfy the same functional equations. Art. 95. By means of the X(M)-function we can make the above theorem more general in that the order of the infinities of F{u) is not restricted. We have noted in Art. 93 that X (m) is zero for the value u = " = c, say. Hence X(m + c)= for u = Xa{^ = 0, 1, 2, . . . ). ^ If we write X(m + c)= Xi(m), it is seen that Xi(w) = for m = 0. We also observe that the function Xi(m) satisfies the two functional equations Xi{u + a) =Xi(w), -— (2 u+a+b + b) Xi{u + b) = e " Xi(m). We have immediately the following relations: -—(2u-2ui+a+b+b) Xi(m - Ui +b) = e ° Xi(m - Ml), --(2m-2u2+o+6 + 6) Xi(w - U2 + b) = e " Xi(m - Mz), „ , ,, -~(2u-2ut + a+b + b) Xi(u - Uk + b) = e " Xi(m - Mfc). CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 117 If we put '*'(w) = Xi(w - til) Xi(m — M2) . . . Xi(w - Uk), it is seen that , -^{u + a) = ■*-(«), ^(m + 6) = e " ^(u), provided that h(a. + h) — 2(mi + u^ + ■ ■ ■ + Uk) = 2 ma; that is, if Wi + M2 + • • • +mj: = A;c — ma, (1) where m is any integer. Hence if A: = n + 1, the function '^(w) becomes zero on any n arbitrary points Ui, U2, . . . , u„, while the other zero must satisfy equation (1). As some of the points wi, U2, . . . , Un may be made equal to one another, it is seen that the zeros are not restricted to being of the first order in ■^(w). We may therefore let'S[' {u) take the place of f{u) in the preceding article and mutatis mvtandis have the same result as stated there. Art. 96. It is convenient to form here a function which becomes infinite of the first order for u = Q, u = a, u = 2 a, ■ ■ ■ . Such a func- tion is the Zeta-f unction (see Art. 97), 7 AA - X'(m + c) X(m. + c) This function Zo(w) is one-valued in the entire w-plane and has an essen- tial singularity only at infinity. By means of this fundamental element Hermite * has given a general method of expressing any one-valued doubly periodic function which in the finite portion of the plane has no essential singularity. We shall so choose the period-parallelogram that F{u) does not become infinite on its boundaries. If the function F{u) is infinite of the ylth order say at ui, the development in the neighborhood of this point is ^(^) = 7^^ + , ^"~\, 1 + • • • + -^^ + ^(« - «i). (m — Mi)^ (m — Ml)^~^ M — Ml the &'s being constants. We shall now give a method of representing this function when for every infinity the complex of all the negative powers is known. This complex of negative powers we have called (in Chapter I) the principal part of the function. We introduce a new variable f and form f($) = F(f) Zo(u - 0, where now u is to play the role of a parameter, being a point within the initial parallelogram, while $ is the variable. We consider in the f-plane * Hennite, Ann. de Toulouse, t. 2 (1888), pp. 1-12, and "Cours" (4th ed.), p. 226. 118 THEORY or ELLIPTIC FUNCTIONS. a period-parallelogram of F{$), upon whose boundaries there is no infinity of F(0. The function /(f) becomes infinite within this period-parallelogram on the points Ui, U2, . . . , Un, the points at which F(f) is by hypothesis infinite; and /(c) is infinite also at the additional point $ = m, since Zo(0) = <^. We form the sum of the residues of /(f) with regard to all the above infinities and have after Cauchy's Residue Theorem VRes/(e) = -^. I'md^, where the integration is to be taken over the sides of the period-parallel- ogram and in such a way that the surface of the parallelogram is always ■p^h ■<: to the left. We therefore have .^ fp+n fp+a+b 2niX^esm = / nOd?+ / f{()d$ ^p ^p+a 'p + p+a /(f)rfc + I node; p+a + b *Jp-^b Fig. 25. or, as in Art. 92, 2 ni 2) Res /(f) = a j f(p + at)dt + b j f{p + a + bt)dt -a f fiP + b + at)dt -b j f{p + bt)dt. Further, since ^^^^ + ^) ^ 2o(^), T^oiv + b)= Zoiv) - ^, a F{$ + a)=FiO, Fi$ + b)=F{0, it follows that /(f + o) = F{$ + a) Zo(w - f - a) = F{0 Zo{u - f) =/(e) and fi$ + &) = FiO \zo{u-0+—\= /(O + ^ F(0. ( a ) a We therefore have 2 m]^ Res /(f) = af\f{p + at) - f(p + b + at)]dt = aJ'l-^F{p + at)]dt; and consequently * .^^ pi 2) Res /(f) = - J F{p + at)dt. There is no infinity of the function F on the path of integration, this being a side of the parallelogram above. Hence the integral on the right * Cf. Hermite, loc. cit., p. 226. CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 119 has a definite value, a value which is independent of u, and as it does not contain f, it is a definite constant. Art. 97. We shall next determine by direct computation the sum of the above residues of /(f)- We had ~^,, , -, X{v + c) The function X('W + c) becomes zero of the first order for v = 0, and is one-valued and finite for all finite values of v. Its development is therefore of the form X(» + C) = yiV + ■f2V^ + ■ ■ ■ y where the fs are constant and fi r^ 0. Through differentiation it is seen that X'(v + c) = ri + 2 r2« + • • • , and consequently Zo(^) = i ri + 2 r2^ + • • • = 1 + rf^ + rfj^ + ^2^2 + . . _ V ri + r2^ + • • • V We note that the residue of Z('u) with respect to w = is unity. This function, as shown in the sequel, has in regard to the doubly periodic functions the same relation as has cot u with respect to the simply periodic functions and as has - to the rational functions. V If for V we substitute w — f , we have Zo(w - f) = ^-7 + rfo + di(M - f) + • • • u — ^ = - -^— + do + (ii(M-f)+ • • • , f - M which is the development of Zo(w — f) in the neighborhood of f = m. We next form the corresponding development of F{^). In the interior of the period-parallelogram the function F(f) becomes infinite at the points Ml, W2, . . ■ , Wn but not at u. Hence we may develop F{^) by Taylor's Theorem in the form ii'(f)=i^(M)+^(f-M)+^(f-M)2+- • •. Further, since /(f) = -P'(f) Zo(w - f), it follows that /(f) = -f^ + doi^(«)+- • •> and consequently Rgg _y(f) = _ 7r(^). We saw above that S Res /(f) is independent of u, but as shown here, the single residues are dependent upon this quantity. 120 THEORY OF ELLIPTIC FUNCTION'S. Aet. 98. We shall next calculate the residues of /(f) with respect to the other infinities Mi, U2, . . . ,Un- Suppose that the function F(f) becomes infinite of the Ath order on the point ui, so that F(f) when ex- panded in the neighborhood of this point is of the form ^(f) = 7F^^+7F-^^^Vi + - • •+^-^^ + co + ci(f-wi) + - ■ •, (f - Ul)^ (f - Ml) ^^ f - Ml where the 6's and c's are constants. For the value f = Wi the function Zo(m — f) is not infinite and may be developed by Taylor's Theorem in the form r» / ^N r7 / \ Zf>'(M — Ml),* s , Zo"(M — Ml) ,» vo Zo(m-0=Zo(m-Mi) ^^-^-jj ^(f-Mi)+ " ^^^ i^(f-Ml)2-- • •. It follows that the coefficient of in the product F{$)Zoiu—i) is b^Zo{u-u,)-^Zo'{u-u^)+^^Zo"{u-m)+- ■ .±-^^Zo(^-"("-Mi), which is the residue of /(f) with respect to the infinity f = Ui. The resi- dues with respect to the other infinities W2, U3, . . . , Un are found in the same manner. The b's and ^, of course, have different values for each of these points. Let the orders of infinity at mj: be ^t (fc = 1, 2, . . . , n) and in the neighborhood of the infinity Uk let the principal part of the function (n — 'Hj.\^k — ^ {if — i/i.'l^fc— 2 (m — ut)"' (m — Ma)''*"! (m — Mt)'''''^ U — Uk It follows at once that k = n X Res/($)=;^ 6fc,lZo(M - Uk) j^Zo'(m - Uk) (i-=l,2, ■.,")?),, r„ , + ^f Zo"(m - w,)- . . . ± _^^^Zo(^*-i)(m - Uk) We also saw that Res /(f) = — F{u), which must be added to the sum just written. On the other hand we had X Res /(f) = -f^FiP + at)dt = C, say, where C is a constant. Equating these two expressions for the sum of the residues, we have Fiu)=C + '^\bk.iZoiu-Uk)-^Zo'{u-Uk) + ^-^Zo"iu-Uk)-- ■ ■ ±(ir%!^°"*""^"--4 which is the required representation of the doubly periodic function F{u). We thus see that a doubly periodic function may be expressed through a finite sum of terms that are formed of the function Zq and its derivatives. CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 121 EXAMPLE Show that two doubly periodic functions with the same periods and the same principal parts differ only by an additive constant. In Chapter XX, several methods of representing a doubly periodic function will be found and the consequences which result therefrom will be derived. All these methods, however, are little other than different interpretations of the above formula. It is seen at once from this formula that we may represent a doubly periodic function when its principal parts are given, the function being completely determined except as to an additive constant. This expres- sion for a doubly periodic function is the analogue of the formula for the decomposition of a rational function into its simple fractions or of the decomposition of a simply periodic function into its simple elements (see Arts. 11 and 25). It may be shown that the latter cases may be derived from the former by making one of the periods infinite for the case of the simply periodic functions, and by causing them both to be infinite for the rational functions. Aht. 99. There is a restriction with respect to the constants that appear in the above development. We saw that Zo(?) + a) = Zo(tO and Zo(v + 6) = Zo(v) - — • a It follows that ZoW is not a doubly periodic function; but all its derivatives are doubly periodic, since we have Zo'{v + a) =Zo'W, Zo'(v + &) = Zo'W, etc. Hence under the summation sign of the preceding article all terms except the first are doubly periodic. Further, since F{u + h) = F{u), it also follows that k=l 2)fci,iZo(M - uk) = ^bk.i'^oiu - Uk + b). Since Zo(u — Uk + b) = Zo(w — Uk) —< it is evident from the equality of the two summations just written that „ .k-n k-n We thus have the very important theorem: The sum of the residues within a period-parallelogram of a doubly periodic function with respect to all of its infinities, is equal to zero. 122 THEORY OF ELLIPTIC FUNCTIONS. If we wish to form a doubly periodic function, when its principal parts with reference to its infinities are given, the restriction just mentioned must be imposed upon the constants. Art. 100. We may prove in a different manner that i:ReaF{u) =0. ° Take any period-parallelogram, upon the sides of which there are no infinities of F{u). Then by Cauchy's Residue Theorem B 7 p a >■ / A / Fig. 26. 2mS, Res F(u) = ( F{u)du. ^ JpACB But from Art. 92 we have / F{u)du = a f F{p + at)dt + b f F(p + a + bt)dt JpACB Jo Jo - a f F{p + b + at)dt -hi F{p + ht)dt. Jo Jo Further, since Fiu + a) = F{u) = F{u + b), it follows that 2 Res F{u) = 0. Art. 101. It follows directly from the above representation of a doubly periodic function that it cannot be an integral transcendental function (cf. Art. 83). In this case all the quantities bk,i, bk,2, ■ ■ ■ , &*, a» would be zero and consequently F{u) = C. It also follows that a doubly periodic function cannot be infinite of the first order at only one point of the period-parallelogram. For if -Uj were such a point, then is F(u)= ^^i— + Co + Ci(m - Ml) + • • • U — Ui in the neighborhood of this point, and consequently S Res F(u) = 6i,i. But as the sum of the residues is equal to zero, it also follows that 6i,i = and consequently F{u) would be an integral transcendent. But an inte- gral transcendental function with two periods is a constant (Art. 83). We have consequently the following theorem due to Liouville: A dovbly periodic function must have at least two infinities of the first order vxithin the period-parallelogram, or it must be infinite of at least the second order on one such point. CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 123 Art. 102. We have then two different methods which may be followed in the treatment of the doubly periodic functions, the one where the two infinities of the first order in the period-parallelogram are distinct, which is the older method employed by Jacobi, say z = snu; while the other method where the function becomes infinite of the second order is the one followed by Weierstrass, and in this case z considered as a function of u is written z — ^. The notation in the two different cases is inserted here, as it is convenient to refer to the two methods by means of this notation before the general treatment of these particular functions is considered. In the next Chapter it will be shown that a doubly periodic function which becomes infinite at n points (the order being finite at each point) is algebraically expressible through either one of the above simple forms z = snu or 2 = ^; and consequently the general theory of doubly periodic functions is reduced to the consideration of the two simpler cases. The Eliminant Equation. Art. 103. We have shown in Chapter III that a one-valued simply periodic function which in the finite portion of the plane has no essential singularity and which takes within a period-strip any value only a finite number of times, satisfies an algebraic differential equation in which the independent variable u does not explicitly enter. In Chapter II we have seen that associated with every one-valued analytic function which has an algebraic addition-theorem there exists an equation of the form just mentioned. We shall see later in Art. 158 that every one-valued doubly periodic function has an algebraic addition-theorem, so that (see Art. 35) the' notion of the doubly periodic function and of the eliminant equation is seen to be coextensive for the one-valued functions. We wish now to show that there is an eliminant equation which is associated with every one-valued doubly periodic function. First, how- ever, it is necessary to consider certain preliminary investigations. Art. 104. Suppose that the doubly periodic function F{u) has n infinities of the first order within a period-parallelogram, or if it becomes infinite of the ylth order on any point, let this point be counted as X infin- ities of the first order, so that the totality of infinities is still n. Let v be any arbitrary quantity and consider the number of solutions of the equa- tion F{u) = V within a period-parallelogram. After the same method by which we constructed a period-parallelogram which had no infinities upon its boundaries we may also construct one which has no zero of the function F{u) — v upon the boundaries. We 124 THEORY OF ELLIPTIC FUNCTIONS. may therefore assume that there are no zeros or infinities of the function F{u) — V upon the boundaries of our period-parallelogram. Consider next the function G{u) = F{u) - V. It is a doubly periodic function with the same periods as F{u), viz., a and b. As it becomes infinite at the same points as F{u), it has n infinities within the period-parallelogram. Form next the logarithmic derivative of G{u), ^ = H(u), say. G{u) The function H{u) has the periods a and b and becomes infinite at the points where G'{u) is infinite and also where G{u) is zero. Let Ml be an infinity of G(u) of the Ath order, so that Giu) = (m - ui)-''Gi(u), where Ci(mi) ^ 0. We then have (Art. 4) in the neighborhood oi ui, H{U) = ^ + P{U - Ml), M — Ml SO that Res H{u) =- X, that is, the residue of H{u) with respect to Ui is the order of the infinity of G{u) at the point Mi tuith the negative sign. Suppose next that Wi is a zero of G{u) of the fi th order, so that G{u) = (m — ■wi)>'G2iu), where G2{wi) j^ 0. We then have in the neighborhood of wi H(m) = —^ — + P(m - wi), or u — W\ Res H{u) = fi, that is, the residue of H{u) with respect to a zero of G{u) is equal to the order of the zero at this point. Further, since the sum of the residues of a doubly periodic function with respect to all its infinities within a period-parallelogram is zero, it follows that -I,X+I,/i = 0, where HX denotes the sum of the infinities of the function G{u) in a period- parallelogram, an infinity of the Ath order counting as X simple infinities, and where 2// denotes the number of zeros oi the first order of G(u), a CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 125 zero of the jith order counting as fi zeros of the first order. Since G{u) = F{u) — V, it follows that the number of roots of the equation F{u) - •« = within a period-parallelogram is equal to the number of infinities of the first order of the function F{u) within this parallelogram. It follows that a doubly periodic function F{u) takes within every period- parailelogram any value v as often as it becomes infinite of the first order within this period-parallelogram.* Art. 105. Let z = F(u) be a doubly periodic function of the nth order with the primitive periods a and b and let w = G{u) be a doubly periodic function of the A;th order with the same periods. Neither of these functions is supposed to have an essential singularity in the finite portion of the M-plane. We assert that there exists an algebraic equation with constant coefficients connecting z and w. For if a definite value is given to z there are n values of u, say ui, U2, . . . , u„, for which F{u) = z. If we write these values of uin w = G{u), we have n values of G{u), say Wi = G(u\), TO2 = G{u2), ■ ■ ■ ,w„ = G{un). Hence the variable z is related to. the variable w in such a way that to one value of z there correspond n values of w and similarly to one value of w there correspond k values of z, and consequently between z and w there exists an integral algebraic equation G{z, w) = 0, which is of the nth degree in w and of the kth degree in z. We may next suppose that z = ^{u) is a doubly periodic function dz with the periods a and b, then w = -—= 6' (u) is a doubly periodic function du having the same periods. Hence from the theorem above there is an dz algebraic equation connecting z and — , say du 4i)^°- dz It is easy to determine the degree of /in 2 and — ; for if 4>{u) = z is of . du the nth degree then — occurs to the nth degree in the above equation. du If Ml is an infinity of the Xi\\ order of {u), then mi is an infinity of the X + I order of {u), the order of infinity of '{u) being one greater on each of these points than is the order of 4>{u) on the same point. If all the infinities of 4){u) are of the first order and if n is the order of (j){u), it follows that ^'(m) is of the 2 nth order and consequently the * Cf. Neumann, Abel'schen Integrate, p. 107. 126 THEORY OP ELLIPTIC FUNCTIONS. degree of flz, — )is at most 2 n in z. This equation flz, — )we have \ duj \ duj called the eliminant eqvMtion. Art. 106. In Art. 104 we saw that any two doubly periodic functions that have the same periods are connected by an algebraic equation. It will therefore be sufficient, if we confine our attention to any doubly periodic function and express the others which have the same periods through this one. This function we shall take of the second order (cf. Art. 92) and consequently either z = sn u ov z = pu (Art. 102). Let 2 be a doubly periodic function of the second order (n = 2), so that the eliminant equation is /I H)'"' dz which is of the second degree in — and at most of the fourth degree in z. du The above equation must therefore have the form (I) 3o(2)f^T + 9l(2)^ + 32(2)=0, \du) du where the g's are integral functions of at most the fourth degree. dz We saw above that z and ~— are infinite at the same points within the du dz period-parallelogram and that — - does not become infinite for values of u du other than those which make z infinite. But from (I) it is seen that rf^ ^ - gi{z) ±'^gi{zY - igoiz) g2{z) du go{z) and becomes infinite for those values of z which make goiz) = 0. It follows that go{z) must be a constant and consequently the equation (I) (J, {^)\gAz)f^ + g.iz)^0, where the constant has been absorbed in the two functions gi{z) and 92 (z)- Art. 107. If 2 is a doubly periodic function, then also v =- is a doubly periodic function. Further, we have at once ^ dz dz dv 1 dv du dv du v^ du Making these substitutions in the above differential equation we have fdvV 1 n\dv 1 , /1\ ^ Since ji(-)andg2 f-lare at most of the fourth degree in-, it follows \v/ \vj V CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 127 that V^gd-j and V^gzIA are integral functions of at most the fourth degree in v, which we denote respectively by g^iv) and 32 (v)- The above differential equation is then v) dv ldvY_ gi(v) \du) v^ + g2{v) = 0. du We saw above that -^ is finite for finite values of z; the same must also dv du be true of — and v. du a ( \ But in the differential equation just written \^' becomes infinite for v^ V = 0. It follows that gi{v) cannot be of the fourth but must be of the second degree in z at most. It then follows from the equation (I') that du or, if we write 4 R{z) = giizy — 4 ^2(2), du where R{z) is an integral function of at most the fourth degree. It follows that p! dz Our problem consists in the treatment of this integral when R{z) is of the third or the fourth degree; when R{z) is of the second or first degree the integral is an elementary one. If we write u = I — > Jo Vl - 22 we have u = sin-^s, where the inverse sine-function is many-valued. We know, however, that the upper limit z considered as a function of the integral and written z = sin m is a one-valued simply periodic function of u. In the more general case above we wish to consider 2 as a function of u. This is the so-called " probleyn of inversion." Possibly the clearest and simplest method of treating this problem is in connection with the Riemann surface upon which the associated integrals may be represented. Before proceeding to the problem of inversion we shall therefore consider this surface in the next Chapter. EXAMPLE 1. If two doubly periodic functions /(2) and ^(z) have only two poles of the first order in the period-parallelogram and if each pole of the one function coincides with a pole of the other, then is ^(2) = Cf(z) + C„ where C and Ci are constants. CHAPTER VI THE RIEMANN SURFACE Article 108. At the close of the preceding Chapter we were left witli the discussion of an integral which contained a radical. Such an expres- sion is two-valued, and we must now consider more closely the meaning of such functions and their associated integrals. Take as simplest case the example s = ± Vz — a = ± (z — a)*, where 2 is a complex variable and a an arbitrary constant. For the value z = a, we have s = 0; but for all other finite values of z there are two values of s that are equal and of opposite signs. The point a is called a branch-point of s. The point 2 = 00 is also a branch-point of this function ; for - = == = for z = 00 . Consequently — and likewise s has s ± \/ z- a s only one value for z = as. There are other reasons why z = a and z = o + 2!t] i(fe So = ro*e ^ = ro*e^e"= - So. We thus see that s^ has taken the opposite sign after the circuit. Art. 109. Consider next the expression s2 = Riz), where R{z) is an integral function of the fourth degree in z. We may write Fig. 28. R{z) = A(z — Oi) (2 — a2) {z — 03) {z — 04), A being a constant. We then have s = ±\/R(z) = ±Ai {z- ai)i iz - 02)* (z - ag)* (2-04)*- The function s has two values with opposite signs for any value of z except Qi, a2, as, 04. When 2 is equal to any of these values, s has the one value zero. The points ai, 02, as and a^ are branch-points. The value \ ai — zo | is the radius of the circle about 20 which goes through ai. Suppose that 2 is any point situated within this circle so that Then, since z I z — 2o| < I ai — 2o|. ai = 2 — 2o — (ai — 2o), we have ai)4 = V {.z (ai 20) S 1 - — 2n 2o Since z - gp ai — zq Theorem in the form < 1, the right-hand side may be expanded by the Binomial (2 - ai)i = V- (ai - 2o) ]l - h zq ai - zo + I \ai - Zo 20 This series is uniformly convergent for all values of z within * the circle. In the same way we may develop (2 — 02)*, (z - as)*, (2 - a^)^ in posi- tive integral powers of 2 — zq. All these series are convergent within circles about zq- We have the development of s in powers of z - zq by multiplying the * When we say "within" we mean within any interval that lies wholly within. See Osgood, loc. cU., p. 77 and p. 285. 130 THEOEY OF ELLIPTIC FUNCTIONS. four series together, the multiplication being possible, since the series of the moduli of the terms that constitute the four seri es are convergent. We thus derive the result: We may develop s = \/R{z) in positive integral powers of z — Zq, if zq is different from, the four branch-points Oi, 02, a^, 04. The series is uniformly convergent within the circle about Zq as center, which passes through the nearest of the points aj, a27 <3-3, «4- Art. 110. We may effect within this circle the same development by Taylor's Theorem in the form VR(7) = So + \^^^ (z- zo) + ■ ■ . 2 So We must decide upon a definite sign of sq = \^R{zo) and use this sign throughout the development. If at the beginning we decide upon the other sign, then in the series we must write — Sq instead of Sq', that is, all the coefficients are given the opposite sign. If the sign of sq has been chosen and if the development of s has been made, then s is defined through the above series only within the circle already fixed. If we consider a value of 2 without the circle of convergence, we do not know what value s will take at this point. To be more explicit we may proceed as follows; Let z' be a point without the circle and join z' with zq through any path of finite length which must not pass indefinitely near a branch- point. Let the circle of convergence about 2o cut this path at 6. Then at all points of the portion of path 20 f the corresponding values of the function are known through the series. Let 2i be a point on this portion of path which lies sufficiently near to the periphery of the circle. We may express the value of the function at 21, that is, Si = \/R{z{) through the series Sl \/B(2i) l«o + - (2 - 2o) + Jz^zi Pi_ 29. Thus Sl is uniquely determined, if the sign of So has been previously chosen. We next take 21 as the center of another circle Ci, which also must not contain a branch-point. Then precisely as we expanded s in powers of 2 — 2o in the circle Co about 20 we may now expand s within Ci in powers of 2 — 2i about 2i. This circle Ci may extend up to the nearest branch- point and is not of an infinitesimally small area, since by hypothesis the path did not come indefinitely near a branch-point. The point 2] is taken sufficiently near f that the circle about z\ partly overlaps the THE EIEMANN SUEFACE. 131 circle about zq. That this may be the case zi must lie so close to f that the distance between the points is less than the radius of the circle Ci, a condition which evidently may always be satisfied. Hence the circles Co and Ci have a portion of area in common. Let the power series which is convergent within Co be denoted by Poiz — zq) while the one in Ci may be represented by Pi{z — zi). As we have already seen in Chapter I the series Pi gives for every value of z which is common to the two circles the same value as does the series Pq. But the development Pj holds good for the entire circle Ci. We thus go in a continuous manner to values of the function which lie without the circle Cq. The series Pj represents the continuation of the function s. It is clear that this process may be repeated and that we will finally come to a circle Cm around a point z^ of the path as center within which the point z' lies. We may develop the function within Cm i n positive integral powers oi z — Zm and may then compute s' = ^^(2') from this development. This process is called the " Continuation of the function along a prescribed path from Zq to s'." Such a continuation is possible in the entire z-plane, since zq may be connected by such a path with any other point z which is not a branch-point. Art. 111. Let B and 5i be two different paths which join zq and z' and suppose that neither of these points lies indefinitely near a branch- point. The question arises whether the value of the function at z' which is had through the continuation of the function along the path Si is the same as the one which is had through the contin uation from zq to / alon^ B. It is clear that if the two values of V'ii(z') thus ob- tained are different, they can differ only in sign. Through the circles which are necessary for the con- tinuation of the function from zq to z' along B is formed a strip (see figure of preceding article) which has every- where a finite breadth. This strip may be regarded as a " one-value realm." The function s remains one-valued within this realm. First suppose that the path B^ lies also wholly within this realm. Since none of the circles contains a branch-point there cannot be one between B and Bi, and it is evident that we come through the continuation of the function along these curves to the point z' with the same value of the function. For let the normal at any point ak on B cut the curve By at ak where Bi is taken very near to B, as shown in Fig. 31, and call ak, ock a pair of neighboring points. We suppose that the curves B and B\ have been taken so near together that one of the circles employed in the continuation of the function along B contains both ak and ak and that all points within this circle are ex- 132 THEORY OF ELLIPTIC FUZSTCTIO^STS. FiK. 31. pressed through Pa(z — Zk); and at the same time we assume that one of the circles used in the continuation of tlie function along the path Bi includes also the same points ak, ock and that all points within this circle are had through the series F)e{z — Z]i). Hence we must have the same value of s at the point z = ak from either of the power series Pk or P^', provided this is true of every pair of neighboring points that preceded this pair. The same is also true of the point z = Uk'- But the first pair of neighboring points was the point zq. We therefore come to z' with the same value of s along either path B or B^. Heffter {Theorie der Linearen Differential- Gleichungen, p. 72] has given a somewhat similar proof which suggested the one given here [see my Calculus of Variations, pp. 15, 16 and 256 et seq.]. If next B and Bi are two curves which are drawn in an arbitrary manner between zq and z' , but which do not include a branch-point, then we may fill the surface between B and Bi with a finite number of curves drawn from zn to z' which lie at a finite distance from one another " z and are so situated that each one lies within the one-valued realm which is formed by the circles that are necessary for the continuation of the func- tion along a neighboring curve. Thus by means of the intermediary curves with their associated one- valued realms it is evident that we come to z' with the same value of s when we make the continuation along either of the two curves B or Bi provided that there is no branch-point between them. It follows also that the value of the function at the point z' is independent of the form of the curve between zq and z' . Art. 112. Let (1) and (2) be two paths between zo and z' which do not ^ include a branch-point. If we go along (2) from zq to 2' and then back again along (1) from z' to Zq, we come to the same initial value of th e function. From this it follows: // the Y^ function s = \^R{z) is continued from the point z = Zq along a closed curve which does not contain a branch-point, we return after the circuit to the point zq with the same initial value of the function. The form of the curve is arbitrary, provided only it does not inclose any branch-point. Hence instead of making a circuit around an arbitrary curve, we may choose a circle which passes through Zo- Fig. 32. THE EIEMAjSTN SURFACE. 133 Art; 113. Suppose next that the closed curve inclvdes a branch-point, for example ai. We again fix the sign of sq for z = zq, and write s = Vz - ai\/Ri{z), where ^1(2) = A{z- az) {z - aa) (z - 04). We may al low VRx{z) to have an arbitrary sign, and so choose the sign of ^2 — oi that s = So will have the same sign for 2 = zq as has been pre- viously assigned to it. If we make a circuit about a^, it is seen that VRiJz) is not affected by it, since ai is not a branch-point of VRi{z). Hence upon making a circuit about aj we need consider only the first factor (2 — ai)^. AVe may make this circuit along a circle of radius r with Gi as center. For the points of the periphery, it is clear that I z - ai I = r, so that z — ai = re'*. It follows that i^ {z — ai)i = r*e^. Let the value of ^ corresponding to z = Zq be 4> = 4>o, so that i£. Fig. 34. (zo - tti)* = rie 2 , where the point Zq of course lies upon the periphery of the circle. When a complete circuit is made about ai, starting from zq, it is seen that (J^q is increased by 2 n, and consequently after this circuit the above expres- sion becomes i(jio+2 r.) i^ t^o j-ig 2 = 7-ig 2 gte = _ rig ^ It follows that after a circuit* about ai h as been made, the quantitj'' (2 — ai)i and consequently also s = VRiz) changes its sign. Further, if we make a circuit about ai along any arbitrary curve B which does not include any other branch-point except ai, then s changes sign with this circuit; for this is the case when a circuit has been made about the circle around ai, and as there is no branch-point between the circle and the path B, it follows that starting from zq we will again return to this point along both of the curves with the same value of the function. Art. 114. We may next ask what happens if the circuit includes two branch-points. First suppo.se that the circuit is made along the path zoaPyzo. Let dez be a closed curve about ai and tjOk a closed curve about 02. It follows immediately from the above considerations that the * Cf. Boliek, Elliptische Functionen, p. 150. 134 THEORY OF ELLIPTIC FUNCTIONS. two curves between which there is no branch-point lead always to the same initial value of the function. Hence instead of making the circuit about ai and a2 along the path ZQa^yzo we may just as well make the circuit along the path zodszzoTjdKZo, there being no branch-point between this curve and the curve zocajS-j-zo- After the circuit ZoSsTZo the function s changes sign as it again does after the circuit zqujOkzo, so that after the two circuits around the points ai and 02 we again come to the point Zq with the initial value of s. We conclude in the same way that if we make an arbitrary circuit around four branch- points we again come to the same value of the function, while if we have encircled three branch- points, we arrive at zq with the other value of s. Aet. 115. We may next see how the function s = Ai V{z — ai)(s — 02) . . ■ {z — a„) behaves when a circuit is made around the point at infinity. When n is an even integer and when a circuit is made so as to include the n points ai, a2, . . . , an, it follows from above that when z returns to its initial position, the value of s has not changed its sign. In the above expression write 3 = — , so that when z = 00 , we have t = Q. In the z-plane the point at infinity corresponds to the origin in the i-plane. We then have s = < ^ AW{\ - ait) (1 - aaO . . (1 - ant). Now take a circuit about a circle with the origin as center and which does not contain one of the branch-points ai, a2, . . . , a„. We must therefore write ( = re*'*, and it is seen that the function s changes sign when n is an odd integer. In this case the origin in the i-plane is a branch-point, and consequently in the z-plane the point at infinity is or is not a branch-point according as n is an odd or even integer. Art. 116. We shall draw lines connecting the points ai with a2 and 03 with 04. The paths along which the function s is continued must never cross these lines aj 02 and a^ a^. They may be called " canals." The z-plane which contains these two canals may be denoted by the i-plane, a dash being put over z (see Fig. 36). THE EIEMANN SURFACE. 135 If once the initial value sq of the function s = "vRiz) is fixed for the point 2o, then s is completely one-valued in the 5-plane; for in whatever manner the continuation from zq to zf may be made, any two different paths will always include an even number of branch-points or none, since the canals cannot be crossed. It follows that s = \^R{z) no longer depends upon the path along which this function is con- tinued from one point to another and is consequently one-valued in the z-plane. The two canals are sometimes called branch-cuts. If further the sign has been ascribed to the initial value Sq of the function s, then we may ascribe to s its proper value for every value in the ^plane. These values we suppose have been written down on a leaf, which represents the 2-plane. Again starting with — sq for the initial point we consider the corresponding values of the function written down upon another plane or leaf. In this second leaf the two canals connecting a I with 02 and as with a^ are also supposed to have been drawn, so that s is also one-valued on it. W e no te that corresponding to the same value of z, the values of s = ±'^R{z) in the two leaves are equal but of opposite sign. If, further, starting from a point ai on the upper bank of the canal we make a circuit tti Tcci Fig. 37. around oi, say, and return to the point 02 immediately opposite on the lower bank, the values of s at these two points are the same with con- trary sign. The same is true for all points* opposite one another along the two canals ai a2 and 03 04. We imagine the two leaves placed the one directly over the other, with the canals in the one leaf over those in the second leaf. The left * Cf. Neumann, Abel'schen Integrale, p. 81. i3(j THEORY OF ELLIPTIC FUNCTIONS. bank of each canal in the upper leaf is joined with the right bank in the lower leaf and the right bank in the upper leaf with the left bank in the lower. If being in the upper leaf we cross a canal we will find ourselves in the lower leaf; and if being in the lower leaf we cross a canal we will come up in the upper leaf. Thus the values of the function s change in a continuous manner when by crossing the canals we go from one leaf into the other; and in this manner we are able to make the two-valued function s behave like a one-valued function by means of the above structure. In this structure there is no crossing from one leaf to the other except in the manner indicated by means of the canals. The structure is called the Riemann surface* of the function s = \/R(z) {cf. Grundlagen fur eine allgemeine Theorie der Funktionen einer kom- plexen verdnderKchen Grosse. Inauguraldissertation von B. Riemann'. Crellc, Bd. 54, pp. 101 et seq.). If the function is continued anywhere in this Riemann surface, the function has always at any definite point a definite value, which is indepen- dent of the path along which the function has been continued. It is thus shown that the function s is a one-valued function of position in the Riemann surface. In this surface, if for a definite value of z the corresponding value of 5 is to be found, we must also indicate whether the value of z is taken in the upper or in the lower leaf. c 7 /a. Oi aj /"\ 1 ya.^ \ 1 <^ \ ai / di j a.\ a... y Fig. 38. In the figures a path that is taken ui the lower leaf is denoted by a broken line ( ), while a path in the upper leaf is indicated by an uninter- * See also Neumann, Theorie der Abel'schen Integrate ; Durtge, Elemente der Theorie der Funktionen. For other references see Wirtinger, Ency. der math. Wiss., Bd. IP, Heft 1 . THE EIEMANN SUEFACE. 137 rupted line ( ). The fact that the function s, when a circuit is taken around no branch-point, or around two branch-points, or around four branch-points, retains its sign, while it changes sign if the path is around one or three such points, is brought into evidence by means of the Riemann surface. It is indicated in the figures on page 136. We note that by a circuit around one or three branch-points we always pass from one leaf into the other, and that at two points situated the one over the other the function s has the same absolute value but different signs. The One-Valued Functions of Position on the Riemann Surface. Art. 117. We have defined a function as being one-valued on the Riemann surface. We may now consider more closely what is meant by such a function. When we say that a function is " one- valued on the Riemann surface," we mean something quite different from what is meant by saying a " function is one-valued." The signification of the first defini- tion is: "If the value of the variable z is given and also the position on the Riemann surface, then the function is uniquely determined"; if, however, only 2 were given, the function would not be uniquely determined. Let w be any function whatever of z which we suppose is one-valued on our fixed Riemann surface. In the upper leaf of this surface the function w has for a given z a definite value, say wi, and in the lower leaf it takes another value, say W2i for t he same value of z. In the special case above^ where w = s = ± \^R{z), we have wi = — W2. In general, however, this is not the case. But if we consider the sum wi + W2, this sum is a one-valued function of z, for if z is given, wi + W2 is completely determined. The same is also true of the product Wi • W2. It follows that w satisfies a quadratic equation of the form u)2 _ (f,(z)w + ■yjr{z) = 0, where {z) and ■yjr{z) are one-valued functions of z, such that wi + W2 = 4>{z) and Wi • W2 = '^iz)- Hence every one-valued function of position on the Riemann surface s = \/R{z) is a two-valued function of z and satisfies a quadratic equation, whose coefficients are one-valued functions of z. In particular, we shall study those one-valued functions of position on the Riemann surface which have a definite value at every position on the Riemann surface. In this case ^{z) = wi + W2 will have a definite value for every value of z, as will also ■^{z) = wi • W2. But one-valued functions which have everywhere definite values (when therefore there is no essen- tial singularity) are rational functions. If then w is to be a one-valued 138 THEORY OF ELLIPTIC FUNCTIONS. function of position on the fixed Riemann surface and is to have every- where on this surface a definite value, then (f){z) and •>/r(2) must be rational functions of z. Art. 118. When we solve the above quadratic equation, we have j^ = ^ + i\/_4t(2) +cl>Hz), where the root is to be taken positive or negative. We have thus shown that w is equal to a rational function of z, increased or diminished by the square root of a rational function. Suppose that the radicand - 4 ^jr{z) + ^{z)^ S(z), say, becomes zero or infinite of the (2 n + l)st order for z = h, where n is an integer. We note that the point h cannot be a branch-point on the Riemann surface, for ai, 02, 03, en are the only branch-points on this surface. We may write ^(3) _ ^^ _ ^y n+is,(z), where Si{z) is a rational function of 2. About 6 as a center describe a circle which does not inclose any other zero or infinity of S(z). We then have 2n+i VS{zj= (z-b) ^ \/Si(z); and if 2 makes a circuit about the circle, the function V')Si(z) retains its 2 w + l sign, while (z — b) ^ changes sign. Consequently the function \/S{z) changes its sign with this circuit, so that w = ^^ H ^— does not 2 2 resume its initial value and is therefore not a one-valued function of position on the Riemann surface. Hence the factor z — b must occur to an even power if it enters as a factor of either the numerator or the denominator of the rational function S{z), so that Siz) must have the form S{z) = 5i(2) { (2 - ai) (2 - 02) (z - ag) (2 - 04) }. We may therefore write w = \ ^(2) -I- \ Siiz) V{z — ai) (2 - a2) (2 — 03) (2 — a^) = p{z) + 5(2) VR{z) = p + 5 . s, where p = p{z) = ^^, q = 9(2) = —~ are rational functions of 2. It has thus been shown * that " Every one-valued function of position, which has everywhere a definite value in our Riemann surface, is of the form w = p + qs, where p and q are rational functions of z." * Cf. Neumann, loc. cit., p. 355. THE RIEMANN SURFACE. 139 Reciprocally, every function of the form w = p + qs is a one-valued function of position on the Riemann surface, since p, q, s taken separately have this property. If then w has this form, it is the necessary and sufHcient condition that w be a one-valued function of position on the Riemann surface. The Zeros of the One-valued Functions of Position. Art. 119. Let 2 = a be a position on the Riemann surface, which is different from the branch-points ai, 02, as, a^. We may then draw a circle around a which lies entirely in one leaf of the Riemann surface. It may happen that w = for z = a, while at the same time p and 5 are infinite for z = a. For suppose that p = ^ "' ., + , ^'-\^ . + • • • + -2J— + Co + Ci(2 -«) + •••, (z — a)* {z — a)'''^ z — n q = , -^" , + , ■^'"l ■ + • ■ • + -=^ + go + gi(z ~ a) + ■ ■ ■ . (2 — a)" (2 — a)"-^ z ~ a We may also develop s for points within the circle in the form s = ho + hi{z — a) + h2{z — aY + It is evident that s is not infinite for z = a, and it is also clear that if X y^ [1, then w becomes infinite for z = a; but il X = fi, then we may so choose the coefficients in the development of p and q that u; = for z =a. This will be the case if in the development of w all the negative powers and also the constant term drop out. The coefficient of {z — a.)~^ in this development is e^ -\- hofp, or, since X = ft, we must have ex + hofx = 0. Further, it is necessary that the coefficient of (2 — a) -(^-D be zero, that is, ex~i + fx-iho + fihi = 0, etc. These conditions may all be satisfied; and consequently w = kriz - ay + kr+i{z - ay+^ + ■ ■ , where the k's are constant and where r is a positive integer greater than 0. Finally we may write iw = (2 — ay[kr + kr+iiz — a) + • • ]. We see that w becomes zero of the rth order for 2 = a. We thus experi- ence no trouble in determining the order of zero for w at any point a, even if at this point the functions p and q become infinite. Similarly if p and q remain finite f or z = a there is no difficulty. 140 THEORY OF ELLIPTIC FUNCTIONS. Art. 120. We shall next study w in the neighborhood of one of the branch-points, ai say. If z makes a circuit about aj, we return with a value of w that lies in the other leaf, and in order to reach the initial point of the circuit we must make a double circle about Oi, since by the second circuit we again come into the leaf in which the initial point is situated. As in Art. 113, we write s = VR{zJ = (2 - ai)i\^R^. Since ai is not a branch-point of VRi{z), we may expand this function in positive integral powers of 2 — fli and have - ai)i[bo + biiz- ai) + bziz - a^Y +•■•]■ We put {z — ai)*= t or t^ = z — Oi. Let a circuit be made about a\ along a circle with radius r, so that z — a^ = fi = re'*, t = Vre 2 If then z describes a circle with radius r around Oj in the 3-plane, then t describes a circle with radius Vr around the origin in the i-plane. If the circuit of z begins with the initial value = 0, then the circuit of t begins with the value i^ = 0, and when /2 increased by -. Hence to the whole circle in the 2-plane there corresponds the half-circle in the i-plane, and to the double circle which z describes in the Riemann surface in order to return again to its initial point, there corresponds the simple circle in the i-plane. Suppose that w vanishes at a branch-point, oi say. Further suppose by the substitution 2 — a^ = fi that p{z) becomes p{t) and q{z) becomes q{t). In the neighborhood of the point t = 0, let pit) =a:„<"-t-a„ + i«'"+i-|-a„+2<'"+2+ • • • , and 5(0 = ;3„<" -f- /3„+i«"+i-|-;fl„+2«''+2_,_ . . . , where m and n are integers (positive or negative including zero). If m and n take negative or zero values, there must exist equations of condition as in the preceding article. Since 2 — Oi = t^, it follows that 2 — 02 = ai — a2 + fi, z — as = ai — 03 + fi, z — a4 = ai — 04 + t^, and consequently Ri{z) becomes V{t), where ¥(1) = (oi — a2 + t^) (ai — ^3 + fi) (a^ — 04 + fi). We note that this function does not vanish for i = 0, so that there is no branch-point of this function within the circle t = 0, ii this circle is taken THE RIEMANN SURFACE. 141 sufficiently small. We may consequently expand Vvlt) within this circle in positive integral powers of fi and have VR{z) = t[bQ + b^fi + b2t^ + • • ■ ]■ It is further seen that if w becomes zero at the point z — a^ = t^, it may be developed in the neighborhood of i = in positive integral powers of t in the form X >+l A+2 = Co(2 - ai)^+ ci(z - tti) 2 + C2(2 - ai) ^ + • • • . It follows also that the function w becomes zero of the /Ith order at the branch-point z = a^. In other words, if w becomes zero at a branch-point z = ttk, then TWICE the exponent of the lowest power of z — ak in the develop- ment of w in ascending powers of this quantity, is the order of the zero on this position. If, however, the zero-position z = a, say, is not a branch- point, we have the development w = Co'{z - a)'''+ Ci'{z ~ a)'''+^+ ■ ■ ■, and here the exponent of the lowest power of z — a in the development in ascending powers of this quantity is the order of the zero of the function at z = a. This difference respecting the order of the zeros seems at first arbitrary, but the significance is evidenced through the following consideration: Let a be a zero which does not coincide with one of the branch-points. We may then develop w in the form w = (z - aY[co' -f- Ci'(z -a) + C2' {z - a)2 + . . ], and consequently log w = /log(z — a) -I- log[co' -1- ci{z — a) -I- C2'{z — a^ + ■ • ]. Since the expansion within the bracket does not become zero for z = a, its logarithm is not negative infinity and the expression may be developed in integral powers of z — a. We then have log w = X' log (z — a) -1- ei -f e2'(z — a) -f • • ■ . If z makes a complete circuit about a, the power series e i' + 62' (z — a) -f- . . . does not change sign; log (z — a) is, however, increased by 2 ni and con- sequently X' log (z — a) is increased by 2 niX'. It follows that 2 ^r-rlogw; 2 TO is increased by X' when a circuit is made about the zero z = a: in other words, the order of the zero of the function w at the point z = a is the 142 THKOKY" OF ELLIPTIC FUNCTIONS. number due to the change in log w when z makes an entire circuit 2 ni about a. This same analytic property must be retained if a is also a branch- point, say ai. From the development above w = (z - ai)^[co + Ci{z - ai)4 4- £2(2 - ai)»+ • • •] it follows that log to = -log (2 - Oi) +log[co + ci{z - ai)4 + 02(2 - Oi)^+ ■ • • ], or log w = -log {z - Oi) + eo + ei (2 — ai)i+ • • ■ . Now to make a complete circuit around a\ we must make a double circle. By this circuit (2 — Oi)* does not change sign. It follows that the change experienced in log w\?,X since log (z — ai) changes by 2 • 2 ni. But 2 Tzi here X is twice the exponent of the lowest power of 2 — Oi in the above expansion of w. The infinities of w may be treated in precisely the same way as its zeros. Integration. Art. 121. We shall next consider the integrals taken over certain paths in the Riemann surface. These are formed in the same manner as are the integrals of functions of the complex variable in the plane. \i w = f{z) = p + gs is a function which for all points of the path of _z\s' integration takes finite and con- tinuous values, and if a definite path of integration is prescribed which is ta ken from the point Zq, where VR{z) takes the v alue So, to the point 2', where VR{z) takes the value s', then the inte- gral \f{z)dz taken over this path Fig. 40. '^ ^ has a definite value. If a portion of the path of integration lies in the lower leaf, the significance is that the function under the integral sign takes values in the lower leaf which form a continuous connection with the values in the upper leaf. aa THE EIEMANN SURFACE. 143 An integral is called closed when the path of integration reverts to the initial point in the same sheet from which it started, as illustrated in the following figures: Fig. 41, Cauchy's Theorem for the plane is also true of the Riemann surface, viz. : // a function f(z) within a portion of surface that is completely bounded, the boundaries included, is everywhere one-vaZusd, finite and continuous, then the integral taken over the boundaries of the surface in su/ch a way that it has the bounded surface alwaijs to the left, is zero. Art. 122. We must consider more closely what is meant by the boundaries of a portion of surface. The simplest case is a portion of surface as shown in the figure. We must make a dis- tinction between an ouier edge and an inner edge. If we have a point a on the inner edge and a point b on the outer edge, it is clear that we cannot go from the point a to the point b without crossing the boundary. We say in general that a portion of surface is completely bounded when it is impos- sible to go from a point on the inner edge to a point on the outer edge without crossing the boundary. Consider next * a closed curve a^j- in the Riemaftn surface. We may go from a point a on the outer edge to a point b on the inner edge without crossing the curve aj^y, "^^ which lies wholly in the up- per leaf. Consequently the Fig. 42. Fig. 43. curve a^y must not be re- garded as the c mplete boundary of a portion of surface. But if we also draw a congruent curve oi'^'f, that is, one imme- diately under the first curve and in the lower leaf as shown in Fig. 44, then it is not possible to go from the point a to the point b without * Cf. Bobek, loc. cit., p. 155. 144 THEORY OF ELLIPTIC FUNCTIONS. crossing one or the other of the two curves a^y or a'/9 y'. Hence a^y and a'^'y' together form the complete boundary of this portion of sur- face of the Riemann surface. By Cauchy's Theorem the integral taken over f{z), where the path of integration extends over both afiy and a'P'y' , must be zero if the direction of integration is taken as indicated above and if f(z) is one-valued, finite and continuous within and on the boundaries of this surface. Upper leaf Lower leaf Fig. 44. Fig. 45. To prove this we note that instead of taking a^y and a'P'y' as the paths of integration we may take paths which lie indefinitely near the branch- cut aia2, this one branch-cut, of course, lying in both the upper and the lower leaf. It is seen that, if the integration is taken in both the upper and the lower leaf (see Fig. 45), /wdz = I [p + qs]dz = / 2 qsdz- I 2 qsdz = 0, the elements of integration taken in the opposite directions over I pdz canceling one another. Art. 123. If a one- valued analytic function be developed in the z-plane in the form fiz) = + + + P{z - a), (2 — a)'^ (2 — a)''-^ z — a where P{z — a) denotes a power series in positive integral powers oi z — a, then we know that the residue oi /(z) with respect to 2 = a is bi = Res f{z), z = a the quantity bj being the coefficient of z — a The same definition is given for the residue of a function of position on the Riemann surface, provided the point o does not coincide with a branch-point. If, however, this point is a branch-point, Oi say, and if the function becomes infinite at this point, then it follows from above that the development oi w = f{z) in the neighborhood of this point is m ; + - ; + + - ;+■ r + P{(2-ai)4| (2-ai)^/2 ' (2-ai)(''-i)/2 ' ' (2-ai)5 {z - oi)i Before we define the residue here, we may consider a theorem which gives the residue in the form of an integral: If in the 2-plane we draw a circle THE RIEMANN SURFACE. 145 about the infinity a of the function f{z) and if /(z) does not become infinite on any other point within or on the circumference of this circle, then is ^//(^)'^^ Res/(z), where the integration is taken over the circumference of the circle. We shall also retain this formula as the definition of a residue on the Riemann surface when the point a coincides with a branch-point, say ax. The integration is to be taken over a complete circuit about the branch- point, that is, over a double circle. We may write under the sign of integration instead of f{z) the power series by which it is represented. The general term is / k (z — ai^dz, where the integration is over the double circle. Suppose that r is the radius of the double circle, so that z — ai = re'*, and consequently also / (z - ai)^dz =1 r2e 2 rie^Mcj} = ir^ ^' j e" ^' dcj). k This integral is always zero, except when 1 + - = 0. In this latter case It follows that J(z — a-S^dz = i I d(f> = 4ot. d'ble-circle "^O Iles/(z) = -^ ff{z)dz = ^M -4:7:1, d'ble-circle where 62 is the coefficient of (z - ai)-', since k = - 2. We thus have finally Res/(z) =262;' or, the residve vxith respect to a branch-point is equal to double the coefficient of {z — ai)-i in the development of the function in powers of (z - Oi)*. Art. 124. Suppose that a portion of surface is given which is completely bounded by certain curves. At isolated points of this surface suppose that the function becomes infinite. We draw around these points small cir- cles, dmple if they are not branch-points, and double when they are branch- points. The interior of these circles we no longer count as belonging to the surface. In this manner we derive a new portion of surface which is completely bounded on the one hand by the original curves and on the other by the small circles. The integral taken over the boundaries of this new portion of surface is zero, since the function is everywhere finite 146 THEOEY OF ELLIPTIC FUNCTIONS. within this surface, boundaries included. The integration is to be so talcen that the interior of the portiou of surface is always to the left. If the direction of integration taken over the small circles is changed so that the interiors of these circles lie to the left of the integra- tion, then the signs of the corresponding integrals must be changed, and we have the following theorem : The integral over the com- plete boundaries of the original -portion of surface is equal to the sum of the integrals over the circles {or double-circles) which are drawn around the infinities (poles). But on the other hand each of the inte- grals around one of the circles is equal to the residue of the function with respect to the infinities in question multiplied by 2 rci. We have therefore the theorem: // a function within a completely bounded portidn of surface, boundaries included, is everywhere one-valued and discontinuous only at isolated points, then the integral multiplied by 1/2 ni and taken over the complete boundary of this surface is equal to the sum of the residues of the function with respect to all the points of discontinuity within the portion of surface. Art. 125. We saw that any one-valued function of position on the Riemann surface s = ^R(z) was of the form * Fig. 46. w = p -{■ qs, where p and q are rational functions of z and where s It follows that w dz ,. ,,.,,m .-.[,. |,|gj] p + qs qs If the numerator and the denominator of the right-hand side of this expression are multiplied by p — qs, we have L ^ w dz Qs, where P and Q are rational functions of z. It is thus seen that the logarithmic derivative of w = p -f 5s is a rational function of 2 and s and indeed of the same form as is w itself. The logarithmic derivative becomes infinite at the points where w vanishes and at the points where w becomes infinite. * See Riemann, Werke, p. 111. THE RIEMANN SURFACE. 147 If /i is the order of the zero of the function w at the point a, then in the neighborhood of a ^-^=-J^ + Piz-a) [see Art. 4]; dz z — a and if X is the order of infinity of the function w at the point /?, then in the neighborhood of j3 It follows that Res^-1^ = /^ 2=a dz and Res^i5^=-1 z=0 dz The above discussion is true when a and /? are not branch-points. If a is a branch-point, say a^, and if w becomes zero at this point, then in the neighborhood of this point we have w =(2- ai)^[go + 9i{z- ai)^ + g2{z - a^)^ + ■ ■ ■ ], and consequently log w = ^log {z ~ ai)+ log[j?o + 9iiz - Oi)* + •••]. It follows that dlOSW 2 : d , r , , .i , 1 / = + —log [go + gi{z - aiy + •••]. dz z — ax dz Since the logarithmic expression does not become infinite for z = ai, it may be developed in the form log[fifo +gi{z- tti)* + • ■ ■] = ho + hx{z- ai)* + ■ • • , and consequently dlogw 2 , hi , „ v-i , — -&- = + -^ (z - ai) + . • . . dz z — ai Z We therefore have (cf. Art. 120) Res ^^ = 2.^=/*. z-oi dz 2 If on the other hand a i is an infinity of the ^th order of w, then is Res ^Mi? = - 1 148 THEORY OF ELLIPTIC FUNCTIONS. Art. 126. We shall now apply Cauchy's Theorem to the function dz As the portion of surface over whose boundaries the integration is to be taken we shall choose a region which contains all the infinities of the function P + Qs. In order to have such a surface, we construct in the Riemann surface a very small circle which does not contain any of the infinities of P + Qs. The rest of the Riemann surface, that is, the entire Riemann surface except- ing the small circle, will then contain all the infinities of P + Qs. The point at infinity may be one of these infinities. In the latter case we make the substitution z = -. The function P + Qs becomes by this substitu- tion, say P + Qs = Pi (0 + Qoi.t) ^/^i - ai) (^ - 02) (y - as) (j - ^4) Pl(t)+Ql{t)^{i - OlO(l - 020(1 - 030(1 - 040, Qo(0 where Qi(0 i^ The functions Pi(0 and Qi{t) are rational functions of t; and in the <-plane the origin is now an infinity. The other infinities in the old Riemann sur- face remain at finite distances from the origin on the new Riemann surface, whose branch-points are the reciprocal of those in the old Riemann surface.* We thus have no trouble in computing the order of the infinity at the point infinity. The boundary of the region is evidently that of the small circle, and the integration is to be taken so that the region without the circle lies to the left. After the theorem of Art. 92, when we remain on the original Riemann surface J_ rdl£gHd,=2;Res^i^=V Res (P + Qs), where the integration is taken so that the bounded region is on the left, that is, so that the interior of the small circle is on the right. Noting that the integral taken over the boundary of this small circle, within which there is no infinity of the function, is zero, it is seen that 2 Res (P + Qs) = 0, where the summation extends over all the infinities of P -I- Qs. * Cf. Neumann, loc. cit., p. 111. THE RIEMANN SUEFACE. 149 These residues fall into two gi'oups: those of the one group have refer- ence to the infinities of the function sJ£ ^ which exist through the van- '^^ rl 1 ishing of w, while those of the other group refer to the infinities of °^^ , which are also the infinities of w. If by "Lfi we denote the sum of the orders of the zeros and by 2 A the sum of the orders of the infinities of w, then for P + Qs the sum of the residues of the first group is S/i, while — 2A is the sum of the residues of the second group. It follows at once that 2 Res (P + Qs) = 2ju - 2>l = 0, or HX = S/i. It has thus been shown that the sum of the orders of the zeros of w is eqvxd to the sum of the orders of its infinities; or, in other words, the function w becomes as often zero as it does infinity in the Riemann surface, if a zero of the /xth order is counted fx-ply and an infinity of the Xth order is counted X-ply. Art. 127. Suppose that k is an arbitrary constant and write p + qs = k. The function p + qs — k is a, rational function in z and s. It becomes infinite as often as p + qs is infinite, and since the relation is true also here, it becomes zero as often as p 4- 5s becomes zero. We thus have the following theorem: The equation p + qs — k has in the Riemann surface as many solutions as p + qs has infinities. Hence also the function p + qs takes every value in the Riemann surface an eqvnl number of times. Art. 128. We have often employed the term " complete boundary " and have in particular considered this expression in Art. 122. We shall again emphasize the fact that it is of extreme importance to under- stand the full significance of this term. If from a portion of surface A a piece is cut out, for example a circle around a point of discontinuity, then in this new portion of surface every closed curve no longer forms Fig. 47. a complete boundary. If P is the small circle that has been cut out of A, then the closed curve B no longer forms a complete boundary, since B and C together constitute this com- 160 THEORY OF ELLIPTIC FUNCTIONS. plete boundary. If from any portion of the surface A we cut out a circle and join this circle with the original boundary by means of a cross-cut, it is then impossible to draw a closed curve in A which does not form the complete boundary of a portion of surface, so long as we do not cross the cross-cut. Every surface which has the property that every closed curve drawn in it is the complete boundary of a portion of surface, is called a simply connected surface.* The Riemann surface on which the function w = ^^' p _|_ qs is represented is not a simply connected one. We may, however, as shown in the figure, easily transform it into a simply connected surface by drawing the two canals a and 6. We note that one- half of the canal 6 lies in the lower leaf. These canals cannot be crossed by going from one leaf into the other as is the case with the canals aia2 and 0304. The Riemann surface con- taining the two canals a and ft we denote by T". The sur- face which does not have these canals is denoted by T. The surface T' is said to be of order t unity. We note that two canals or cross-cuts were necessary to make it simply connected. One may easily be convinced by trial that every closed curve in T' forms the complete boundary of a portion of surface, so long as the curve does not cross the canals a and 6. Art. 129. Anticipating some of the more complicated results of the next Chapter, we may consider here the simpler case of the function s2 = r{z), where r{z) = A(z — a^) {z — a2). The associated Riemann surface con- sists of two leaves connected along the canal 0102. The integrals P * Cf. Neumann, loc. cit., p. 146. t In general, if N denotes the number of branch-points belonging to any function, n the number of leaves in the associated Riemann surface, and p the class or order of the Riemann surface, then (see Forsyth, Theory of Functions, p. 356) N = 2p + 2 n — 2. (Cf. Riemann, Werke, p. 114.) The name deficiency was introduced by Cayley, On the Transformation of Plane Curves. 1865. The deficiency of a curve is the class or order of the Riemann surface associated with its equation; that is, y'= Rix) is a curve of deficiency unity, if s'= R(z) is a Riemann surface of order unity. Fig. 49. THE EIEMANN SUEFACE. 151 where the paths of integration are taken over the two curves (1) and (2), 1 are equal since the function Vriz) for all points of the surface between these two curves. If we let the path of integration (1) approach indefinitely near the canal a 102, then, since the values of Vr(z) on the right and left banks of this canal have contrary signs, we have p = c '^^ 4- r —^==2 r"* ^^ ' 'faKaVr(z) J^s's/riz) J „^ Vr(z) is one-valued finite and continuous ' afi$ Fig. 50. where in the last integral the integration is taken along the upper leaf and the left bank. It foUows that P is dififerent from zero and consequently also the integral taken over a curve such as (2) is not zero. This two-leaved Riemann surface T we next cut by a canal so that the integral r dz r{z) will be a one-valued function of position in the surface where the cut has been made. This integral wiU then be independent of the path of integration, which we have just shown by going around the canal ai02 is not the case in the Riemann surface before the cut has been made. From a point C on the upper bank of the canal we draw a line CA which goes off towards infinity and this line is indefi- nitely continued from C in the other direc- tion in the lower leaf. We thus form a cut or canal AB which is not to be crossed. The surface with this new canal we call T'. From the figure it is seen that we may go from any point a on the bank of the canal AB to a point /? immediately oppo- site on the other bank without crossing either the canal AB or the canal ai02, but it is impossible to make a circuit around the canal a 102 or around either of the branch-points a-i or 03 without crossing one of these canals. It follows that the above integral in T' is one-valued. 'B Fig. 51. 152 THEORY OF ELLIPTIC FUNCTIONS. Art. 130. u{z, s) and let Next let ''•" dz f :oVr{z)' uiz,s)= I where the path of integration is in T'; where the path is in T. 22,S~ „sr,\/r{z) The integration in both cases is always counted from a fixed point Zq, So, which as a rule may be arbitrarily taken, but when once taken must be retained as the lower limit for all the integrals that come under the dis- cussion. We know that if the function Vr{z) is one- valued, finite and continuous within the area situated within the two curves (1) and (2) of the figure. =j =j +j ' Zo,ao '-'Zo.so t/Zo.So ^2|.Si dz Fig. 52. It follows that in T' (1) (2; where the integrand Vr(z) stood with every integral. is to be under - /"''"= P'"- / " ' = ^(22, sa) - ■u(zi, Si). ^'zi,si ^z^,sq t/zo. So Next take the integral from zq, Sq to z, smT where there being no canal vli? we go by the way of the two points A and p. We have a u{z,s) = C' + p + p'°, where the distance between X and p being indefinitely small the middle integral on the right may be nag- [ lected. But from above J Zr., So and X' m(z, s) — u{X), rig. 53. where both of these integrals are in the T surface. From this it is seen *^^* w(2, s) = u{z, s) + u{p) - u{k}, where u{p) and a (A) are the integrals from zq, sq to p and from zq, sq to / in the T' surface, the path of integration being taken in any manner so long as neither of the canals a^az and AB is crossed. THE RIEMANN SUKFACE. 163 On the other hand, •J I or, from the figure, ^p Op i/ayB U a dz in T'; and since we have Op Oay^ O 02 where the integration of the last integral is taken in the upper leaf and the lower bank of the canal aia2 . Fig. 54. We have finally u{z, s) = u(z, s)+P, where P is a quantity which does not depend upon the path zq. sq to zi, Si. The quantity P is called the modulus of periodicity. If the path of integration is taken so that we pass from the right to the left bank of the canal AB, then is u{z, s) = u{z, s) — P The integral in T differs from the integral in 7" only hy a positive or negative multiple of P. this multiple depending upon the number of times and the direction the canal AB has been crossed [see Durege. Elliptische Functionen (2d ed.), P- 370]. EXAMPLE Show that P=2x in the case of r dz U= I — == J Vi- Realms of Rationality. Aet. 131. Let 3 be a complex variable which may take all real or complex, finite and infinite values. Consider the collectivity of all rational functions of z with arbitrary constant real or complex coefficients. These functions form a closed realm, the individual functions of which repeat themselves through the processes of addition, subtraction, multiplication and division, since clearly the sum, the difference, the product, and the quotient of two or more rational functions is a rational function and con- sequently an individual of the realm. 154 THEORY OF ELLIPTIC FUNCTIONS. This realm of rationality we shall denote by (2). Consider next the one- valued functions on the fixed Riemann surface. If we denote any such function Ijy Wi = pi + qis and any other such function by 102 = P2 + 92s, then the sum, difference, product and quotient of the two functions Wi and ■W2 are functions of the form w = p + qs. It is evident that if we add (or adjoin) the algebraic quantity s to the realm (z), we will have another realm (2, s), the individual functions or elements of which repeat themselves through the processes of addi- tion, subtraction, multiplication and division. This realm we shall call the elliptic realm. It includes the former realm. We note that every element of this realm is a one-valued function of position on the fixed Riemann surface. In the present Chapter we have proved that every element of the realm (2, s) takes every arbitrary value that it can take an equal number of times. It also follows that within this elliptic realm there does not exist an element that becomes infinite of the first order at only one point of the Riemann surface. This latter statement is left as an exercise (see Thomae, Functionen einer complexen Verdnderlichen, p. 94). CHAPTER VII THE PROBLEM OF INVERSION Article 132. We have seen (Chapter V) that every one-valued doubly periodic function of the second order which has no essential singularity in the finite portion of the plane, or Riemann surface, satisfies a certain differential equation in which the independent variable does not exphcitly appear. This equation may be written where -piz) is an integral function of at most the second degree and R{z) is an integral function of th e fourth degree. We saw in the preceding Chapter that p{z) + 'VR{z) is a one-valued function of position on the fixed Riemann surface. We are thus led to the study of the integral dz -k viz) + VR(z) As the lower limit of this integral we tak e any point zq of the Riemann surface, at which s has the value So=+^R{zo). Throughout the whole discussion this point Zq, sq will be taken as the initial point. The integral is taken along any path of integration to the point z, s. It follows then that Pz, i dz Ja„^„P{z) +VR(^ is a definite function of the upper limit, a function which is dependent upon the path of integration. We may also consider the upper limit z as a function of u; and we shall now raise the question; Under what conditions is the upper limit z a one- valued function of u? It is possible that the point z, s lies in the neighborhood of a branch- point ai, say. We then have the following development: p(.) = p(ai)+£^) (.-«:)+ 2:^ (z-ax)2+ .... VR(^ = bi{z - ai)i + biiz - ai)i + ■ • • ; and consequently piz)+VR(^)=p(ai) + bi(z-ai)i+J^^^(z-ai)i + b2{z-a,)i + -■ . We thus have a series which proceeds in ascending powers of (z — oj) . 155 156 THEORY OF ELLIPTIC FUNCTIONS. Art. 133. Suppose that p(ai) does not vanish. We may then develop • in integral powers of (z — ai)* in the form P(^') +'^Ri^) ^ ^ = -^ + Ci(2 - ai)i + C2(Z - oi)3 + • ■ • . p{z) +VR(z) p(ai) If we put p, rfa it is seen that dz «/2(,,s„l)(z) +\/i2(2) ./zo,»o«(2) +^72(2) J a. Piz) +VR{z) Jz„.s„p(z) +\/R{z) J a, p(z) +VR(z) p.« dz J a, p{z) +VR(zj We have here assumed that the point z, s has been so chosen that there is no point of discontinuity of the integrand within the triangle aizzo- It follows that r^-' dz — a = I ^=^: J a, V(Z) +VR(z) p{z) +\/R{z) ^' By hypothesis the point z, s lies in th/e neighbor- hood of ai, that is, on the inside of a circle within which the series developed above is convergent. We may therefore integrate this series and have \~, — : + Ci(z — ai)*+ C2(a — ai)' + • • -[diz — ai) di (p(ai) ) z — a i , 2 , ^J! , piai) 3 If we put z - ai = t^, we have t^ 2 u- a^ -— - + -ci<3 + . . . P(oi) 3 It follows that {u or (w-a)l=_l- +/2«2 + . p(ai)4 where of course the quantities ci, C2, . . . , /2, etc., are constants. By the reversion of this series we have ' = ?i(w - a)* + 32(m - a)^ + But smce t^ = z ~ ai, it is seen that z is two-valued and not one-valued in the neighborhood of m = a. THE PROBLEM OF INVERSION. 157 Art. 134. If p{ai) = 0, the above development becomes -1(2 - ai)-i + eo + ei(2 - Oi)^ + • ■ p(s) +VR{z) We then have u — a= I [e-i(z — ai)-^ + Co + ei{z — ai)i + ■]d{z~ai) = 2e-i{z — oi)i + eo(e — ai)i + = 2e-i< + eof^ + ■ ■ From this we conclude that t is developable in positive integral powers of 14 — a and consequently is one-valued in the neighborhood of it = a. It follows also that z is one-valued in the neighborhood of this point. Hence in order that 2 be a one-valued function of u, it is necessary that p(ai) = 0. In the same way it may be shown that p(a2) = = P(a3)= Pidi)- On the other hand, p{z) is an integral algebraic function of at most the second degree in z. Such a function cannot vanish at more than two points without being identically zero. It follows that p{z) = 0. We therefore have the theorem: In order that z be a one-valued function of u, it is necessary that p{z) =0 and consequently also that du Art. 135. The last investigation would be true even if a^r ^-^= ^2n.so p{z) +\' R{Z) We may prove, however, as follows that this integral is 1 were infinite. never infinite. •We saw above that p{z) +\'R{z) which is convergent within a certain circle. Let this circle cut the path of integration at the point 2', s'. We then have p. dz r^'-" dz «/2o. So P (2) -I- V .R (2) J20, so p (2) -I- VR (2) dz , is developable in a power series ,p{z) +\'R{z) Fig. 56. p{z) +VR{z) The first integral on the right is finite, since it does not become infinite for any value between zq, sq and 2', s'; while the second integral, as shown above, may be expressed through the series [2 e-i(2 - ai)i + eo(z - a^)i + ■ ■ ■ Yj^,- 158 THEORY OF ELLIPTIC FUNCTIONS. This series is finite for the values z' , s' and a\. dz It follows therefore that p(2) +VE(2) has a finite value even when p(ai) =0, and at the same time it has been shown that the integral / dz zo.B.'piz) +VR{z) is finite when the upper limit is a branch-point. Art. 136. We may now confine ourselves to the consideration of the integral p.« dz Jz«,soVR(z) oVR{z) This integral is called an elliptic integral of the first kind. We have seen that the integral u remains finite when the upper limit coincides with a branch-point. We shall next see that this integral remains finite when the path of integration goes into infinity. In one of the leaves of the Riemann surface, for example the upper, draw a circle with the origin as center which in clude s all the branch-points. On the outside of this circle the quantity \/R{z) and consequently also , is one- valued; for if we make a closed circuit without this circle VR{z) it includes either none or all the branch-points and consequently — =^ VR{z) does not change its value. We have 1 _ 1^ ^2 VR{z) 1 'iVi'-m'-m^-T Since — > -^. — , — are proper fractions for all z z z z values of z without this circle, each of the above factors is developable in positive integral powers of -, so that z Fig. 57. VR{z) 9o 1 , 1 , 9.- + 92-,+ which series is convergent for all values of z without the circle. Let z', s' be the point where the path of integration starting from the point Zq, So and leading to infinity, cuts the circle. We have dz r^'-^ dz , r" dz 'h„.SoVR{z) J ^,80 VR(z) J 2!,^ VR{z) THE PROBLEM OF INVERSION. 159 We have seen that the first integral on the right is always finite, whether the path of integration goes through a branch-point or not. For the second integral we have = r_2o_l2l T , L 2 222 j^/ an expression which is finite for both the upper and the lower limit. We have thus shown that the integral "'■ " dz f 'z«.soVR{z) is finite everywhere, even when the upper limit is indefinitely large or if it coincides with one of the branch-points. Art. 137. We represent by T' the Riemann surface of Art. 128 in which the canals a and 6 have been drawn. We noted that any closed curve on this surface formed the complete boundary of a portion of sur- face. If on this surface the curve C includes one or several branch-points, for example aj, we isolate them by means of small double circles. If K denotes the double circle about ai, and if the curve C includes only one such branch-point, then by Cauchy's Theorem we have r^+ r^ = 0, where s = \/ft(^. Jc s Jk s Note that in this second integral the integration is over two circles lying directly the one over the other in the two leaves of the Riemann surface. In these two leaves the quantity s has opposite signs, while at points the one over the other the absolute values of s and z are equal. It follows J" dz ■ ■ ■ ■ — the elements of integration cancel m pairs, so K s . Cdz that this integral is zero. We have thus shown that the mtegral / — Jc s taken over any closed curve in T' is zero. ^^ If in T' we draw any two curves (1) and (2) between the points zq, sq and z, s, without crossing either of thn canals a or b, the two curves will form a closed curve, and from what we have just seen 2o, So S >Jz,a S (1) (2) C'-' dz ^ C''' dz 1/ «o. So S i/zg, «„ S CD (2) Fig. 58. the numbers in parentheses under the integral signs denoting the paths along which the integration has been taken. 160 THEORY OF ELLIPTIC FUNCTIONS. Hence if we write u{z, s) » dz where the dash over u signifies that the integration is to be taken in the Riemann surface T' , in which the canals a and 6 are not to be crossed, it follows from above that u{z, s) is entireli/ independent of the path of inte- gration. It follows also that the integral u{z, s) is a one-valued definite function of the upper limit. Art. 138. We shall consider next the integral u{z, s) = I — ! «^20, So S where the path of integration is taken in the Riemann surface T, which does not contain the canals a and 6. We shall show that here the integral u{z, s) is not a one-valued function of the upper lim'it 2, s, but depends upon the path of integration. In the T-surface the inte- gral corresponding to u{z, s) is dz s u{z,s)= I •-'zo. So . pdz^ p.» + dz_ s Fig. 59. a \^ h y The points p and X are supposed to lie indefinitely near each other, so that the middle integral to the right is zero. We con- sequently have ... , . Tp dz , C'-'dz (A) u{z, s)= f — + / — «/zo, So S ^X S We have seen that in the Riemann surface T" every integral is indepen- dent of the path of integration. We note that (see Art. 130) J'^^'^^dz C'^'^'dz C'"^^dz -, s -, s — =/ / — =w fe, sz) - m(2i,Si). Si.Si S tJzri,3o ^ 1/20, So 5 Returning to the equation (A), it is seen that neither of the canals a or 6 is crossed between zq, sq and p, so that «^2o, - = uip) in r. „ s THE PROBLEM OF mVERSlON. 161 Further, there is no canal between ^ and z, s. It follows from what we have just shown that J Ti(z,s) - ti{X) in T', where we go in T' from zo, so to r, s by crossing the canals 03 04 and fli (1-2 as shown in the figure. We have to make the same crossings to go from -0. So to X. We therefore have from the equation (\) ii(z. s) = Ti(z, s) + u{p) — u.{X). If the canal a had been crossed at any other point pi, Xi instead of at p,X, we would have had u(z, s) = u(z. s) + u{pi) - TiiXi). Consider the difference {Ti{p) - »i;,)}-{iz(pi)- «,(>ii)}. or {(7(,o) - «(;0i)(- {7/U) - »(^i)}. The points ,0 and pi are both on the same side of the canal a, while the point X and ^i are both on the opposite banlc. It is seen that — in T' , \. s where the path of integration in 7" may be quite arbitrary, provided only it does not cross the canals a and b. \Ve xaa,\ therefore take the path of integration from p to ,01 indefinitely near the right bank of the canal, while the path from X to Xx is taken indefinitely near the left bank. Since these two paths differ from each other by an infinitesimal quantity, the integrals over them are equal. It follows then that { uip) - uiX) \-\u (px) - U(Xi) \ = 0, and consequently (/(,") - "(^) li:^* the same value at whatever point tlio crossing has taken place. .\rt. 139. If we cross the canal a from -o. So to ^, s in the opposite direction from that gone o^•er in the previous case, we have «/Jti, So S .'.-,,,■(>, S i/p S = Ti(X) + Ti(z. s) - u(p) (in T'), = u(z, s) + u{X) — u{p). 162 THEORY OF ELLIPTIC FUNCTIONS. We note that in T' we must go from Zq, sq to the canal joining ai and a2 and after crossing this canal into the lower leaf come out again ^ J into the upper leaf by crossing the canal 03 a4 and then pro- ceed to 2, s. We thus see that when we cross the canal a in the opposite direction to that fol- lowed in the previous article we have to subtract the quantity u{p)—u{i.) from u{z, s). If the canal a is crossed fi times in the first direction and V times in the second direction, we will have u{z, s) = u{z, s) + ifi- v){u{p)-u{^)]. Fig. 60. We have precisely the same result if we cross the canal 6. Of course, the constant u{p) - u(X) is different here from what it was in the previous case when we crossed the canal a. We shall write for the canal a : m(A) — u{p) = A, for the canal 6 : u{p) — u{X) — B. We therefore have in general u{z, s) = u{z, s) -\- mA + nB, where m and n are positive or negative integers and where u{z, s) is the integral in which the path of integration is free, u{z, s) being the integral in the Riemann surface T', in which the canals a and 6 cannot be crossed. The quantities A and B are called the Moduli of Periodicity. Art. 140. We have seen that if a and b are two quantities whose quo- tient is not real and if the coefficient of i in the complex quantity - is a positive, we may determine a function (w) which satisfies the two func- tional equations 4>(m + a) = ^(u), 4>(w + 6) = e ° (w). This function is (cf. Art. 86) *(m) = .6 Trt- whereQ = e " and Bm+k= B„ X BmQ m2 2jrl -r Witt THE PROBLEM OP INVERSION. 163 If the two moduli of periodicity A and B have the property that the A jefficient of i in — is positm form a function 4>(w) so that A coefficient of i in — is positive, then we may write a = B and h = A and B •m= +00 77l2 2 Trt* *(w) = X^-^^o*^" .A '5. where Qo = e ^and Bm+k= B^. We then have ^, , _. , , , (m) = [m(z, s)] = '^{z, s), say. It is seen that ^{z, s) is a function of position in the Riemann surface and is not a one-valued function; that is, when z, s are given, ^{z, s) does not take one definite value. For u{z, s) depends upon the path of inte- gration, so that (cf. Art. 139) u{z, s) = v,(z, s) + mA + nB. Hence the complex of values ^{z, s) which belong to one position z, s is expressed through ■^(z, s) = ^[u{z, s) -\- mA + nB], where m and n are integers. Since (w) has the period B, the above complex of values reduces to 4>[m(z, s) + mA]. We saw in Art. 91 that the following relation existed for the general ^-function: -^(2mu+m'b) 4>(m + wb) = e " $(w). Consequently the complex of values above becomes - ^[2mSfe3)+mM] ^[u{z, s) + mA] = e ^ *[«(«, «)]• It is evident that ^[u{z, s)] = ^{z, s) is a one-valued function of position on the Riemann surface T'. It also follows that between ^{z, s) and ■^(z, s) there exists the relation The integer m is positive or negative depending upon the number of times the path of integration has crossed the canal a and upon the direction at the crossing. 1C4 THEORY OF ELLIPTIC FUNCTIONS. Abt. 14L We saw in Art. 94 that ^{u) = Bo^oiu) + Bi^iiu) + • • ■ + Bk-i'^k-iiu). Let the corresponding ^-functions be denoted by We then liave, for example, '''-[2mS(z,s)+m-'4] — It follows that -~[2mu(^z,s)+m^A]— ^ ^i(z,s) ._ N^i(z,s) . '^■2(2, S) ^2(2, s) ' and since ^i(z, s), '^2(2, s) are both one- valued functions of position on the Riemann surface, it is also seen that — ^^ ' ' is a one- valued function ^2fe s) of position on the Riemann surface. The functions "^{z, s) = ^[u(z, s)] are infinite series which are conver- gent for all values of the argument u{z, s) which are not infinitely large (Art. 86). We have proved, however, that 'dz - 2o, So is infinite for no point of the Riemann surface, including the point at infinity. It follows that '^1(2, s), '^2(2, s) are everywhere convergent and consequently the quotient ^ has definite values everywhere on the ■*'2(2, s) Riemann surface. But a one-valued function of z, s which has every- where a definite value is a rational function of z, s. It follows then that ^2(2, s) where R denotes a rational function. Art. 142. Let us next study more closely some of the subjects which we have passed over rather rapidly. We had on the canal a: u{X) — u{p) = A, on the canal 6 : u{p) — u{_X) = B. It made no difference where the point X, p was situated on the canal. We may therefore take the point a, a' where the canal 6 cuts the canal a and have accordingly M(a')— u{oi)= A, or ' dz in T' , (cf. Neumann, loc. cit., p. 248], pa.' %J a THE PEOBLEM OF mVEESION. 165 the integration being in the negative direction. In the T'-surface we may, starting with a, follow the canal b around to the point a', and conse- quently have 9^ f— i Jb s iuT', the integration being in the negative direction; i.e., the quantity .4. is the closed in- tegral around the canal b. In the same way B = u{p)- u{X) = u{a)-uip) Fig. 61. Jb S Ja dz the integration being in the negative direction. Fig. 62. in T', We have thus shown that B is the closed in- tegral over the canal a. Art. 143. In the previous discussions we have assumed that R{z) is of the fourth degree in z. When R{z) is of the third degree, we have only three finite branch- points, Oi, a2, Os, say. But here the point at infinity is also a branch-point (Art. 115). We may therefore connect ai and 02 by a canal and 03 with the point at infinity. The Riemann surface may then be represented as in the former case (see figure). Art. 144. In the derivation of the function (m) the ratio — cannot be real. Following the methods of Riemann* we shall show that this ratio is imaginary and that the coefficient of i must be positive, a result which was also necessary in the previous discussion. We saw that w(2, s) was a one-valued function of position on the Riemann surface T'. All functions of the complex variable are in general also complex, and we may consequently write (t(z, s) = p -f- iq. * Riemann, Theone der Abel'schen Functionen, CreUe, Bd. 54, p. 145; see also Koenigs- berger, Elliptische Functionen, pp. 368, 369; Fuchs, Crelle, Bd. 83, pp. 13 et seq. 166 THEORY OF ELLIPTIC FUNCTIONS. The quantity u{z, s) is everywhere finite in T', and from the developments by which it was shown always to be finite, it is readily proved to be also continuous. If we write z = x + iy, then p and q are everywhere one-valued, finite and continuous functions of X, y. Noting that ^"^'''^ = ?^^^^ ^ = ^^^^^ . 1 = ^ = -J=, ^ ax dz dx dz dz VR{z) it is seen that —is infinite for 2 = aj, 02, 03, or a^. On the other hand, ax _ 9x ax ax and consequently either ^ or -2- or both of these derivatives are infinite dx dx for z = tti, 02, a-i, or 04. Form next the integral where the integration is to be taken over the whole boundary of the Riemann surface T". This surface, see figure in the preceding article, is bounded by the two banks ^ and p of the two canals a and 6. It is seen that we may go over both the banks X and p of a and 6 with a single trace. The integral / pdq taken over tljis trace may be divided into several integrals as follows: /rip) f(.i) fu) f.») pdq = I pdq + I pdq + j pdq + I pdq, where {p) as an upper index means that we are on the right bank, ayfi means the portion of curve gone over, and + a means on the canal a in the positive direction. Abt. 145. We saw above that du = — , = dp + idq, VR{z) dz dx + idy , , ., or , == — , = dp + idq, VR(z) VR{z) If we write = 4>{^, v) + ^'"/'(x, y), then is 0(x, y)dx - -./r(x, y)dy + i\'^{x, y)dx + (f>{x, y)dy} = dp + idq. It follows that {x,y)dx-^{x,y)dy = dp, ^{x, y)dx + ^(x, y)dy = dq. THE PROBLEM OF INVERSION. 167 The function >jr(x., y), which is the coefficient of i in , will have at two opposite points on the left and right banks of the canals values which are different only by an infinitesimal small quantity, since the canals « and b are indefinitely narrow. The same is true of the function 4>{x, y). It follows that dq will have at two points opposite each other on the canal a the same values, but the signs will be different, since the integration at these points has been taken in the opposite direction. We may therefore write the above integral in the form /r w (» r ((.) (« Pdq = J^JP - P}dq +J_^JP - P}^^- In Art. 139 we put A = u{^) — u{p) on the canal a; or A = p + iq — {p + iq} = p- p + i{q - q}. If further we write A = a + i^, then is a = p — p on the canal a. We also had ^ _ (^, (^, (,, ^^■^ B = u(p) — u{X) = p + iq — {p + iq\ (p) (A) (P) U) = p-p + i{q-q\, and writing B — r + id, (p) u) , , , it follows that y = p — p on the canal h. It is seen at once that the above integral may be written — dz Since du = ■ > it is clear that J pdq = -aj dq + rj *?• \/R(/)' A == J^ = £idp + idq) Further, since A = a + il3,we have fi = I dq; and similarly 8 =Jdq. The integral above is finally / pdq = T^ — aS- 168 THEORY OF ELLIPTIC FUNCTIONS. Art. 146. We shall calculate the same integral in another manner. Suppose that P and Q are real functions of the real variables x and y; then the curvilinear integral Updx + Qdy), /< where the integration is taken over the complete boundary of a region within and on the boundary of which P and Q together with their partial derivatives of the first and second order are one-valued, finite and continuous, is equal to the surface integral sm - f )-*• taken over the same region.* Consider the curvilinear integral fpdq-J(p'^dx+p^^dy), where as above the integration is to be taken over both banks of the two canals a and b in the Riemann surface T'. We have seen that p is one-valued, finite and continuous within this surface, since it is the real part of u{s, s). But (see Art. 144) -2^ and -2- become infinite at the points 1 dx dy '^it 0,2, o-s and 04. Hence to apply the theorem just stated, we must cut these points out of the surface by means of very small double circles. The resulting Riemann surface call T". In this surface the conditions required are satisfied. The curvilinear integral must now also be taken over the double circles. But as shown in Art. 137 the integrals over these double circles are zero. If then we write in the formula instead of Pdx + Qdy the quantity p-^dx + p-^dy, we will have to substitute ^ ^ for — the expression ~ —^ + p — ^ 9x dx dy dxdy and for ^ the expression ^ ^ + p -9!3_- dy dy dx Sx9y and consequently * Forsyth, p. 23; see also Casorati, Teorica delle funzioni di variabili complesse, pp. 64-69; Neumann, Abd'sche Integrate, 2d ed., p. 390. Schwarz, Ges. Werke, Bd. II, has shown that there are certain limitations of this theorem; and Picard, Traite d' Analyse, t. 2, pp. 38 et seq. THE PROBLEM OF INVERSION. 169 Butsince ^ = ^ ^nd ^ = - ^ dx dy dy d< (being the conditions that u{z, s)= p + iq have a definite derivative), and since / pdq = /?^ — ad, it follows that As the elements under the sign of integration are essentially positive, it is seen that Py — ai5 is a positive quantity. But we have B ^ r + i^ _ (r + id ) (a - ifi) _ ay + ^d . ad - fir Since ad — Py is different from zero, the ratio -—is not real* and the B ■ A coefficient of i in — is negative; hence the coefficient of i in ^is positive. We may therefore (see Art. 86) form functions (m + S)= $(w), (^{u + A)=e " (m). Art. 147. In the expression - ^[2mii(z, s) +mM] — ■>I'(z, s)=e ^ ^(Z, S), since u{z, s) is always finite, the exponential factor is always finite so long as m is finite. Further, since is only infinite for infinite values of its argument, it follows that ^{z, s) = *[m(3, s)] is never infinite. Hence also ^{z, s) is only infinite when m is infinite. It is also evident that '^(z, s) can only be zero when "i^{z, s) = 0. We shall now see how often the function '^{z, s) becomes zero on the Riemann surface T'. In Art. 92 we saw that if a function f{z, s) is discontinuous at isolated positions within a portion of surface, but otherwise is one-valued and finite, then _!_ fd\o^.fiz,s)^^^ 2 nij dz where the integration is taken over the complete boundary of the portion of surface, is equal to the sum of the orders of the zeros of the function * Cf. Thomae, Abriss einer Theorie der FuncHcmen, etc., p. 102; Falk, Acta Math., Bd. 70; Pringsheim, Math. Ann., Bd. 27. 170 THEOEY OF ELLIPTIC FUNCTIONS. diminished by the sum of the orders of its infinities within the portion of surface in question; i.e., IniJ dz As the portion of surface we shall take the surface T' which is bounded by the canals a and 6, and ior f{z, s) we have here '^(z, s). There being no infinities, Il/< = 0, and consequently IniJ ^d l og ^(z, s) ^^^ 27zi.f dz where the integration taken over both banks of the canals a and 6 is equal to the sum of the orders of the zeros in 7". Now on the canal a we have u{^)— u(p)= A, or u{}.) = u{p) + A. It follows that *[«(/)] = ^{u{p)+ A] = e"^'^"'''^"'''^'[w(p)], and consequently that On the canal b we have u{p) — u{k) = B, or u{X)=u{p)~B. It follows that ^[u{X)] = <^[u{p) - B] = ^[u{p)l or From the figure in Art. 142 it is seen that r rflog^(z,s) ^._ rw d\og^{z,s) ^_ ^ p» d\ofr^(z,s) ^. Jt' dz Jar0on+a dz Jpifon-b dz ^ P^ d log ^(2, S) ^^ _^ rW d log ^(2, S) ^^ J ? r'a' oa-a dz Ja'a'ao-a+h dz j^.dzy ^(W^)j j^^dzy ^(j«^) J which owing to (M) and (N) ^ 2nih r dz THE PROBLEM OP INVERSION. 171 But from Art. 139 it is seen that L VR{z) We therefore have finally 27ti J dz '^' =B. dz = Su = k. It is thus seen that the intermediary function ^(z, s) has k zeros on the surface T' ; and since '^{z, s) vanishes on the same points as ^(2, s), it follows that ^(z, s) has k zeros on the Riemann surface T. Art. 148. We saw (Art. 87) that when fc = 2 ^{u + a)= (m), -— (2«+6) a)(M + 6)=e " (w). Further, write Q = 9*, and it follows that 4>o(m)=0i(u)= ^5-^5" , ^ = — 00 i(w)=Hi(M)= 2 g^ 2 ^" U= —00 If in 0i(w) we write — /^ in the place of /i, the summation is not thereby changed, and we have u = + OC AtzI — pU 0i(m)= '% q'^e From this it is seen that @i{u)= ®i{ — u), or &i{u) is an even function. Similarly writing —fi—l for /z in the formula for Hi(m) we have = +00 /2li+l\- 2irt ^=+00 / 2)i+i \- _; Hi(M)= 2?^ 2 ''e'" ■^2^+l)u or Hi(m)= Hi( — m), so that this function is also even. Art. 149. If in @i(m) we write m(z, s) instead of m, then @i(m) becomes — -^[2mu(z.s)+m.'A] %(3, s) = ^0(2, s) . e ^ (cf. Art. 140). Suppose that, starting from a point zq, sq in the upper leaf of the Riemann surface T', a path of integration is taken to the point z, s, which may cross the canals a and 6 as often as we choose. The point z, s may lie in either the upper or the lower leaf. Next starting from the point zq, -sq, which lies immediately under the point zq, sq, let us construct a second path, which , is everywhere congruent to the first path, that is, which lies in the under 172 THEORY OF ELLIPTIC FUNCTIONS. leaf when the first path is in the upper, and is in the upper leaf when the first path is in the under. If further we form the integral of the first kind u{z, s) for each of these two paths, and add the two integrals, it is seen that the elements of integration cancel in pairs, so that J^'-^dz r^'-^ dz ^ Q zo,so S t/zo, -So S (I) (H) where (I) and (II) are used to denote the paths of integration. Suppose that zq, So coincides with one of the branch-points, for example with aj, then Zq, So and Zq, —sq coincide, and we have dz ^ ai " •J at = 0, s (f) (li) or u (z, s) + gA ^- hB + u{z, - s) + g' A + h'B = 0, where g, g', h, h' denote integers. It follows that m(z, s) + u{z, - s) = yA + dB, where y and o are integers. // then we take a branch-point as the initial point of the path of integration, the function Ti{z, s) has at two points situated the one over the other in the Riemann surface T' , values whose sum is equal to integral multiples of A and B. _Art. 150. If we write w(2, s) for u in @i(u), we have the function ■^o(z, s); similarly let ^i(z, s) denote the result of substituting u{z, s) for u in Hi(m). Then noting the relations existing between '^o, ^o and between '^i and ^i, it is seen (cf. Art. 141) that 3V?^ = lo(£^ = fifes), Ti{z,s) ^iiz,s) where R{z, s) denotes a rational function of its arguments. It will be shown in the following Chapter that R{z, s) = g{z) + s • h{z), where g{z) and h{z) are rational functions of z alone. We form next _ ^^(,, _ ,) ^ Qj^(.,-s)l ^i(2, -«) B.,[u{z,-s)] _ @i[- u(z,s) + rA + dB] @i[- u(z,s)] Hi[- u{z, s) + rA + 8B] Hi[- u{z, s)]' as is seen from the functional equations which @i and Hi satisfy. Since ©1 and Hi are even functions, it follows that Hi[m(2, S)] THE PROBLEM OF INVERSION. 173 We therefore have g{z) - s • h{z) = g{z) + s • h{z), and consequently s • h{z) = 0. Since s is not identically zero, we must have h{z) = 0; and finally Riz,s)=g{z), or R(z, s) is a rational function of z alone. Art. 151. Since fc = 2, it follows that Hi and @i have two zeros of the first order on the Riemann surface; and since the quotient of these two functions is a rational function of z, it is evident that (>I) Hi(m) ^ Aiz + A2 ^ @i(m) A3Z + Ai where the A's are constants. This function has the two zeros of the first order z = -^. s Ai and the two infinities z Remark. — If the zero z = — --^ is a branch-point, say Oi, then (see A', Art. 120) twice the exponent of the lowest power of 2 — ai = 2 H = in the development in ascending powers of z — ai is the order of the zero. But as the development of the numerator of the above expression i^ simply .li 2 + -p h it is seen that 2 is the order of the zero for 4 L ^iJ z = — "—^ ■ Such a zero is therefore to be counted as two zeros of the first ■^^ A. order. The case where * is a branch-point may be treated in an analogous manner. Art. 152. It follows directly from equation (M) above that ^zQiW - ^44Hi(m) A3Hi(w)-.4i@i(m)' from which it is seen that z is a one-valued doubly periodic function of u with periods A and B. We call 2 the inverse of the elliptic integral u, where dz « = / VA{z — Oi)(2 — 02) (2 — a3)(z — 04) Although u is not a one-valued function of z (Art. 139), the inverse function z is one-valued in u. The constant A under the radical is of course not the same constant as the period A . 174 THEORY OF ELLIPTIC FUNCTIONS. We may also note that ^^ du is a one-valved function of v; for the derivative of a one-valued dovbly 'peri- odic function is one-valued and doublij periodic. Akt. 153. The following remarks of Lejeune Dirichlet {Geddchtniss- rede auf Jacobi; Jacobi's Werke, Bd. I, pp. 9 and 10) are instructive and historical: " Es ist Legendres unverganglicher Ruhm in den eben erwahnten Entdeckungen die Keime eines wichtigen Zweiges der Analysis erkannt und durch die Arbeit eines halben Lebens auf diesen Grundlagen eine selbstandige Theorie errichtet zu haben, welche alle Integrale umfasst, in denen keine andere Irrationalitat enthalten ist als eine Quadratwurzel, unter welcher die Veranderliche den 4ten Grad nicht tibersteigt. Schon Euler hatte bemerkt, mit welchen Modificationen sein Satz auf solche Integrale ausgedehnt werden kann; Legendre, indem er von dem gliick- lichen Gedanken ausging, alle diese Integrale auf feste canonische Formen zuriickzufiihren, gelangte zu der fijr die Ausbildung der Theorie so wichtig gewordenen Erkenntniss, dass sie in drei wesentlich verschiedene Gat- tungen zerfallen. Indem er dann jede Gattung einer sorgfaltigen'Unter- suchung unterwarf, entdeckte er viele ihrer wichtigsten Eigenschaften, von welchen namentlich die, welche der dritten Gattung zukommen, sehr verborgen und umgemein schwer zuganglich waren. Nur durch die ausdaurerndste Beharrlichkeit, die den grossen Mathematiker immer von neuem auf den Gegenstand zuriickkommen liess, gelang es ihm hier Schwierigkeiten zu besiegen, welche mit den Hiilfsmitteln, die ihm zu Gebote standen, kaum iiberwindlich sheinen mussten. . . . " Wahrend die friiheren Bearbeiter dieses Gegenstandes das elliptische Integral der ersten Gattung als eine Function seiner Grenze ansahen, erkannten Abel und Jacobi unabhangig von einander, wenn auch der erstere einige Monate friiher, die Nothwendigkeit die Betrachtungsweise umzukehren und die Grenze nebst zwei einfachen von ihr abhangigen Grossen, die so unzertrennlich mit ihr verbunden sind wie der Sinus zum Cosinus gehort, als Functionen des Integrals zu behandeln, gerade wie man schon friiher zur Erkenntniss der wichtigsten Eigenschaften der vom Kreise abhangigen Transcendenten gelangt war, indem man den Sinus und Cosinus als Functionen des Bogens und nicht diesen als eine Function von jenen betrachtete. " Ein zweiter Abel und Jacobi gemeinsamer Gedanke, der Gedanke das Imaginare in diese Theorie einzufiihren, war von noch grosserer Bedeutung und Jacobi hat es spater oft wiederholt, dass die Ein- fiihrung des Imaginaren allein alle Rathsel der friiheren Theorie gelost habe." THE PROBLEM OF INVERSION. 175 Art. 154. If we had not wished to stu dy th e one-valued functions of position on the Riemana surface s = VR(z), we might have shown immediately that /^ \2 For in the differential equation (cf. Art. 106) or Ao{z)(^J+ Ariz) ^+1=0, when a definite value is given to z, say zq, then the sum of the two roots of the equation is ,^. +/^\ = - ^'(^o) UA W/2 Ao(zo) ' ^'' On the other hand, corresponding to the value Zq there are within the initial period-parallelogram two values of u say Wi and M2- Also, since Ml -I- M2 = Constant, it follows that ^ + ^ = 0. (ii) dz dz But the left-hand side of (i) is the same as the left-hand side of (ii), and consequently* Ai{z) = 0. Art. 155. A Theorem due to Liouville. Suppose that w = F{u) is a doubly periodic function of the fcth order with periods a and b ; also let z = f{u) be a doubly periodic function of the second order with the same periods. There exists then (see Art. 104) an integral algebraic equation of the form g^^^ ^) _ 0, which is of the second degree in w and of the fcth degree in z. This equation may be written Lu)2 + 2 Pw + Q = 0, L, P and Q being integral functions of degree not greater than k in z. It follows that -P±\/P2_LQ - P + a w = === — — = ) where o=±VP^ - LQ. We therefore have ^ ^ j^^ _^ p so that a is a one-valued function of w. We saw above that j. . f- = ±VR{z). du * Cf. Harkness and Morley, Theory of Functions, p. 293, where numerous other references are given. 176 THEORY OF ELLIPTIC FUNCTIONS. It is also seen that corresponding to one value of z there are two values of a differing only in sign, and corresponding to this same value of z there dz are two values of — which differ only in sign. du Hence T{z)= a -^{dz/du) is a one-valued function of z with periods a and h. It follows also (see Art. 104) that an algebraic equation exists between a -i-{dz/du) and z; and consequently cj -^(dz/du) is inde- terminate for no value of u. But a one-valued function which has no essential singularity is a rational function (Chapter I). Hence T{z) is a rational function of z. It is also seen that j -P + T(z) ^ du , w = = p + qs, p and q being rational functions of z. dz We have thus shown* that w may be expressed rationally in z and s = — ; du or w = R{z, s), which theorem is due to Liouville. Art. 156. A Theorem of Briot and Bouquet {Fonctions Elliptiques, p. 278). Suppose that w = F{u) is a doubly periodic function of the kth order with primitive periods a and b and let t = fi (u) denote any other doubly periodic function with the same periods. We shall show that t is a rational function of w and There exists (Art. 104) between w and w' = — an integral algebraic du equation (I) G{w, w') = 0, which is of the ^th degree in w'. Hence corresponding to one value of w there correspond in general k values of w' in a period-parallelogram. Suppose that for the value wq there correspond the k values Wi',W2', . . . , Wk'. (1) Further, since w is of the kth order, there correspond k values of u to wq in the period-parallelogram, say Ml, W2, . , Uk. (2) We also know that between the functions t and w there is an algebraic equation (II) Gi{w,t)=0 of the A;th degree in t, so that corresponding to the value wq there are k values of t, say t f , /ox fi, f2, • ■ • , tk. (3) * Liouville, Crelle, Bd. 88, p. 277, and Comptes Rendus, t. 32, p. 450. THE PROBLEM OF INVERSION. 177 We note that the system of values (3) correspond to the system of values (1) in such a way that to every system of values (w, w') there corresponds one definite value of t and only one. The functions tw',tw"^, . . . , tw'^-i enjoy the same property. It follows that the sums '{u), 4>{v), '{v), say 4>{u + v)= Ri[cl>{u),,j,'{u),'{v)l (1) where R with a suffix denotes a rational function, and consequently also cl>'{u + i;) = R2mu), {v), cl>'iv)]. (2) For the present admit the above statements. By Liouville's Theorem it follows that w = F{u) is a, rational function of <^(M) and cf>'{u), or p^^^ ^ Rs^u), {u + v), 4>'{u + v)] = Rl4>{u),'iu),c{>{v),cj>'{v)]. (3) Also from Briot and Bouquet's Theorem {u)=Rs[F{u),F'{u)] ^""^ '{u)=RAF(,u),F'{u)]. t Hence from (3) we see that Fiu + v) = RrlFiu), F'(u), F{v), F'{v)l It has therefore been proved, since w satisfies the latent test expressed by the eliminant equation, that this function has an algebraic addition- theorem, and in fact is such * that F(u + v) may be expressed rationally in terms ofF{u), F'{u), F{v), F'{v). This property, see Chapter II, also belongs to the rational functions and to the simply periodic functions. It has thus been demonstrated that to any one-valued function 4>{u) which has everywhere in the finite 'portion of the plane the character of an integral or {fractional) rational function, belongs the property that (f>{u + v) is rationally expressible through (f>{u), 4>'{u), 4>{v), 4>'{v). As it was shown in Art. 74 that a one-valued analytic function cannot have more than two periods, it follows (cf. also Art. 41) that a one-valued analytic function which has an algebraic addition-theorem is either I, a rational function of u, II, a rational function of e " , III, a rational function of z and — • du The first two cases (Art. 41) are limiting cases of the third. Every tran- scendental one-valued analytic function which has an algebraic addition- theorem is necessarily a simply or a doubly periodic function. * See Schwarz, Ges. Math. Abhandl., Vol. II, p. 265. THE PROBLEM OF INVERSION. 179 Art. 159. We have seen that any rational function of z and s is a one- valued function of position on the Riemann surface s. Hence the function w of the preceding article, which is the most general one-valued doubly periodic function, is a one-valued function of position on the Riemann sur- face.* The quantity s is the root of the algebraic equation and by adjoining this algebraic quantity to the realm of rational quanti- ties (z) we have the more extended realm (z, s) composed of all rational functions of both z and s. This latter realm includes the former. Since all functions of the realm (2, s) are one-valued functions of position on the Riemann surface T and since this surface is of deficiency or order unity, we may say the realm (s, z), the elliptic realm, is of the first order, the realm of rational functions {z) being of the zero order. We thus see that the study of functions belonging to the realm of order unity is coincident with the study of the doubly periodic functions and in fact the study of one necessitates the study of the other. The elliptic or doubly periodic realm (s, z), where s = V A (z — a{) {z — 02) (2 — as) (2 — 04) = --, du degenerates into the simply periodic realm when any pair of branch- points are equal and into the realm of rational functions (2) when two pairs of branch-points are equal (including of course the case where all the branch-points are equal). Thus the elliptic realm (2, s) includes the three classes of one-valued functions : First, the rational functions, Second, the simply periodic functions. Third, the doubly periodic functions. All these functions, and only these, have algebraic addition-theorems. In other words, all functions of the realm {z, s) have algebraic addition- theorems, and no one-valued function that does not belong to this realm has an algebraic addition-theorem. We have thus proved that the one-valued functions of -position on the Riemann surface s2 = R{z), R denoting an integral function of the third or fourth degree in z, belong to the closed realm {z, s) of order unity, and all elements of this realm and no others have algebraic addition-theorems. * Cf. Klein, Theorie der elliptischen Modulfunctionen, Bd. I, pp. 147 and 539. CHAPTER VIII ELLIPTIC INTEGRALS IN GENERAL The three kinds of elliptic integrals. Normal forms. Article 160. At the end of the last Chapter we saw that the most general elliptic function could be expressed as a rational function of z, s. We shall now consider the integral of such an expression.* Let Ri{z, s) denote a rational function of z, s. This function may be written in the form n (z s) = ^o + ^ig + ^2S^ + • ■ ■ + AkS'' ' Bo +Bis + B2S2 + ■ • • + B,s^ ' where the A's and B's are integral functions of z. Owing to the relation s^ = A(z — fli) (z — a2) (z — as) {2 — a^), it is seen that the even powers of s are integral functions of z, while the odd powers of s are equal to an integral function of z multiplied by s, so that R (z s)= ^0 +^i'^ = Uo' + A,'s){Bo' ~ Bi's) ' Bo' + Bi's Bo'^--Bi'2s2 _ C + Ds ^' where C, D, and E are integral functions of 2 as are Aq', Ai', Bo' and Bj' Writing f = P^'^) ^"^^ f = ?(^)' it is seen that ^^^^^ ^^ ^ ^^^^ ^ ^^^^,^ ^ ^^^^ ^ Q(^^ s where q{z)'S^ = Q{z) and where p{z), q{z), and Q(z) are rational functions of 2. (See also Arts. 125 et seq.) Consider next the integral CRi{z,s)dz = j'p{z)dz + C^^dz. The first integral on the right may be reduced at once to elementary integrals, so that we may confine our attention to the integral /^^^dz, which may be written j -^-^L^dz, « -^ Vr{z) f(z) denoting a rational function of z, and s = \/R{z). * Legendre, Mimoire sur les transcendantes elliptiques, 1794. See also Legendre, Fonctions Elliptiques, t. I, Chap. I. 180 ELLIPTIC LNTEaRALS IN GENEEAL. 181 Art. 16L Suppose * in general that R{z) = Co2" + Ci2"-i + . . • + C„, where the C's are constants. When n is greater than 4, the integral /: /(g) dz VR{z) is no longer an elliptic but a hyperelliptic integral; when n = 3 or 4 we have the elliptic integrals, and when n = 2 we have the integrals that are connected with the circular functions. The rational function f{z) may be written /(.) = £lM = G(.) + ^, g(z) g{z) the g'a and G's denoting integral functions, and say g{z) = B{z - ^i)-*. {z - h2)^-{z - bs)^' ■ • • . Hence when resolved into partial fractions f{z) = G{z) +]£ — ^^^>—, (A^< constants), i {z - bi)^' and also ffmL^ f-^s^dz+x^x.f '^ — ■ J VR{z) J VR{z) T 'J (z-biY'VRiz) Since G{z) is an integral function, the first integral on the right-hand side may be resolved into a number of integrals of the form f-L^dz. J VR(z) We thus have two general types of integrals to consider, , dz VR{z) and Hk = / ; • J (z-b)''VR(z) Art. 162. Form the expression -^[zWR(z)] = kz'^-^VR(^) +l-PgL^^= ^]— [2kR{z) +zR'iz)] dz'- 2VR{z) 2\/R{z) - ^^~^ [(2A;+n)Co2"+(2A;+n-l)Ci2"-i + (2A;+n-2)C22"-3+ • • 2\/R{z) + {2k + \)Cn-lZ+ 2kCn]. * Briot et Bouquet, Fonctions Elliptiques, p. 436; see also Koenigsberger, Ellip- tische Punctionen, p. 260; Appell et Lacour, Fonctions Elliptiques, p. 235. 182 THEORY OF ELLIPTIC FUNCTIONS. It follows through integration that 2z*\/fi(^) = i2k + n)CoIk + n-i + {2k + n - l)Cih+n-2 + ■ ■ ■ + {2 k + l)Cn-lIk + 2kCnIk-l. If in this expression we put A; = 0, it is seen that I„-i may b e exp ressed through In-2, ln-3, ■ ■ ■ , Iq, I-i and through the function \/R{z); when fc is put = 1, we may e xpress In through In-i, In-2, ■ ■ ■ , /o and through the function zVR{z). If further we write for In-i its value, we may express /„ through In-2, In-z, ■ . ■ , lo, I -i and an algebraic function. This algebraic f unct ion is an integral function of the first degree in z multiplied by VR{z). Continuing in the same manner, we may express In+X through In-2, In-3, ■ ■ ■ , Iq, I -I and an algebraic fun ction which is an integral function of the ^ + 1 degree in z multiplied by ^R{z). Art. 163. We consider next the integrals of the type Hk- Form the expression d r v^i ^ _ k ^p-. 1 R'{z) dzl{z-h)'^\ U-6)'= + i 2 (2_6)*Vi2{2) 1 2\/R{z){z- 6)«=+i [-2kR{z) + R'{z){z-h)]. If we write - 2 kR{z) + R'{z) {z-b)= {z). It follows, since 4>{z) = ) + • • • , that ' ^- ^■ {z) = -2kR{b) + 1-=^ R'{b){z -b) + ?—^R"(b){z - 6)2 •3! (n — 1)! + ^~ ^^ fiW(fe) (3-6)". n! We therefore have ^[-^^1=^ \_2kRib) + ^~^^^R'{b){z-b) + . ■ d2L(2-6)*J 2VR{z){z-b)'' + ^\ 1! _^n_-J_-2fc^(„_i)^^^^^_^^„_i _^ "-2fe jg(„)(^)(^_b)nf (n — 1)! n! ELLIPTIC INTEGRALS IN GENEEAL. 183 Integrating it is seen that '^^^^=. - 2kR{b)Hk^, + '^-=^R'{b)Hk ^"^^^ R"{h)Hk-^+ ■ ■ ■ (n — 1)! n\ If we put k = 1, we see that H2 niay be expressed through Hi, Ho, H-u H-2, . . . , //-(„_2), and ^^^ . z — b This is correct only if R{b)7^ 0; i.e., if b is not a root of the equation R{z)= 0. This case is for the moment excluded. We note that Ho= f~^=Io; H-,= f(A,zM^^i,-M,;-.-; J VR{z) J VR{z) „ r (z-b)--^dz W-(n-2)= I ;= . From this it is seen that the integrals Ho, ff-1, H-2, • • • , H-(7!-2)niay be expressed through integrals of the type /*. Hence the integral Hi alone offers something new. We note that H2 may be expressed through Hi, Iq, /],..., In-2 and through an algebraic function of z. If we put k = 2, we niay express H3 through H2, Hi, . . . , H_(„_3) and through ^ ; or, if for H2 we write {z-by its value just found, H3 may be expressed through Hi, Iq, Ii, . . . , In-2 and an algebraic function of 2. In general, we may express H^ through Hi, /o, •'^i, • ■ • , In-2 and an algebraic function of z. We thus have to con-sider only the integrals Zq, /i, . . . , 7,1-2 and Hi = 7-i, since /-i is a special case of Hi, viz., when 6 = 0. If 6 is a root of the equation R{z)= 0, then the term with H^+i drops out. Since R(z) cannot have a dou ble root, as otherwise it could be taken from under the root sign in \/R{z), we may in this case express Hi through the integrals Ho, H_i, . . . , H_(„_2), -^; and conse- z — quently through integrals of the type Ik alone. Art. 164. We have therefore to consider the integrals J _ C z^dz where k = 0, I, . . . , n — 2, where n is the degree of the integral func- tion R{z), and in addition the integral fj = f d^ 184 THEOKY OF ELLIPTIC FUNCTIONS. where 6 is a root of the equation g{z) = 0. We note that there are as many integrals of the type Hi as there are distinct roots of the equation g{z) = 0. The quantity b is called the parameter (Legendre, Fonctions Elliptiques, t. I, p. 18) of the integral Hi. Art. 165. For the elliptic integrals, if n = 4, we have the integrals h, h, h, Hi;iin = 3, there are the integrals Iq, Ii, Hi. In the first of these cases we shall see that /i reduces to elementary integrals; and with Legendre we call /dz — an elliptic integral of the first kind, \/R(z) 'h VR{z) ^ ^ an elliptic integral of the second kind, and VRiz) H\= I — an elliptic integral of the third kind. J {z-b)VR{z) Legendee's Normal Forms. Art. 166. In the expression dz dz VR{z) VA{z - Oi) (z - 02) {z - as) {z - 04) let us make the homographic transformation at + h 2 = ; • ct + d It follows that g - a, = (" ~ "°^^^ + h-dak (fc = 1^2,3,4) ct + d ^"^^ , ad -he dz = (ct + d)2 We then have dz ^^^^ (ad - bc)dt \/ A[{a—ca-i)t+b—da-i] [(a—ca2)t+b—da2][{a—ca3)t+b—da3][(a—ca4)t+b—dai] We note that the expression under the root sign is not essentially changed, since we still have an integral function of the fourth degree, the branch-points, however, being different. Legendre * conceived the idea of so determining the constants a, b, c, d that only the even powers of t remain under the root sign. If we neglect- the constant A, the radicand may be written [got^ + git + g2][hol^ + hit + h2], * Legendre, loc. cit., Chap. II. ELLIPTIC INTEGRALS IN GENERAL. 185 where go = {('■ — c«i) (« — ca2), gi = (a — coi) (6 — da2) + (a — ca2) (6 — dai), g2= {h - dai) {b - da2), and where /iq, 'ii, ^2 are had when we interchange ai with 03 and 02 with 04 in the expression for the ^'s. That the coefficients of t^ and t disappear, we must have hogi + gohi = 0, 91^2 + hig2 = 0. These two equations are satisfied if we put gi = and hi = 0. From the expression gi = it follows that 2 ab — (ad + be) (oi + 02) + 2cdaia2 = 0; and from hi = we have 2 ab — (ad + be) (03 + 04) + 2eda3ai = 0. These two equations may be written - + -I («! + 02) + 2 - - aia2 = 0, b a\ b a 2 - Ir + -! («3 + 04) + 2 ^ -0304= 0. lo al b a From them we may determine — I — and considered as unknown ..,- b a b a quantities. If as + 04 = Oi + 02 and a^ • a^ = ai • a2, the two equations reduce to -one and then we need only determine the quantities 7 + - and - • - b a b a so that they satisfy the one equation. When these two quantities have ■ ■ ■ d c been determined, the quantities - and - may be found from a quadratic ,. b a equation. When these conditions have all been satisfied, then in the expression [got^ + git + g2][hofi + hit + h2] the coefficients of t in both factors drop out. We have finally dz (ad — be)dt VR(z) VAigot^ + 92) (hot^ + ^2) Legendre further wrote 2P = _ p2^ ^ = _ ^2 92 h2 so that ^2 (ad - bc)dt VR{z) VAg2h2{l - p2<2) (1 - q2t2) 186 THEORY OF ELLIPTIC FUNCTIONS. If finally we write t = - (the Gothic z being a different variable from P the italic z), we have - (ad — bc)d2 dz V VR{z) >jAgM^-^'}\^-f2^'\ If we put ^ = ^^ and C = p2 '" p VAg2h2 the above expression is dz Cdz VR{z) V(1 - z2) (1 - /c2z2) The quantity k is called the modulus (Legendre, loc. cit., p. 14). In theo- retical investigations it may take any value whatever, real or imaginary; but in the applications to geometry, physics, and mechanics we shall see in the Second Volume that it is necessary to make this modulus real and less than unity. Art. 167. If we make the above substitutions the general integral of Art. 160 fm± becomes • f ^^^^^^ J VRiz) J V{1 - Z2) (1 - fc2z2) where /(z) denotes a rational function of z. We may write this function in the form f, . ^ <^(Z2)+Z(/.1(Z2) ■'^ ' >|r(z2)+Z^^l(z2)' where <^, <^i, ^Ir, yjri denote integral functions. If we multiply the numer- ator and the denominator of this last expression by •^(z2)— zi|ri(z2), it is seen that f{z) = fo{z^)+ z/i(z2), where /o and /i are rational functions of z. The above integral correspondingly becomes zA(z")rfz f /(z)rfz ^ r /o(z2)dz ^ r J Va-z^)n-k^z2) -' \/ri - z2ui - ifc2z2-v j^ \/(l-z2)(l-/(;2z2) J \/(l - Z2) (1 - jfc2z2) J ^(1 - z2) (1 - ifc2z2) The second integral on the right-hand side may be reduced to elementary integrals by the substitution z2 = (^. Proceeding as in the general case above and noting that iLrz2fc+l x/(l _ z2)Q _ ;(.2z2)] = £o? + ^'^ + '^ -2^ dz \/(l-z2)(l-/(;2z2) and d_ r z\/(l-z2)(l-Fz2) 1 _ gp + ai(z2 - b) + a^jz^ - 6)2 + a^jz^ - 6)3 dzl (z2-6)ft J (z2 - b)'fc+i ^(1 - z2)(l - A;2z2) ELLIPTIC INTEGRALS IN GENERAL. 187 it may be shown that the integral fl fo(z^)dz V(l-z2)(l-A;2z2) is dependent upon the evaluation of the integrals r dz r -' V(l-z2Kl-fc2z2)' JVd-: V(l-z2)(l-fc2z2) J \/(l-z2)(l-fc2z2) dz I-, {z? - 6)V(l-z2)(l-A;2z2) These integrals are known as Legendre's normal integrals of the first, second, and third kinds respective^. Art. 168. The name " elliptic integral " is due to the fact that such an integral appears in the rectification of an ellipse. Writing the equation of the ellipse : -1;+ ji; = 1 , the length of arc is determined through a-^ o2 If the numerical eccentricity is introduced: p / a2-e2x2 f «^-^^^^ rf^ Jo V a2 - x2 Jo V(a2 - x2) {a^ - e^x^) ' If further we put x = a sin (f>, it is seen that s = / Vl — e2 sin2^ dd>. Jo This is also taken as a type of normal elliptic integral of the second kind,* being in fact composed of the normal forms of the first and second kinds as above defined. Regarding the forms of the integral of the second kind see Chapter XIII. Art. 169. If the integral which we have to consider is of the form mdz k Vaz^ + 3 b22 + 3 cz + d where f{z) again denotes a rational function of z, we may by writing z = mt + n "^al^6 a23 + 36z2 + 3c3 + d = 4i3_52«-S'3, where 92 ^^nd gz are constants. This is effected by writing n = , am? = 4. a * The elliptic integral of the second kind was considered by the Italian mathema- tician Fagnano (1700^-1766) and was later recognized as a peculiar transcendent by Euler (in 1761). 188 THEORY OF ELLIPTIC FUNCTIONS. The above integral then becomes Fit)dt /; V'4 <3 - g2t - 93 where F{t) is a rational function of t. The evaluation of this integral (cf. Art. 165) depends upon that of the three typical integrals dt r dt r tdt r J V4:fi -got - 03 J Vifi - 02« -93 '^ I 92« -93 J Vifi - 921 -93 'J {t-h) s/i t^ - 92t - 93 which correspond to the normal forms employed by Weierstrass. Art. 170. In the expression (1) R{z) = A{z - ai) {z - aa) (z - as) (z - a^), make the homographic transformation (2) .= «-±^, l-flZ and so determine the coefficients * that to z = ai, z = a2, z = as, 2 = 04 correspond z^-i. z = -l, z = 4-l, z = + |. It follows immediately from (2) that ,o\ P(l + kz.) .,< 0(1 + z) (3) z - ai = i^ '-, (4) z-a2=^f '-, \ — (IT. 1 — /iz /t-N r(l — z) /o\ s(l — kz) (5) z-a3 = -^ i, (6) z-ai=^ ^-, 1 — /iZ 1 — flZ where p, q, r, s are constants which may be determined as follows: In (4) write z = as, z = 1, and in (5) put z = a2, z = — 1. We thus have = -A^, - - 2'- as — a2 = — ^ — I tto — as = 1 - u " 1 + fl Equations (4) and (5) thereby become z — ao (7) ^ ~ "-2 ^ 1 — /( _ 1 + z ,n, z — as _ 1 + /t _ 1 — z 03 — 02 2 I — fjiz 02 — as 2 1 — uz In the same manner we derive from equations (3) and (6) the following : 1-'- 1+^ (Q) z — ai _ k 1 + fcz /.„, z — a^ k_ 1—kz 04 — tti 2 1 — ,uz ' ai — a4 2 1 — fiz * Koenigsberger, Elliplische Fundionen, p. 271. ELLIPTIC INTEGEALS IN GENERAL. 189 Equations (7) and (8) become through division Z — a2 ^ ;U — 1 _ 1 + z — as /i+1 1 — z Writing in this equation 2 = 04, z = - . we have k 04— 02 _ n — \ fc + 1 . 04—03 /i+l k — 1' and similarly for the values z = ai, z = , the same equation gives k ay — a2 _ ^ — 1 fc — 1 «i — 03 ^+1 k + \ The quantities A; and n may be determined from the last two expressions in the form (11) / I - /c V_ ai - a2 _ 04 - 03 \1 + /c/ Oi — 03 ai — 02 (12) 1 + /t ^ «! — as _ 1 — fc 1 — ^ Oi — a2 1+fc From the equations (9) and (7) we have ^ ^ (l-g)fc(a4-a:) ^^ ^ (l-;.2)(,3_^^) dz 2(1 - /iz)2 ' dz 2(1 - /iz)2 and consequently dz 01 --"')(! -fi)^'t(«4-«i)(a3-«2) ^^^^ dz ^ 2(1^;;^^ Through the multiplication of (7), (8), (9), (10) and (13), it follows at once that dz J VR{z) MJ V(l-z2)(l-fc2z2)' M = I4 / ^(«3 - a2)(a4 - ai) , where — „ ., 2 » fc and where z and z as determined from (7) and (8) are connected by the relation 03 + 02 03 - ^2 Z - /i 2 = -h " ' — } 2 2 1 - ,«z the quantities ,u and fc being determined from equations (12) and (13). 190 THEORY OF ELLIPTIC FUNCTIONS. Art. 171. If in the equation \l+k) ai- — a2 a^— as - • y a3 04— a2 we put the right-hand side = t, then the six different anharmonic ratios which may be had by the interchange of the a's are denoted by 1 ,_ _^. _j^ T-1 . T 1 — r T— 1 T and corresponding to each of these values there are two values of k, in all twelve values of k. Denoting any one of these values by k, it is seen that all twelve may be expressed in the form ±,. ±1 ±(i^y, ±(i±^v Ji^)\ ±(i±i^l k Kl+Vk/ \l-Vk/ \l+iVk/ \l-iVk/ (Cf. Abel, (Euvres, T. I, pp. 408, 458, 568, 603; Cayley's Elliptic Func- tions, p. 372.) Remark. — We may make use of the above results to transform the expression dz VA{z - ai){z - 02) (z - as) into Legendre's normal form. Noting that A(z — ai)(z — 02) (2 — 03) = Limit (z — ai){z — 02) (2 — 03) (z — 04) , 04=00 L ^4 J 4 we have to write in the formulas above — — in the place of A, and let 04 04 become infinite. We then have r dz _ _i r J VA(z- ai)(z- a2)(z- as) MJ- dz \/A{z-ai)iz-a2)iz- a^) MJ V {I - z^) (1 ~ k^z^) where m = i v/^ii^^^^-^ 2 V k 2 V k ^ ^ ffl3 + ^2 _j_ as - 02 2 2 1 -kz and /L:^y=^i:z^, ^ = fc. \1 + k/ oi— as ELLIPTIC INTEGRALS IN GENERAL. 191 Art. 172. In the expression \/R{z) = \/ A{z — ai) {z — 02) (z — as) (2 — a^) = {z- a,Y J A '-^^ . ^-^^ . '-^^^, y z — a-i z — ai z — tti write = t — e, or z = — ^ -> z — ai t — e — 1 dt = ^±^=-^^dz. (Z - Oi)2 If we put V(ai — 02) (fli — as) A = 2 M, we have ai-a4 y ( \ ai - a2/ ) ( \ Oi - as/ Choose e so that 3 e - "^ ~ "^ - "^ ~ "^ = 0. Oi — a2 fli — a2 Let ei be the value of e that satisfies this equation, and write e, = e,- "^^^^^ and e^ ^ e, - ^^Sl^. ai — a-i o-i — 0.3 We finally have dz 1 dt VR(z) 2M V{t- ei) it - 62) (i -ea) ^_1 dt M V4t^-g2t-gs' where 6162 + 6263 + 6361 = - ig'2, 616263 = igs. It also follows that rPiz)dz _ r p(t)dt J VR(zj J V4t^-g2t-g3 where P and p denote rational functions. The quantities ^2 and ^3 which occur in Weierstrass's normal form are called invariants, their invariantive character being especially evidenced in the Theory of Transformation. We may now consider more carefully their meaning. Art. 173. Write u = f-^' J VRiz) where the function R{z) may be written R{z) = Ooz* + 4aiz3 + 6032^ + 4032 + 04. 192 THEORY OF ELLIPTIC FUNCTIONS. Write z = ^, X2 J _ X2dXi — XidX2 X2 where the variables xi, xj individually are not determined, but only their quotient. We then have R{z)= — -Ao-oXi* + 'iaiXi^X2 + 6a2Xi^X2^ + 403X13:2^ + 04X2*} -4fiXl,X2). X2^ J_ X2* It is seen that/(xi, X2) is a binary form* of the fourth degree. We have at once dz _ X2dxi — Xidx2 VR(zj Vf{x,,X2) If next we write xi = oi/i + by2, X2 = cyi + dy2, 2 _ aj!/i + hy2 ^ cy\ + dy2 it is seen that/(xi, X2) becomes another binary form 4>{y\, j/2) of the fourth degree. Art. 174. In general make the above substitutions in the binary form of the nth degree f(xi, xn) = aoJTi" + niaia:i"-ia;2 + n2a2Xi-^-^X2^ + • • • + a„2:2", where n\, n2, ■ ■ . are the binomial coefficients. We thus derive another binary form of the nth degree 4>{,y\, 2/2). It is seen at once that — r/(a;i, X2) = Oo2" + niajz"-! + w2a20"-2 + • • ■ + a„ 3:2 = ao{z - ai) {z - a2) . . . {z - a„), say. It follows that /(xi, X2)= ao(xi - aiX2) (xi — 02X2) . . . (xj - a„X2), and correspondingly iyu 2/2)= ao'Cvi - /3i7/2) (2/1 - P2y2) ■ ■ ■ (?/l -^nVi). Further, since Xj - a 1X2 = ayi + 61/2 - ai(c!/i + d)/2) = {a- aic)\yi i 1/2 \ > ( a — aiC ) ♦ Bocher, Introduction to Higher Algebra, p. 260. ELLIPTIC INTEGRALS IN GENERAL. 193 it is evident that one of the /9's, say o aid — h Pi = , a — ttic and similarly o a'yd — b , P2 = -^ ' etc. a — a^c From this it is clear that, if some of the a's are equal, some of the /9's are also equal, and that there are just as many equal roots in the equation ^(2/i> 2/2) = as there are in/(a;i, x^) = 0. Art. 175. The above correspondence gives rise to the following con- sideration: Suppose we have given the quadratic form aoz^ + 2 ttiz + 02. The roots of the quadratic equation Oqz^ + 2 a\z + a2 = o-\ , 1 « / 9 are 2 = *■ ± — \/ a^^ — a^a^. ao ao If we write aj^ — ao02 = D{ao, a^, 02), we know that the two roots of the quadratic equation are equal if D is equal to zero. The quantity D after Gauss is called the discriminant of the quadratic equation. Also for forms of higher order we may derive such discriminants, whose vanishing is the condition that the associated equation have equal roots.* The quantity D(ao, a-i, 02, ■ ■ ■ , an) is an integral rational function of ao, Oi, • • • , dn and is homogeneous with respect to these quantities. If next we form the discriminant D{ao', a\' , 02', • • • , a»') of the form 0(2/1, 2/2)= ao'2/1" + niai'2/i"-Vv2 + n^p.^y^'^-'^yi^ + • ■ ■ + a„'?/2", then the vanishing of this discriminant is the condition that 4>^y\, j/2) have equal roots /?. But we saw that ^(j/i, 2/2) had equal roots when the roots of/(xi, X2) are equal. It follows that D(ao', fli', • ■ • , o„') = C'I>(ao, oi, . . . , a„), where C is a constant factor. This constant factor * is | ad — 6c |»('»-i). * Cf. Salmon, Modern Higher Algebra, p. 98; Burnside and Panton, Theory of Equa- tions (3d ed.), p. 357; etc. * Cf. Salmon, loc. cit., p. 108; Bocher, loc. cit., p. 238. 194 THEOEY OF ELLIPTIC FUNCTIONS. Art. 176. If the function /(2;i,X2) =aoXi" + niaia;i"-*X2+ • • • + o„X2" becomes through the substitutions xi ayi + by2, X2 cyi + dy2, iyi, 2/2) = 0,0'yi" + Wiai'2/i"-i!/2 + • • • + a„'!/2", and if / is a function of the coefficients such that I(ao',ai', . . . , a„') = (ad - bc)>'Iiao, a^, . . . , a„), where fi is an integer, then / is called an invariant of the form/(xi, X2)- It may be shown * that, if / is an invariant, ft must be equal to J np, where p is the degree of / with respect to the coefficients Oq, ai, . . . , a„. The quantity /t is sometimes called the index of the invariant. The following theorem is also truerf All the invariants of a binary form f{xi, X2) may be expressed rationally through a certain number of them which are called the fundamental invariants. For the form of the fourth degree, f{xi,X2) = a^xi* + 4aiXi^Z2 + 6o2Zi2a;2^ + 403X1X2^ + a4^2*, there are only two fundamental invariants (cf. Sylvester, Phil. Mag., April, 1853). The one of these is % I2— a^a^ — 4oia3 + 3a2. If by the given transformations we bring /(xi, X2) to the form ^(2/1, 2/2) = ao'i/i* + 4ai'!/i32/2 + ■ ■ • + a^y2^, then it is easy to show that ao'tti - 4ai'a3' + 3 02'^ = {a^ai - 4aia3 + Za2^){ad — be)*. In this case p = 2, n = 4, // = J n,o = 4. We thus have 1 2' = hiad - bey. The other fundamental invariant § is I3 = 000204 + 20102^3 — 0,2^ — OqOs^ — 0401^. It is seen at once that h' = hiad - 6c) 6. * Cf. Salmon, loc. cit., p. 130; Buraside and Panton, loc. cit, p. 376. t Cf. Salmon, pp. Ill, 132, 175; Bocher, loc. cit., Chap. XVII, and Buraside and Panton, p. 405. t Salmon, loc. cit., p. 112. Cayley, Cambridge Math. Journ. (1845), Vol. IV, p. 193, introduced this invariant. § To Boole, Cambridge Math. Journ. (1841), Vol. Ill, pp. 1-106, is due the discovery of this invariant; see also Cambridge Math. Journ., Vol. IV, p. 209; Cambridge and Dublin Math. Journ., Vol. I, p. 104; Crelle, Bd. 30, etc.; and Eisenstein, CreUe, Bd. 27, p. 81; Aronhold, Crelle, Bd. 39, p. 140. ELLIPTIC INTEGRALS D^ GENEEAL. 195 Art. 177. The discriminant D of the binary form /(xi, X2) may be rationally expressed (cf. Salmon, loc. cit., p. 112) in terms of I2 and 1 3 in the form It is evident that D = U^ 27/32 jy = j^'Z _ 27/3'2 = Diad- bc)^^. Art. 178. The functional-determinant or Jacobian of the two forms •^i(xi, X2), ' [3^2] [3^?] Suppose next that We then have dxl dXi 9Ai dXi 8yi a,¥2 dX2 = [F]s d!/l a-«2 ^2/2 d>(2 dyi a.V2 aj/i 32/2 a;i = Ai (2/1, 2/2) = aj/i + by2, X2 = '^2(2/i> 2/2) = cyi + dy2. * = [F]s a, b c, d = [FUad - be). 196 THEORY OF ELLIPTIC FUNCTIONS. Art. 179. Let/(a;i, X2) be a binary form of the nth degree. It is seen that ^/^^i' ^^^ and M^it^are binary forms of the degree n - 1. dxi 9x2 The functional-determinant F of these tv/o functions ay 3X12' 9¥ ay 6x1 ax2 _ay 8x2^ i:^(/), say, Suppose that by the ax 18x2 is called the Hessian covariant * of the form /. substitution xi = ay\ + by 2, X2 = cyi + dy2, the function /(xi, X2) becomes 4>(yi, 2/2) and form the Hessian covariant for this latter function, viz., n Vaj/i/ H() We have ai/i ai/2 \9y d /a^ ay2 \a.//2, or and similarly ayi \dy2/ dyz \a//2/ LdXiXa.vi Laa:2jsa!/i' a*/i La^ijs Laa;2j Idxijs Id d4> a?/i dy2 Laa^ijs Laa;2j When these values are substituted in the above determinant, it follows that a,'/iLaa;iJs dyi\_dx2js a;/2LaxiJs a!/2Lax2j d ~ H{4>)- 1 = {ad — he) ! Laxi J a^iLa^iJs a!/iLdx2js a?/2Laxijs a2/2La3;2i at/iLaa^iJ, a?y2LaxiJ, ^r^i Am a!/iLaa;2js a,i/2Lax2j; Further, since -^[^1 = T^l a + [^^1 c, etc ai/iLaxiJ, LaxiH La^iaa;2j3 Laxi^j, Laxiaxgjs Laa^rjs L axiax2j, a\^L-'\^-c\m, ,\^f--\+d\m Laxiaxzjs \jix2^M Laa;iax2j, \_dx2^}s we have H{) = {ad-bcy\H{f)\ *H2 X2= CVl+ dW2 Art. 180. We may consider more closely the meaning of the covariant. Suppose we have a binary form /(xi, X2) of the nth degree. With its coefficients Oo, fli, • • • , a„ and with Xi, X2 we form an expression C\ao,ai, . . . , a„; a;i,a;2}, C denoting a functional sign which with respect to xi, x^ is of the vth degree, and in regard to the a's it is of the joth order. Suppose further that by the substitution Xi ayi + by2, X2 cyi + dy2, the function /(xi, X2) becomes ^(j/i, 2/2)- With the coefficients a^', a/, . . . , a„' of 0(2/i, 2/2) and with yx, j/2 we form the same function C[ao', tti', . . . , a„'; 2/1,2/2}- If then C[ao',ai', . . . ,a„'; j/i, ?/2 } = (ad-6c)''[C{ao, fli, where p. = \{np — v), we say that C is a covariant * of the binary form /(a; 1, X2)- Art. 181. In the theory of covariantsit is shown that for every binary form f{x\, X2) there is a finite number of independent covariants, through which all the other covariants mxiy be expressed, t If /(xi, X2) is a binary form of the fourth degree, say /(xi, X2) = aoxi^ + 4aiXi3x2 + Qa2XiH2'^ + 403X1X2^ + 04X2*, there are two fundamental covariants (Salmon, loc. cit., p. 192): The one is the Hessian, where , an'jXi, X2 \^xi = ayi + bV2 ' ■'Xi'-cyi+dy-j and consequently fi n-2+n-2=2n i[2n-2n + 4] = 2. 2; * Salmon, loc. cit., p. 135; Burnside and Panton, toe. cit., p. 376. t Salmon, loc. cit., pp. 132, 175, 176; and see also Clebsch, Theorie der bindren alge- braischen Formen, pp. 255 et seq. 198 THEOEY OF ELLIPTIC FUNCTIONS. This covariant is H{f) = iaoa2-ai^)xi* + 2{aoa3- aia2)xiH2+iaoai+ 2aias-3a2^) Xi^X2' + 2(aia4 - a2a3) x-^^x.^ + (0204 - a^) X2*. The other fundamental covariant is the Jacobian of the quartic and its Hessian: 8 bx 1 bX2 dHU) dHjf) dxi 3x2 For this covariant it is seen that v = n-l + 2n-5 = 3n-6, p = 1 + 2 = 3, fi = i{np - v) =3, and therefore so that r = [Tl (ad - 6c)3, Art. 182. Between the two covariants T and H{f) there exists the relation * -T2 = hp - hpHif) + 4H(/)3. This formula is given by Cayley in Crelle's Journal (April 9, 1856, Bd. 50, p. 287). The formula, however, as stated by Cayley, is due to a communication from Hermite.f We have at once 2^2 P = 2/, J //(/) //(/)3. or 2 H{ /"I writing ^"^ = <^, it is seen at once that 2J2 / -^=2/3-/2C + C^- Art. 183. Consider next the determinant A = H(f),f dH(J), df H{f), f dxi dX2 fdx^+^dX2 dxi 6x2 The functions / and H{f) being homogeneous of the fourth degree in xi, X2, it follows that -^ Xi -\ ^ X2 = 4/. dxi 8x2 dx 1 dX2 .x^+--^^x2 = AH{p. * Cf. Salmon, loc. cit., p. 195; Halphen, FoncHons Elliptiques, t. II, p. 362; Clebsch, toe. cit., §62. t Similar relations have been derived by Hermite for the quintic and for every form of odd degree (cf. Salmon, p. 249). ELLIPTIC INTEGEALS IN GENERAL. We therefore have 199 A = i ox I dx2 dxi dX2 miidx, + '-miidx.. ^ dxi dx2 dHjf) dHjf) dxi dxi dX2 dX2 ax I Xl, X2 dxi, dx2 dxi + -i—dx2 dX2 = 2 T{x2dxi — Xidx2). On the other hand A It follows that H{f), f dH{f), df = H{f)df-fdH{f) or hPd^ = - 2 T{x2dxi - xidx2) fidi: = - 4 {X2dxi - xidxz) {^J = - -^-^^{X2dxi - xrdx2)(zl^^ = - , \,, ix2dxi - xidxz) {2 Is - /aC + C^)*. (- 2)* From this it is evident that X2dxi — X\dx2 _ (— 2)i ^f{^UX2) dC 4 V2/3-/2C + C' Since 2 = — , it follows that ^2 3:21^3:1 — Xidx2 _ dz ^f{xi, X2) Vfl(7) ' where R{z) = oqz* + 4 ajz^ + 6 022^ + 4 032 4- 04. We finally have C dz ^ (- 2)^ r dr J V^Tzi - 4 J - 4 J Vc3 - /2C + 2/3 This is practically the transformation given by Cayley * in Crelle. « Journal, Bd. 55, p. 23. * See also Cayley, Elliptic Functions, p. 317; and Burnside and Panton, loc. cit., p. 474; Brioschi, Sur une formule de M. Cayley, Crelle, Bd. 53, p. 377, and Crelle, Bd. 63, p. 32. The Berlin lectures of the late Prof. Fuchs have been of great assistance in the derivation of this transfommtion. 200 THEORY OF ELLIPTIC FUNCTIONS. The mode of procedure, however, as noted above, was suggested by Hermite (cf. Hermite in "Lettre 123 " of the Correspondence d'Hermite et de Stieltjes; read also letters 124 and 125 of the above correspondence and Hermite, Crelle, Bd. 52; Cambridge and Dublin Math. Journ., vol. IX, p. 172; and t. I of the Comptes Rendus for 1866). If we write 2 t ior t^ in the above formula, it becomes f^^^-if. J Vr(z) J ; dt VR(z) J 2\^4:t^ - ht + h Art. 184. Weierstrass employed a somewhat different notation. He put h = 92, h = - 93, and consequently introduced as his normal form of the elliptic integral of the first kind, dt /; \/4i3 _ g^t - gs He further wrote 4^3 _g^t-gs = 4{t- ei) (t - 62) {t - 63)= S{t), so that (cf. Art. 172) between the e's and the g's we have the following relations: ei + 62 + 63 = 0, 6162 + 6263 + 6361 = - ig2, 616263 = i 93- Art. 185. We may show as follows how the Hermite- Weierstrass nor- mal form may be brought to the Legendre-Jacobi normal form. In the expression dt write t = A -\ — - , where A and B are constants. It is seen at once that dt - Bdz VS{t) \/{(^-6i)z2 + B}{(^ -e2)2.^ + B\{{A-e3)7? + B] Under the root sign there is an expression of the sixth degree which con- tains only even powers of t. But by writing A = 63, this reduces to dt - Bdz VS(t) Vfi I (63 - ei)z2 + B } { (63 - e2)z2 + B ] ELLIPTIC INTEGRALS IN GENERAL. 201 If further we give to B the value B = ei - 63, and put ^ 62 - 63 ^ ^2 61-63 ^ebave ^^ 1 ^^ V^!(0 Vei - 63 V(l-z2)(l-/(;2z2) It has thus been shown that through the substitution t — 63 H — , !r{t)wmt, * Salmon, loc. cit., p. 111. t Cf. Klein, Math. Ann., Bd. 14, p. 116, and Theorie der Elliptischen Modulfunc- tionen, Bd. I, p. 25. ELLIPTIC mTEGEALS IN GENERAL. 203 where the integrand is a rational function of t. For example, put , fi — b a = waz + 6 = i, then z = • In this case the reahn {z, a) is evi- a dently the same as the realm (i), since {z, t) is the same as (i), the pres- ence of z within the realm adding nothing to it, as 3 is a rational function of t. Consider next the integral R{z, a)dz, P where a = \^{z — a{){z — 02) = (z — ^2) v ^ ■ ' 2 — a2 By writing fi = ^ ~ "S it is seen that a ={z — a-i)t and 2 = — ^. 2 — 02 \ — t^ We note that both a and 2 are rational functions of t and that t is a rational function of a and 2. Hence every rational function of a and 2 is a rational function of t and any rational function of t may be expressed rationally through 2 and a. In this case we may say that the two realms (2, a) and {t) are equivalent and write (2, a) ~ it). In the case of the integral I R{x, Vax^ + 2bx + c)dx, if we put y^ = ax^ + 2bx + c, we have the equation of a conic section. This conic section is cut by the line y - Tj = t{x ~ $), where t is the tangent of the angle that the line makes with the x-axis, at the point f , tj, say, and at another point a$ + 2b - 2T)t + $fi "" ~ f-a T]fi -2a$t -2bt + at) ^ fi - a Hence as above In the case of the integral {X, y) ~ it). f R{z, s)dz, where s is the square root of an expression of the third or fourth degree in 2, it was shown by both Abel and Liouville that the integrand cannot be expressed as a rational function of t. This we know a priori from our previous investigations; for we saw that an elliptic integral of the first 204 THEORY OF ELLIPTIC FUNCTIONS. kind nowhere becomes infinite, while the integral of a rational function must become infinite for either finite or infinite values of the variable. In Art. 16 6 it is seen that z and s may be rationally expressed through z and s = >/(! —7?)(\ —k"^"^) and at the same time z and s may be ration- ally expressed through z and s so that {z, s) ~ (z, s), and consequently any element of one realm is an element of the other. It is also seen that if r = v'4 t^ — g2t — gz, then (2,s)~(z,s)~ {t,r). We note that by these transformations the order of the Riemann surface remains unchanged. The above three realms of rationality being equivalent, the name elliptic realm of rationality may be applied indifferently to them all. EXAMPLES 1. In the homographio transformation, a + pt + tz + dtz = Q for = Oj, z = Oj, z = a^, z = a^, let t =0, t = \, t =-, i = 00. We thus have a+yay = 0, a + P+la^+Sa^ = 0, «.< + /?+ 7^03 + ^aj = 0, P+Sa^ = 0. The vanishing of the determinant of these equations gives 0,-03 0,3 -o-i X = 02 -o^ dz Show that — = is thereby transformed into Riemann 's normal form. -yR{z) dz 3. In a similar manner transform — into Legendre's normal form and from ■^R{z) the resulting determinant derive the 12 values of k given in Art. 171. [Thomae.] 3. Show that the substitutions z - a, 02 - a^ 0.3 - a^ a^ - a, { = • , Kr = • transform dz J , into ± '^{a^ - a^) (0, - Og) / 0-^t{\-t){\-kH) Ja, n/Cz- Oi) (Z- Oj) (z- O3) (Z- O4) [Riemann-Stahl, Ell. Fund., p. 16.] ELLIPTIC INTEGRALS IN GENERAL. 205 4. Show that the substitution z— Oj Oj— Oj < — a, a^ — a2 3 — OjOg— Cj t— Oj'a^— a, transforms / , "^ == into / - «^ ^A(z—a,)(z — a,)(z ~ a,)(z — a.) ^ ^ d< V A iz — Oi) (2 — Oj) (2 — Og) (z — 04) «^ "^ A (J, — Oj) (< — Oj) (< - a,) (< — aj [Burkhardt, Ell. Fund.] Derive two other such substitutions. 5. Show that the substitution t = e^ + fe~gi)fa~gi) transforms Weierstrass's integral into itself. 6. If a is a root of az' + 3 62^ + 3 C2 + d = 0, by writing z — a = z^ transform dz into Legendre's normal form. N/a2' + 3 62^ + 3 cz + d 7. If /(z) = X* + & mx'^ + 1, show that 4 r dx _ r d? where ,_ m(x-+ D + (1 -3 ^V _ ^ . [Appell at Lacour, ii'oTic. Ellip., p. 268.] CHAPTER IX THE MODULI OF PERIODICITY FOR THE NORMAL FORMS OF LEGENDRE AND OF WEIERSTRASS Article 189. The Riemann surface for the eUiptic integral of the first kind in Legendre's normal form, /: -^, where Z =(1 - z2)(l - /c2z2)= s^ VZ has the branch-points +1,-1, + -' — -• fC ic ~\ In the figure * we join the points + 1 and — 1 with a canal and also the points + - and — - with a canal which passes through infinity. Here we have taken the modulus k, which may be any arbitrary complex quantity, as a real quantity, positive and less than unity. In the follow- ing discussion we make no use, however, of this special assumption. In Art. 142 we saw that A= f^, B= f^. The corresponding quantities here are, say, . 1 and A{k)=2f '^ J\ vZ J-iVZ * Cf . Koenigsberger, ElKpt. Fund., pp. 299 et seq. 206 MODULI OF PERIODICITY. 207 For any integral in the T'-surface we shall take as lower limit the point Zo = 0, So = + 1 ; that is, the origin in the upper leaf. We then have w(z, s) = r"-% in r. Jo.i vZ If we let the upper limit coincide also with the point 0, 1, then, however the curve be drawn in the T"-surface, we have always (I) M(0,1)=0. Art. 190. In Art. 139 we saw that on the canal a, u{X) — u{p) = A{k), and on the canal b, u(p) — w(A) = B{k). We form the integral between arbitrary limits, Z2, S2 and Zi, Si, where the path of integration is free, that is, taken without regard to the canals a and 6. If the path of integration crosses the canal a (see Fig. 63) we have = /+/+/ ' Z2.S2 *-'Z2.S2 Op *J X the integrand for all these integrals being Z2.S2 dz \/(l -Z2)(l - /C2z2) Noting that the second integral on the right-hand side is indefinitely small in T, it is seen that ^>'^_dz^ = ^(p) _ u{x2, S2) + m(zi, Si) - u{k) in T' Z2.S2 VZ = m(Zi, Si) - W(Z2, S2) - A(fc). If, however, the integration is taken in the opposite direction, we have % = m(Z2, S2)- w(Zi,Si)+ A{k). r zi.s. VZ We may form the following rule: If the path of integration for the integral !z /: V(l -Z2)(l-fc222) crosses the canal once in the direction from p to k, this integral with free path is equal to the integral taken in T' decreased by A{k); but if we cross the canal a in the direction from X to p, then this integral with free path is equal to the integral in T' increased by A(k). Upon crossing the canal b we have the opposite result : If b is crossed in the direction from p to X, then B{k) is to be added to the integral in T'. 208 THEORY OF ELLIPTIC FUNCTIONS. We may apply this rule in order to derive a number of formulas, which give the value of u{z, s) at certain points. In Fig. 63 it is seen that in the upper leaf of T' /r^z-<-"-K-i)- (fc). But in the lower leaf where the path of integration is taken congruent to the one in the upper leaf, there being no canal between the points —1 and - -, f-i rfz *' /:^ =«"<-"-«(- 9' * If we add these two integrals and note that the elements of integration are equal in pairs and of opposite sign, it is seen that the two integrals on the left-hand side cancel, so that K-i)i- = 2\u^-l)-u(-^\\-A{k), or A{k) (II) ^ = . (_!)_,(_!). Consider further the integral from -1 to +1 in the upper leaf and on the upper bank of the canal from — 1 to +1 (the upper bank being the one nearest the top of the page) '^'^ = M(+ 1)-M(- l)+B{k). ■1 V2 /: The same integral in the lower leaf and on the upper bank of the canal is /. ^ = u(+l)-w(-l). 1 V^Z It follows, as above, that (HI) -^ ^u{+l)~u{-l). Next forming the integral from + 1 to + i in the upper leaf and upper bank, we have "' and in the lower leaf, upper bank. /:'^-(4)-'-'- MODULI OF PEEIODICITY. 209 We therefore have (IV, -Afil.,(,i^_,,^^,. We then form in the upper leaf, upper bank, k and on the lower leaf, upper bank, / k Adding these two integrals we have (V) B(fc)= w(oo, +oo)+ m(qo, -oo)- 2wA Art. 191. If we form the integral X 1 * dz V{1 - z2)(l - A;2z2) in the upper leaf of T' and take the integration along the upper bank of the canal, it is seen that the path of integration is congruent to the one from + - to + 00 . At two corresponding points of the paths the abso- lute values of z are the same, but the signs are opposite. This difference of sign, however, does not appear in the expression (1 — z2)(l — k^z^). Thfe differential dz is the same along both the paths and positive, and consequently the elements of integration are equal in pairs and we have J-oo \/(l - Z2)(l - fc2z2) Jl V(l - Z2)(l - A;2z2) k In a similar manner we have (N) r-i dz r'^^ dz J_i \/(l - z2)(l - A;2z2) J+i V(l - z2)(l /c2z2) k We form the integration over the path indicated in Fig. 64 which lies wholly in the upper leaf and passes twice through infinity. The integral / — ^ taken over this path must be zero, J V'(l - z2)(l - /b2z2) since the path of integration does not include a branch-point. 210 THEORY OF ELLIPTIC FUNCTIONS. We therefore have 1 r\ r\ p% r\ r-% r \ r\ r=o. +r ~fo "^T upper baak upper bank upper lower bank bank lower bank lower bank We note that the two integrals r^ and f upper bank VZ lower bank are equal, for the sign of dz is different in both integrals, and as both inte- grals are in the upper leaf but upon different banks, there is a difference Fig. 64. in sign and also a difference in sign due to the limits of integration. On the other hand the two integrals f'^ and r^ J+i VZ J^i vz are equal with opposite sign, since there is no canal between the two paths over which the integration is taken. It follows that the sum of the above integrals reduces to J~i VZ J ^i \/Z J-00 \/Z where the integration is on the upper bank for all the integrals. MODULI OF PERIODICITY. 211 Owing to the relation (M) above, this sum of integrals further reduces, after division by 2, to ^^4 = 0. It follows at once that -i^^ J-i VZ J^i u{+ l)-ui-l)+B{k)+2'[u(a,, +00)- w/+i\-B(fc)1=0, or, owing to (III), (VI) |B(fc)=w(oo,+oo)-M(+|Y If we take the congruent path of integration in the lower leaf, we again have, since no canals are crossed, u{+ 1)-M(- l)+2 L(oc,-oo)-mQ1 = 0, We have thus the formula (VII) ^ = «(«,_^)_^Q. Art. 192. We compute the integral from to 1 in the upper leaf of T' on the upper bank of the canal and then the integral taken over the congruent path in the lower leaf. It is clear that 'JoA vZ Jo.-iVZ upper leaf lower leaf It follows that u(+ l)-u{0,l)+ B{k)+u{+\)-u{0, -1)=0, or, since u(0, 1) = from (I), we have (VIII) 2 m( + 1) - m(0, - 1) = - B(fc). Further, it is seen that /•o.+i_dz__ pi dz_ J-\ VZ Jo,+iVZ upper bank upper bank and consequently, multiplying by 2, we have J-i VZ Ai vz upper bank upper bank 212 THEOEY OF ELLIPTIC FUNCTIONS. From this it follows that w(+l)-a(- 1)+B{k)=2{u{+1) -M(0, 1) +B{k)], or, owing to (I) and (III), ^^ =2h{+l)+B{k)']. We thus obtain (IX) -|B(A;)=w(+ 1). We have thus derived the following nine formulas: (I) m(0, 1)=0, (V) M(oo,+ oo)+w(oo,-oo)-2Mf- = B{k), (II) u{- l)-u (-^ = ^, (VI) M(«,+ ^)-uQ = |5(ft), (III)u( + l)-w(-l) = -^, (Vll)u{<^,-'^)-u(^^ = \Bik), (lY) u(+^\-u{+l)= - ^, (VIII) 2ui+l)-u{0,-l) = -B{k), (IX) «( + !) = -iB(/c). From these formulas we have at once : u(+l) = -iB(k), u{-l) = -iB(k), M(0,1)=0, w(0,-l) = -^. Art. 193. Legendre * and Jacobi t did not use the quantities A{k) and B{k) but instead two other quantities K and K'. These quantities are connected with A{k) and B(k) as follows: J_i V(i 4K = B(k)-- or, since -z2)(l-/c2z2) r'^=2p'^, (Art. 192). «/-i vZ t/o,+ivZ 4 '>'o.+lV(l - Z2)(l - )t2z2) * Legendre, Fonctions Elliptiques (1825), t. I, p. 90. t Jacobi, Werke, Bd. I, p. 82 (1829). MODULI OF PERIODICITY. 213 A;'2 = 1 _ jfcZ, If further we write VI - k'^v^ it is seen that 1 Aik)=2 f ^ ^^ =.=2i r ^ ^" . [Jacobi.] ^1 \/(l - Z2)(l - yc2z2) Jo V(l -•y2)(l - h'H"^) If then we write dv ♦ --X" 'O V(l - V2)(l - /C'2v2) we have Aijt) = 2-iK'. The quantity A;' is called the complementary modulus. Since jB(fc) = 4K and A{k) = 2iK', the formulas of the preceding article become m(+ l) = -3ifi:, u{-l)=-K, w(oo, + oo) = -tif', ui^,- oo) = - 2K -iK', w(0, 1)=0, m(0,-1) = -2K. Anticipating what follows, if we write dz Jo V(l Z2)(l - ;c2z2) and if z considered as a function of u is written z = snu, we have from the above formulas sn(-3K)=l, sn{-SK-iK')=l, sn{- iK') ^ oo , etc. k Art. 194. We shall consider next the moduli of periodicity for Weier- strass's normal form of integral of the first kind. We note that the point at infinity is a branch-point (Art. 115) for the integral r dt r dt J 2V(t-ei)(t-e2){t-e3) J VS(i)' where S{t) = S = i{t - ei) (t - 62) (< - 63). 214 THEORY OF ELLIPTIC FUNCTIONS. In the Riemann surface T without the canals a and 6 let and let u(t, Vs) denote the corresponding integral in T'. u{t, vs)= \ ■ Fig. 65. We may here write (cf. Art. 139) u{X) — u{p) = A' on the canal a, and u{p)— u{X)= B' on the canal 6. The quantities w(ei), u{e2), u{ez) may be computed as follows. In the figure we note that, when the integration is taken in the upper leaf, In the lower leaf along the congruent path of integration, dt r Vs w(ei). (I) Through subtraction it follows that* dt_^ Vs upper leaf Similarly along the upper bank of the upper leaf of T', r^= r-^+ r-^ = u{p)-uie,)+u{e2)-E{X)=B'+uie2)-u(.ei); Je, Vs Je, Vs J)i Vs while for the congruent path in the lower bank, "'^ dt r i/e, Vs u{e2)- u{e-i). * Cf. Riemann-Stahl, ElKpt. Fund., p. 134. In comparing the results given by different authors it must be noted that in most cases the sign of equality may b"' replaced by that of congruence. 2 . vs It is also evident that MODULI OF PEEIODICITY. 216 Hence through subtraction it is seen that in the upper leaf We may therefore write (II) 2 p * = 2 r^ + 2 r^ = -A' + B'. Jao Vs Je, Vs Jx vs We further have in the upper leaf of T", = A' + w(e3)- ^(62); while in the lower leaf / '"7= = "(63) - uiez). Je2 Vs Through subtraction we have in the upper leaf ; r^ = A'. 2 p-4 - 2 p^ + 2 r4 ^ -A'+B' + A^, Jo,VS J«^VS Je^VS or (III) r^ = ^'. Art. 195. It follows at once from (I), (II) and (III) that in 7" -, , r«> dt A' Joe Vs 2 -, , r^^ dt -A' + B' „ J 00 Vs 2 _, , r^' dt B' J 00 Vs 2 From these definitions of w, cu', w", it is seen that W" = CO + w'. Again (cf. Art. 185), if we write * P dt Jx Vs and write the upper limit, considered as a function of the integral u, t = p{u), * The sign of the integral is changed in order to retain the notation of Weierstrasa. It is seen in Chapter XV that jpu is an even function. It is called the Pe-function. 216 THEOEY OF ELLIPTIC FUNCTIONS, we have (IV) 62 = ^{oj"). Art. 196. In Art. 185 we derived the relation dt ^ ^/- dz ^ Vs Vz' where 1 = 63 + — ,• £2^ If we write dt r dt - du = -—^, or — w = / — =) Vs J 00 Vs then also du dz Q^ u r dz Ve Vz V'e Jo.iVZ It follows that t-s>{u), z-sn(-\ and It is also evident that K /"I dz . 1 p> dt CO Jo.iVZ Vs Jo, Vs V'e OT at = VeK, and similarly tu' = VeiK'. Art. 197. The conformed representation of the T' -surface. In Chapter VII we saw that if dz »=/ ^„,,VR{^) then 3 is a one-valued function of u. We also saw that if the path of integration is unrestricted, more than one value of u correspond to every value 2, s. The collectivity of these values was expressed by u — u(z, s) 4- mA + IB, where u{z, s) represented the above integral in the simply connected surface T" and m and I were integers. If we write z = 4>{u), then ' Fig. 67. We draw the canals a and b so that they lie indefinitely near the real axis and indefinitely close to the points — 1, -I- 1, 4- -, as shown in the figure.' ^ We had in T Jo.i VZ The differential dz is here real, being taken along the right and left banks of the canals which are supposed to lie indefinitely near the real axis. MODULI OF PERIODICITY. 219 For the bank ^ of a we have 1 — z^ > for all points except z = 1, or z = — 1, and consequently also 1 - k^7? > 0. It follows that a^ is real in the w-plane, since u is real for all points on the bank X of a. Hence in the TT-plane Ui coincides with the real axis. On the bank ^ of 6 we have 1 - z2 < and 1 - k^z^ > 0. The elements of integration are therefore all pure imaginaries along this bank and consequently u is purely imaginary along this bank. It follows that 6i in the w-plane is a straight line that stands perpendicular to the axis of the real. Since a^ is parallel to a^ and bp to hi, the conformal repre- sentation of T" on the w-plane is a rectangle with the sides fid = B{k)==4:K, dy = A(/b)= 2iK'. We may represent the inte- gral in Weierstrass's normal form conformally in a like manner, student. As another exercise derive the results of this Chapter by taking the Riemann surface as indicated in Fig. 68. This is left as an exercise for the EXAMPLES 1. Show that /-co, -00 rfZ _ ^ 2. 3. Show that Prove that Jo Vz 4. The substitution e, - 63 < - e, Co Co Co T dt into ds V'4 (t - e,) (« - 62) {t - 63) J 63 V4 (s - e,) (s - c^ (s - e^ How does this result compare with the one derived by the methods of this Chapter? 5. Derive by means of the Riemann surface the formula Jea VS Je^ VS Jez VS CHAPTER X THE JACOBI THETA-FUNCnONS Article 199. We saw in Chapter V that the -f unctions of the second degree satisfied the two functional equations (m+ a)= (^(u), ^"?(2«+i) .6 If Q = e"", we saw in Art. 87 that jn = + CO Ani - mu Xl :^-— I (2m+l) « m = — 00 We have now to write: a = B{k)= 4:K, h = A{k)=2iK'. -1 E. It follows that Q =e ^ ^ . If further we write we have * 9 = ^2=6""^, Wl = + 00 Trt X-r=z. mu 7n.= — 00 ^ = + 00 /2^y ^^^ When K and K' are introduced into the functional equations, they are $(M + 2iK')=e ^ (m). In ©i(m) the term which corresponds to m = is unity. If we take this term without the summation and then combine under the summa- * Cf. Jacobi, Werke, Bd. I, pp. 224 et seq.; and in particular Hermite, Coura ridigi en 1882 par M. Andoyer, p. 235 (Quatrifeme edition). 220 THE JACOBI THETA-FUNCTIONS. 221 tion the term which corresponds to + m with the term that corresponds to — m, we have itimu Ttimu m = l m = co or ei(M)= 1 + 2 2) i(M)0(M). We may therefore write fHi(w+ii5C')= >l(it)0i(M), (V) 0l(M+iK')= >i(M)Hi(M), H(w+iK')=t/!(w)0(M), 0(m + iii:')=*i(w)H(w). It follows from (I) and (V) that Hi(m + if + ii!:') = -^'^(m)0(w), 0i(w + i!C + iK')= i-l(M)H(M), H(m + X + iX') = /i(u) 0i(m), 'd{u + K + iK')= /1(m)Hi(m). (VI) 223 * Hermite, foe. ct<., p. 236. 224 THEORY OF ELLIPTIC FUNCTIONS. It is clear that Hi(m + 2iK')= Hi[(m + iK')+iK'] If we put it follows that -^(u + iK') B.i{u + 2iK')= /z(w)Hi(m). We have the following formulae : Hi(M + 2tX')= /i(M)Hi(M), (VII) ei(w + 2iK')= ;«(m)0i(m), H(u + 2tif')=- /i(w)H(M), 0(M + 2iK') = -//(w)0(u). It is seen that H and © satisfy the functional equations (w + 4K)= 4>(w), (M + 2iK')=-/jL{u)^{u); while Hi and ©i satisfy <^iU + 4K)= (m), 4)(m + 2iZ') = +//(m)4>(m). We note in particular that the four theta-functions belong to two categories of functions of essentially different nature. Art. 203. The Zeros. — The ©-functions being -functions of the second degree vanish at two incongruent points {congruent points being those which differ from one another by multiples of 4 K and 2 iK'). We saw in Art. 200 that H(m) was an odd function and therefore vanishes for M = 0. We also had H(w + 2K)=-H(u), and consequently H(2ii:) = -H(0) =0. The points 0, 2 K are therefore the two incongruent zeros of this func- tion; i.e., the function H(w) vanishes on all points of the form + mi4:K + h2iK', 2K + mziK +l22iK', where Wi, m2, li, h are integers. Hence all the points at which H(m) vanishes are had for the values of the argument u = m2K +n2iK', where m and n are integers. THE JACOBI THETA-FUNCTIONS. 225 Further, since @{u + iK')=i;i{u)liiu), when M = 0, we see that @{iK')=0; and since e{u + 2K)=@{u), we also have @{iK' + 2A')=0. The zeros of 0(m) are consequently m2K +(2w + l)iK'. By definition we have so that the zeros of Hi(m) are (2 m + 1)K + n2iK'. Finally, since @{u)=@iiu- K), the zeros of ©i(w) are {2m+l)K +{2n+ l)iK'. Art. 204. Write m= +00 iti — R- if' - JT where q = e ^ and 771= +00 Tcimu X qo^e -K' =@,iu;K',iK), where Qo = ^ ^' • It is seen that the latter series fulfills the requirements of convergence given in Art. 86. We also note, cf. formulas (II) and (VII), that 0i(m; K', iK) and e ^^^'Siiiu; K, iK') satisfy the same functional equations ^{u + 2K')=^{u), <^{u + 2iK)=e ^ <|)(m). The two functions have also the same zeros M = (2 m + 1)^' + (2 n + l)iK. It follows that the ratio of the two functions is a constant. 226 THEOEY OF ELLIPTIC FUNCTIONS. We therefore have (ef. Jacobi's Werke, Bd. I, p. 214) e **^'0i(ro; K, iK') = C0i(w; K', iK), e *^^'B.-i{iu;K,iK')= C@(u;K',iK), e ^^^B.iiu;K,iK') = iCIiiu;K',iK), TZU' e *^^'e{iu; K, iK') = CHi(m; K', iK). Expression of the Theta-functions in the Foem op Infinite Products. Art. 205. With Hermite * consider the two functions $(m) = 9S(m + iK') 4>(u + 3iK') 4>{u + 5iK') ■ ■ ■ 4>{ - u + iK')4>{- u + 3iK')4,{- u + 5iK') ■ ■ ■ and i(M) = (f>{u){u + 2iK')(f>{u + 4iK') ■ ■ ■ 4>{- u + 1iK')^{- u + 4iK') • • • . It is seen at once that if ^(w) has the period 2 K, then (« + 2 K) = $(w), i(w + 2/<:) =4>i(m). It is also evident that $i(M + iK') =^{u), ^(u^iK') =4.i(u)^i=liii; 4>{u) and consequently If next we put we have and also 4>(u + 2igo=i'W '^i7^~.iy , 9>(m + %K') ^iju + 2iK')= {u) =1 +e^, cj){u + niK')4>{- u + niK')= 1 + 2g»cos— + gZ". K * See Note sur la tUorie desfonctions elliptiques placed at the end of Serret's Caleul Differentiel et InUgral, pp. 753 et seq.; (Euvres, t. 2, pp. 123 et seq. THE JACOBI THETA-FUNCTIONS. 227 It is thus seen that *(w) = II (1 + 2 ?" cos ^ + g2») (« = 1, 3, 5, . . . ) and 1 + e^jn (1 + 25" COS ^+ 52"). (n = 2, 4, 6, ■ • • .) These products are convergent (cf. Art. 17) if | 5 | < 1 (see Art. 81). Art. 206. The two functions $(m), 4>i(m) both have the period 2K and they satisfy the functional equations ^i{u + iK')=^{u), iriu ^iu + iK')=e ^i(M). Let us introduce a function ^{u) defined by the equation We have at once '^{u + iK') = e 2«^" ^'^{u), ^{u + iK') = e 2XV" 2>'-^(^). *(m + 2K) = 4>(m), ^{u + 2K) = - -^(m). It is evident from formulas (II) and (V) that we may write (cf. Art. 83) Hi(w)=4^(i.), where .4 is a constant. Noting also that Q(u) = @i{K-u), H(m)= RiiK-u), it is seen that @J?^\=A0.+2qcos2u + q^)(l+2q^cos2u + q^)(l+2q^cos2u+q^°)- ■ ■ , nj^^^^ 2Ay/qcosuil + 2q^cos2u + q^){l+2q*cos2u + q») ■ ■ ■ , q/2Km\ ^^ _25cos2w + 52)(i - 2g3 cos2m + g^) . . . _ jj/2_Km\ 2^ .^^ginM(l-2g2cos2M + g*)(l-254cos2M + g8)- • •, where .4 is a constant. 228 THEOEY OF ELLIPTIC FUNCTIONS. Art. 207. To determine the constant A of the preceding article, we follow a method due to Biehler.* Consider the product composed of a finite number of factors (1) fit) = (1 + qt) (1 + qH) . . . (1 + q^^-H) r,2n-l\ (-?)(-?)- ■•^^)- This expression developed according to positive and nega,tive powers of t is of the form (2) fit) = Ao+A,(t+fj + A2(t^ + ^l) + • • • + ^»(<" + ^^)- The following identity, which may be at once verified, fiq^t) (g2" + qt)=fit) (1 + q^^+H), gives between two consecutive coefficients Ai and Ai-i the relation Aiil - 52n + 2i)= ^._j (^2i-l _ 52»+l). We thus have qi\ - g2") ,3(1_ 2. 2) ^ 2_(j2n + 4 4i = ^i_l n2i-l(l _ g2n-2i+2) \ _ Q2«+2i 1 - g*" If these equations are multiplied together, we find that ■U -l„g" (1 - <7^") (1 - 9^^"-^) ■ . ■ (1-g^) °^ (1 - g2™ + 2) (1 - g2n + 4) . . . (1 _ g4«) But it follows directly from (1) and (2) that ■n-n — q ■ We therefore have A = (1 - g27.+ 2) (1 _ ^27^ + 4) ... (1 - gin) ° (1 - 32) il-qi) . . . il- q2n) When n becomes indefinitely large, it is seen that 1 Ac (1 - 52) (1 _ g4) (1 _ ^6) * Biehler, Crelle, Bd. 88, pp. 185-204; see also Hermite, loc. cit., pp. 770-772; Appell et Lacour, Fonctions Elliptiques, pp. 398-399. Jacobi gives two methods of deter- mining this constant (Werke, I, p. 230, § 63 and § 64) and a third proof (Werke, II, pp. 153, 160). THE JACOBI THETA-FUNCTIONS. 229 Further, since ^. _ ^^^^^ it follows from the equation (2) that (1 + 50(1 + qH){\ + qH) . . . (l + 2V1 + ^^^1 + ^) . . . l+,(^ + ^) + ,f +!)+•• -^g^f +!) + ••• (1 - 5^) (1 _ 54) (1 _ g6) . . . Writing t = e^*", this formula becomes (1 + 2 5 cos 2 M + 52) (1 + 2 53 cos 2 M + g6) . . . ^ 1 + 2 7 cos 2 M + 2 (7^ cos 4 M + ■ • • (1 - ?2) (1 _ qi) (1 _ g6) . . . From this we conclude that the constant A of the previous Article is A = (1 - 52) (1 - qi) (1 - 56) ... ; and at the same time it is shown that 61 as defined in the last Article as an infinite product is 01 l?^}i\ =1 + 2 g cos 2 M + 2 g4 cos 4 M + • • • , or @i(")= X ^ ™=+'" 5(mu + »B«Jr) which is the original definition of this 9-function. Exam'ple. — By means of the infinite products prove the formulas (I), (II), (III), (IV) and (V) of this Chapter, and therefrom derive the expressions in infinite series of Hi/^-^Y B.!^^^^ and o/i^V The Small Th eta-Functions. Art. 208. Jacobi (Werke, Bd. I, pp. 499 et seq.) introduced a notation similar to the following (see Art. 210): 7re= +00 6(2 Ku) = ??o(m) = X ^~ l)m^m=g2mriu m= - 00 m=+oo / 2ire + i y H(2Kw)=??i(M)=i V (-l)™g^ 2 jg(2m + l)«« m= —00 m= -1-00 / 27n.4-l \- Hi(2 7^m)=!?2(m)= X ? ^ ''e(2™+»"», m= —00 771= -I- 00 © 1 (2 A'?<) = ?93 (m) = 2) 9"'' <^'^'""" m= —00 230 THEORY OF ELLIPTIC FUNCTIONS. (I') It follows at once [cf. formulas (I) and (V)] that t?3(w + i) = t?o(w); and if T = -— ; K 1 4 (V) 1 1 Mu + iT)=q ^e-^'H 2{u). The other formulas given in the Table of Formulas, No. XXXIII, are left as examples to be worked. Art. 209. For brevity we may write m = ao m=co Co = n (1 - ?'"). Qi = n (1 + ?^'")' m = l 771 = 1 m = co ^2 = 11(1 +9""-')' ^3 = 11(1 -9""-')- 7/1=1 m=\ It follows at once that m = oo t?o(M)=QoII(l -2 92'"-icos2 7rM + g*'"-2), m=l 1 m = t?i(w)= 2Qog sinrrw JJ(1 - 2 g^m ^os 2 ttw + g^m)^ m=l 1 7)l = co t92(M) = 2 Qo9* cos ;rM JJ (1 + 2g2m cos 2 TTM + g*'"), m = l m = a) «?3(m)=QoII(1 + 2 52m-lcos2TO +5*'»-2). m = l If we write z = e™", we have 1 - 2g2"'+'C0s2 7rM + y4m + 2 ^ 1 _ ^2m+lg-2 ^ 1 _ q2m+l^2 (1 - g2m+l)2 Sin ;r /2 m + 1 \ • /2 m + 1 , \ ( r — M sin n [ t + m ) 1-9 sin 27n+l 1 72m+ 1 ■ 2to + 1 Sm TTT 2 sin^ Ttu ■ 2m + 1 sin TTT ,/2m+l \ We therefore have m=oo Oo{u)=QoQ3^Jlll THE JACOBI THETA-FUNCTIONS. 231 /2m - 1 di{u)=2qhmnuQoQo^Tl(l - r^^^]' ^= A sin^ {niTZT)/ m = A C0S2 (mTTT)/ m=oo ^3{u)=QoQ2^Tlh - COS^ /2m- 1 Art. 210. Jacobi's fundamental theorem. If we write nu = x on the right-hand side of the equations above, the theta-functions as given by Jacobi* are m=+oo m= —00 = 1 — 2 g COS 2 x+ 2 5^ cos 4 a; — 2 g® cos 6 x + • • • , m= +00 ■&^{X, q)= l2^(-l)'»gi(2m+l)2g(2m+l);ri m= — o) = 2 g* sin X — 2 gS sin 3 X + 2 q" sin 5 X — • , m = + 00 «^2(a;, g)= ^ ^l(2m+l)V2'»+l)« m== — 00 = 2 5* cos X + 2 gS COS 3 X + 2 g'i''' cos 5 x + 2 gV cos 7 x + • - • , m=+oo Mx, q) = X ^""e^""' m= — 00 = 1 + 2 5 cos 2 X + 2 g* cos 4 X + 2 5^ COS 6 X + • • We have at once i}o{x + i w) = daix) ■9o{x +n)= ■&o{x) ^i{x + \n)=Mx) ^i(x + ;r)=-??i(x) t92(x-|-4?i:)=-t9i(x) «92(:c +7r)=-«?2(a:) ^3(x + ?:) = ^3(:r) ??3(x + i;r)=«?o(x) «?o(a; + ^ logg . i) = -ig-ie^^iCx) !?i(x + J log g • i) = — ig- V^o(a;) ^2(x + ilogg-i)=g-V%(x) T?3(x + i log g • i) = g-ie^h?2(a;) ?9o(-a;)=t9oW t?i(-x) = -??i(x) «?2(-a;) = ^2(:E) ^o{x + logg • i) = - g-»e2^'??oW ??i(a; + logg . i) = - g-ie2^^i(x) ??2(x + logg • t) =g-ie2xi,j2(a;) ,93 (x +logg-i)=g-ie2^3W 7?o(a; + i TT + i log g • i) = g- V^2(a;) ^9i (x + i TT + i log g • i) = g-V^sW «92(a: + ^ TT + i log g • i) = ig-le^^oW r?3(x + j TT + ^ log g • i) = - ig-^e^^^i (x) * Jacobi, Werke, I, pp. 497-538. 232 THEORY OF ELLIPTIC FUNCTIONS. We next observe that if the quantities a, b, c, d; a', h', c', d' are con- nected by the equations a' = ^(o + 6 + c + d), 6' = J(a + 6 - c - d), c' = i(o — b + c — d), d' ^ \{a-b - c + d), (1) it follows that (2) and also that a = ^{a'+b'+c'+d'), b =\{a'+b'-c'-d'), c = i(a'- b'+c'- d'), Id = K«'- b'- c'+ d'), (3) a2 + 62 _,. c2 + d2 = a'2 + fe'2 + c'2 + d'2 We shall next show that if a', b', c', d' are either all even integers or all odd integers, then also a, b, c, d are aU either even or odd integers. ' This may be seen at once from the following table.* We note that all integers, positive or negative, belong to one or the other of the four forms 4p, 4p + l, 4p + 2, 4p + 3, where p is an integer or zero. For four even integers we may write a = b = c = d = 4a 4/? 4r 4d 4a 4/9 4r 4^ + 2 4a 4/3 4r + 2 4,J + 2 4a 4/? + 2 4r + 2 45 + 2 4a + 2 4/? + 2 4r + 2 45 + 2 where the numbers in any column may be permuted among one another. If for brevity we put a + /? + 7- + d =a' a + ^ - r -5 = /?' a-i^ + r ~d=r' a-^~ r + 5 = 5' it follows from equations (1) that a' = b' = c' = d'= 2 a' 2/?' 2r' 2 5' 2a'+ 1 2/?'- 1 2r'- 1 2 5'+ 1 2a'+2 2/?'- 2 2f 2 5' 2a'+3 2p'- 1 2f-l 2 5'- 1 2a'+ 4 2/?' 2r' 2 5' * See Enneper, ElKptische Funktionen, p. 136. THE JACOBI THETA-FUNCTIONS. 233 For four odd integers we may write a = b = c = d = 4a + 1 4/3 + 1 4r + 1 4:3 + 1 4a + 1 4/?+ 1 47-+ 1 4 X^ ' where N = [a ilogq + wif + [b ilogq + xif + [c i^ogq + yif + [d ilogq + zi]^, the summation to be taken over all systems of four even integers a, b, c, d -plus the summation over all systems of four odd integers a, b, c, d. We note that N may be written in the form (5) N = a + b + c + d log q w + x + y + z ■ a + b ~ c 2 _d log 7 2 , W + X — V — z . b + c — d log q 2 2 b — c + d log q w X + y — z ^ + JLtii (C) We define w', x', y', z' through the equations \w' = \{w + X ^ y + z), y' = ^{w — X + y — z), x' = i{w + X — y — z), zf =\{w — X — y + z). It follows at once that y,'2 + ^'2 + y>2 + 2'2 = y,2 + a;2 ^. 2^2 + 32. If further we put accents on all the letters in equation (4) and note that the summation taken over all systems of four even integers a', b', c', d' plus the summation over all systems of odd integers a', b', c', d' is in virtue of (1) and (5) the same as those above over a, b, c, d, it follows that = Mw')^3{x')&3{y')d3{z') + '&^{w')-92{x')^2{y')Mz')- Jacobi Q.OC. cit.) made this formula the foundation of the theory of elliptic functions. Art. 211. If for w we write w + ;r, we have Mw + tt) = Mw), Mw + tt) = -Mw), while at the same time w', x', y', z' are increased by ^ :: so that t?3(u)'+ J) becomes ??o(wi) and j92(«''+ i) becomes — ??i(w')- The formula above becomes Mw)Mx)My)Mz) - ^2{w)d2{x)My)Mz) = Mw')Mx')My')Mz')+Mw')Mx')di{y')^r{z'). The number of formulas which we may derive in this manner is thirty- five, which fall into two categories, namely, changes in w, x, y, z which THE JACOBI THETA-FUNCTIONS. 236 produce corresponding changes of ^ tt and J log q • i in w,' x', y', z' and secondly changes in w, x, y, z which cause changes of ^ ;r and i log 5 • i in w'. x', y', z' . The following eleven formulas belong to the first category, where for brevity we write {,Xp.vp) for t9,(w)^.W«?.(2/)'9.(3) and (^Ip.vp)' for t9,(«;')'?;.(a;')«?.(2/')'?.(2')- (A). (1) (3333) + (2222) = (3333) ' + (2222)' (2) (3333) - (2222) = (0000)' + (1111)' (3) (0000) + (1111) = (3333)' - (2222)' (4) (0000) -(11 11) = (0000)' -(11 11)' (5) (0033) + (1122) = (0033)' + (1 122)' (6) (0033) -(1122) = (3300)' + (2211)' (7) (0022) + (1133) = (0022)' + (1133)' (8) (0022)- (1133) = (2200)' + (3311)' (9) (3322) + (0011) = (3322)' + (0011)' (10) (3322) - (0011) = (2233)' + (1100)' (11) (3201) +(2310) = (1023)' -(0132)' (12) (3201) - (2310) = (3201)'- (2310)' Equations (11) and (12) are counted as one equation, since (11) becomes (12) when x, w, z, y are written for w, x, y, z. We also note that the equations (5) (7) (9) (11) are transformed into (6) (8) (10) (12) and vice versa, when — X, — y are written for x, y, and consequently also w' becomes z' and x' becomes y'. If we put w = X + y + z, it follows that w'=x + y + z, x'=x, y'=y, zf=z; while if we write w = — (x + ?/ + z), we have w'= 0, x'=-{y + z), y'=-{x-\-z), z!=-{x + y). Equations (A) may then be combined into double equations. If for brevity we denote ■&o{0)^x{y + z)'&p.{x + z)'&v{x + y) by \ OXfiv | and ^j(2; + y + z)^i,(x)'&^{y)&p{z) by {}.fivp], the five most interesting of these double formulas are given in the following table. 236 THEORY OF ELLIPTIC FUNCTIONS. (B). 0000 1= {3333} - {2222 j = {OOOO} + {llll } 0033 I ={0033}- {1122} = {3300} + {2211} 0022 1= {0022} - {1133} = {2200} + {3311} 0011 I = {3322} - {2233 } = {OOll } + { llOO} 0123 1= {3210} + {2301} = {1032} - {0123} We may derive a more special system of formulas if in the formulas in table (A) we put w = X, y = z, w' = X + y, x' = X — y, y' = 0, z'= 0; or if we put w =- X, y =- z, „/= 0, x'== 0, y'=-{x- y), z'=- {x + y). Similar formulas, making in all thirty-six, are had by writing W'= y, x= z; w =x + y, x' = 0, y =-{x-y ), z=o, w=-y, x= —2; w' = 0, x'=x~y, y'=o, z'=-{x+y) w= z, x= y; w' = y + z, x'=0, 2/' = 0, z'=-{y-z). w= —z, x= -y, w' = 0, x'=-{y + z), y'-y-z, z' = 0. Using the notations * [XfjLvp] = ^x^^Mx + y)^,{x - y), {Xfivp) = &,{x)^,{x)§.{y)§,{y), these thirty-six formulas are included in the following table. (1) (2) (3) (4) (5) (6) (7) (8) (C). [3333] = (3333) + (1111) = (0000) + (2222) [3300] = (0033) + (2211) = (3300) + (1122) [3322] = (2233) - (0011) = (3322) - (1100) [3311] = (1133) - (3311) = (0022) - (2200) [0033] = (0033) [0000] = (3333) [0022] = (0022) [0011] = (3322) (1122) = (3300) - (2211) (2222) = (0000) - (1111) (1133) = (2200) - (3311) (2233) = (1100) - (0011) * Koenigsberger, Elliptische Functionen, p. 379. THE JACOBI THETA-PUNCTIONS. 237 (9) (10) (11) (12) (13) (14) (15) [2233] = (3322) + (0011) = (2233) + (1100) [2200] = (0022) + (3311) = (1133) + (2200) [2222] = (2222) - (1111) = (3333) - (0000) [2211] = (1122) - (2211) = (0033) - (3300) [0202] = (0202) + (1313) [3232] = (3232) + (0101) [0303] = (0303) + (1212) [0220] = (0202) - (1313) [3223] = (3232) - (0101) [0330] = (0303) - (1212) [0231] = (1302) - (0213) [3201] = (0132) - (3201) [0321] = (1203) - (0312) ij, we have from (1), (2) and (11) (16) [0213] = (1302) + (0213); (17) [3210] = (0132) + (3201); (18) [0312] = (1203) + (0312); If in the above formulas we put x the following: !93%(2 x) = ^3*{X) + &,\x) = ^oHx) + ^2\X) ^2^M2 x) = d2Hx) - ^iHx) = ^sHx) - ^oHx). If we write t/ = in the formulas (C), (1), (2) and (11), we have the formulas of the following table. (D). (1) ^i^3^{x) = §o'^o^{x) + ^2^2''(X) (1') ^Z^oHx) = ^O^sHx) + ^2^iHx) (2) »93'^22(:r) = ^2^3''ix) - t9o^i2(x) (3) ^s^^iHx) = i}2^oHx) - ^0^2HX) If in equation (1) we put a; = 0, we have ,934 = ^^4 + ^^4^ or [1 + 29 + 29*+ 2g9 + ]*=[! ~2q + 2q^-2q^ + 16g[l + gl-2+ g2.3+ ^3.4 ^ . . ]4 Art. 212. We have defined and developed the theta-functions by- means of infinite power series. These functions being integral transcend- ents are susceptible of the treatment indicated in Chapter I and per- formed there for sin u. It will be shown later (Chapter XIV) that these theta-functions are to a constant factor the same as the Weierstrassian sigma-f unctions. In order to observe the general theory from another point of view and at the same time study Weierstrass's presentation of the subject, we shall develop the sigma-functions by means of infinite binomial products as has been suggested in Chapter I for sin u. It is therefore superfluous here to express the theta-functions through these infinite binomial products. 238 THEOEY OF ELLIPTIC FUNCTIONS. EXAMPLES 1. Show that Tu' ■ r(M+2roiX')' e*^^'0(u) = (-l)'"e *^^' @{u+2miK'), itm' K[u+{2m+l)iKT e*^*^'H(M) = (-i)2'n+ie iKK' @[u +(2m + l)iK'l e*^^'H(u) = ( - l)"" e *^^' H(w + 2 miZ^), -functions in the form , , ■ dz where If we put ^^ J Jz<,,8oVR(z) Jo \/(l-z2)(l-A;2z2) and study a quotient of -functions, it is seen that ^ \ \ must = 0, for 4>,(m) z = in both the upper and the lower leaves of the Riemann surface; and further for z = oo , we must have ^ "^ = oo in both leaves. It follows that ^^^^> $^ = for z = 0, s = + 1 and for z = 0, s = - 1; $i(w) and — ^^ = 00 for z = oo, s = + oo and for z = oo, s = — oo. *i(w) In Art. 193 we saw that m(0,+ 1)=0, w(0,-1) = -2K; and consequently H[m(0,+ D] = 0, H[m(0,- D] = H(- 2is:)= 0. Hence it is shown that H(m) becomes zero for z = 0, s = + 1 and for z == 0, s = — 1. We may therefore take H(w) as the numerator in the quotient of -functions. On the other hand we have w(oo,+ oo)=- iK', w(oo,- oo) = - 2K -iK'; and since e(-iK')=0, @{-2K-iK')=0, we may use © (m) as the denominator of the above quotient. If then for u we write Legendre's normal integral of the first kind, it is evident that 239 240 THEORY OP ELLIPTIC FUNCTIONS. the quotient SIh) has the desired zeros and infinities, and has besides ©(m) no other such points. It follows that &{u) where C is a constant. To determine the constant C, write z = 1 and we have ©[m(1)] But since (Art. 193) w(l) = -3 K, we have ^ H(-3g) In Art. 201 we saw that B.iiu + 3K)=li{u), or Hi(0)=H(-3/i:). In a similar manner it may be shown that 0i(O) = 0(-3/!C). We thus have It therefore follows that (M) cM, or C- ei(o) m = + oc /-^ __ m = — CO ~ m= + 00 /2m-j-_iy 9i(0) Hi(0) (i) S'' This transcendental expression, however, may be expressed algebraically in terms of A:. If we write z = - in the formula z = C _ / > we have H : = C. e "©] _^ H[-3g-tg'] &[- 3 K - iK'] It follows that ^ H[3 K + JK'] _ (^ H[g + ^•A''] ^ r^(u)L=o9i (0) e[3 K + iK'] @[K + iK'] [><(w)]„=oHi(0) ' r_ 1 Hi(0) /c ei(o) (n) THE FUNCTIONS sn u, en u, dn u. 241 But from (i) n Qi(0). Hi(0)' so that C2=i or C = -k, k Vk where the sign is to be taken positive siace it is definitely determined from the expression (M) above. We thus have ^ ^ 2^7 + 2^?+2^^+. . . ^j^^^^. g^_ 236.] l+2q + 2q^ + 2q9 + --- If in the integral of the first kind dz 1/0,1 we write z = sin ^, it becomes »=X' \/(l - z2)(l - A;2z2) Jacobi * wrote so that \/l - k^ sin2 = sin am u. If the modulus /c is zero, it is seen that am u becomes u and consequently z becomes sin u. Somewhat later z = sin am u was called the modvlar sine and written by Gudermannt 2 = sn u. Art. 214. Consider next the quotient e(M) We have (cf. Art. 140) Hi (it) _ Hi(M(z,s)+m4g + n2iK'] 0(m) @[u{z,s)+miK + n2iK'] Since Hi(m) and ©(u) have the period 4 K, it follows that Hi(w) 'H.i\u{z,s)+n2iK'] @{u) e[u{z,s)+n2iK'] If we take n = 1, we have Hi(m) _ -u.(u)lli{u) ^ __ Hi(w) 0(m) fi{u)&(u) @{u) * Jacobi, Werke, Bd. I, p. 81. Here Jacobi retained the word amplitude of Legendre [Fonct. Ellip., t, I, p. 14]. t Gudennann, Theorie der Modularfunctionen, Crelle, Bd. 18. 242 THEORY OF ELLIPTIC FUNCTIONS. Since we have the negative sign on the right, it is well to take the square of the quotient, so that a formula which is true for any value of n. Art. 215. All the Theta-functions have the property of becoming zero of the first order upon only two incongruent points. It follows that the quotient rHi(w)-|2 L0(M)J becomes zero of the second order upon two incongruent points, and upon two incongruent points it becomes infinite of the second order. Since H,(m) = ior u ={2m + 1)K + n2 iK' , it is seen that Hi(w) = for w = - K and M = - 3 K; and from above e(w) = for M = - iK' and M = - iK' - 2 K. In Art. 193 it was found that when u = — K, then z = — 1, when u = — 3 K, then z = + 1, when M = — iK', then z =oo, s = oo, when M = — iK' — 2 K, then z =oo, s = — oo. It follows from Art. 150 that \ '/v ] is a rational function of z. It be- L0(m) J comes zero of the second order on the positions z = — 1 and z = + 1, and infinite of the second order on the positions z = oo, s = oo and z = oo, S = — 00. We note that the function z^ - 1 has the same properties. We may therefore write Vl -z2 = Ciiii^). e(M) The function Vl - z^ is consequently like z a one-valued doubly periodic function of u. It has the period 4 K but not the period 2 iK'; for when u is changed into u + 2 iK', the above quotient changes sign. Hence the other period is 4 iK'. We have V 1 — z2 = Vl — sn^u = cos am u = cnu, or cnu = Ci H,(m) 0(m) We shall so choose the sign that en u has the value + 1 when z = 0. THE FUNCTIONS sn u, en u, dn u. 243 This function en u is called the modular cosine. The analogue in trigo- nometry is naturally the cosine, where cos u = Vl — sin^ u. In order to determine the constant Ci, we may write z = 0, s = 0, so that l = C,5^^i^) or c, Q(Q^ l-2q + 2q^-2q^+. . . 0(0) Hi(0) 2^q + 2^q^ + - ■ ■ Again, if we write z = - . then, since ui-\ = — 3 K — iK', it follows that ./73T = c Hi(-3g-tg0 ^ iJiiZK + JK') V A;2 ^ e(-3K-iK') ' 0(3 K + i/^') _ _ ^ Hi(g + tKQ _ ^ [^(m)]„=oi:0(O) _ .^ _0W ' 0(K + iX') '[.1(m)]„=oHi(0) 'Hi(O) But, since Ct = ^ , we see that iCi2 = v/— (L_^, or Ci = ^ (see Art. 193), '^ K^ Vk the sign being definitely determined through Ci = ^ ' ■ Hi(0) In the preceding Article we saw that ^/k was definitely determined and consequently here V'k' is also definite. We may therefore write _ Vk' Hi(m) cnu = — = '^ / • Vk &{u) dz Art. 216. We saw in Art. 152 that — is a one-valued function of u and du from above it is seen that Vl — z^ is also one-valued. It therefore follows from the expression — == V(l - z2)(1 - A;2z2) du that Vl — k^z^ must be a one-valued function of u. This function is called the delta amplitude u and written A am u, dn u or Acf). Since — = — , it follows, since z = sin cj), that du = ^- d-z V(1-z2)(1-A;2z2) A^ To investigate this function dn u, let us study the quotient V@i{u)-f r@iiu)J l@{u)\ l&{u)j 244 THEORY OF ELLIPTIC FUNCTIONS. The zeros of the numerator are expressed through u ={2m + 1)K +{2n + l)iK'. We may therefore take as the two incongruent zeros the values u =- 3K - iK' and u =- K - iK'. In Art. 193 we saw that uiz,s)=~3K -iK' for z = |. and m(z, s) = — K — iK' for z = — -• Hence the above quotient becomes zero for z = ± -. and it becomes infinite for z = oo, s = + o o and for z = oo, s = — oo. The function Vl — k^z^ has the same zeros and the same infinities. We may therefore write 0(m) We shall choose the sign so that when z = the root has the value + 1. Hence for z = we have If further we write z = 1, we have , , _ ^ 0i(-3g) _ ^ 01(3 g) _ ^ ©(0) ^ 0(-3^) ^ 6(3 /C) 0i(O)' It follows that k' = €2^ or C2 =\//c', and consequently VF= Q(0) l-2y + 2g4_2Q9 + . ■ ■ ©i(0) l+2g + 2g4 + 2g9 + (Jacobi, Bd. I, p. 236.) Finally we have 0(m) Art. 217. We may write* the three elliptic functions of u 1 H(m) snu = — ^ — !^^ , Vk^Qiu) (VIII) ]cnw = ^Hi(«)^ Vk 0(m) 0(w) * Cf. Jacobi, Werke, Bd. I, pp. 225, 256 and 512; Hermite, he. cit., p. 794. THE FUNCTIONS sn u, m u, dn u. 245 The first of these functions is odd, the other two are even. It follows at once that sn -=0, (VIII') en = 1, dnO = 1. The zeros of sn u are 2 mK + 2 niK', the zeros of en u are . (2 m + \)K + 2 niK' , the zeros of dn u are . . . (2 m + \)K + (2 n + l)iK'; the infinities of ail three functions are . 2 mK + (2 n + l)iK', where m and n are integers including zero. We will derive nothing new by forming other quotients of Theta-func- tions. Art. 218. It follows at once from the above formulas that , , K^x 1 Yi.{u + K) 1 Hi(m) Vk Q{u + K) Vk @i(u) Hi(h) 1 Q(m) ^ 1 Vk Vk\ Vk ®iiu) Vfc Vk' dn u e(M) or sn{u -\- K) = en u dnu We may consequently write (IX) , , Tj'\ cnu sn{u + a) = > dn u cn{u+ K) =-k'^, dn u ax') dn{u + K) k' dn u sn K = I, cnK =0, dnK= k'. When the argument u is increased by 2 K, it follows that , ^„„ 1 YL{u + 2K) _ 1 H(m) _ sn(M + 2 a) = — = — ^^ '- = -^ — — - - - snu, \/k@{u + 2K) Vk @{u) We thus have (X) sn(u + 2 K)= — snu, cn(u + 2 K) = ~ en u, dn(u + 2 K)= dnu. 246 THEOEY OF ELLIPTIC FUNCTIONS. Noting that _ _ ou(u I iin ^^' ^'(^ + '^'^ ^^- m = am u. we have d± A4>' I am u du A(^ = dn u. THE FUNCTIONS sn u, en u, dn u. 247 It follows that -^,, a = snu = cnudn u, du cn'u = — snudnu, dn'u = — k^sn u en u. The foUowmg two relations are also evident: srfiu + crfiu = 1, dri^u + k^sn^u = 1. Further, from the relations ^ =\/(l - z2)(l - A;2z2) and 2 = snu, du we have and similarly sn ^u = (1 — sn'^u) (1 — k^sn^u), cn'^u = (1 - cnH) (1 - k^ + k^cn^u), dn'^u = (1 - dn^u) (dn^u - 1 + k^). Art. 220. Jacobi's imaginary transformation* — If we put sin sin i/r = — i tan 0, 1 cosV^ = di/r= — ^ cos^ . drf. cos^ zrf>|r If next we write Vl - F sm2 ^ Vl - ifc'2 sin2 -./r I: # Jo rf^r = iM, say, Vl-Fsin^^ Jo \/l - A;'2 sin2 -f then '^ = am(M, A;') and <^ = am(w, A-). From the relations above we have (XVI) sn(iu, k)= I cniiu, k) = dn{iu, k) = ■ sn{u, k') cn{u, k') 1 cn{u, k') dn(u, k') cn{u, k') * Jacobi, Werke, Bd. I, p. 85. sn{u, k') = ■ cn{u, k') = dn{u, k) = ■ sn{iu, k) cn{iu, k) 1 cn{iu, k) dniiu. k") cn{iu, k) 248 THEORY OF ELLIPTIC FUNCTIONS. Art. 22L As a definition Jacobi wrote coam u = am(K — u). We have at once (XVII) sin coam u cos coam u = A coam u dnu k' sn u dnu ¥ dnu It also follows that sin coam {iu, k) ■- (XVIII) dn{u, k') ik' cos coam(iM, k)= — cos coam(M, k'), k A coam {iu, k) = k' sin coam {u,k'). Art. 222. From the two preceding Articles it is seen that 1 (XIX) sn (u + iK') = en (u + iK') = — k snu i dnu ik' and also that (XX) k snu k cos coam u dn(u + iK') = — i cot am u ; sin coam (m + iK') = cos coam {u + iK') ■- k sin coam u ik' k en u A coam (m + iK') = ik' tan am u. Art. 223. Linear transformations. — If with Jacobi (loc. eit., p. 125) we put t ~ kz, we have i dt .^''-'i'-'i) •>'oV(1 dz z2)(1 - A;2z2) THE FUNCTIONS snu, cnu, dnu. If further we write Jo V{1 - 22)(1 -P22)' we have z = sn{u, k), t = snlku,-\, and consequently* sniku, j)= k sn{u, k), (ku, -A = dniu, k), Iku, - j= cn{u, k). (ku, -]= — > k/ sincoam(zt, fc) Iku, -| = ik' tan am (u, k), 249 (XXI) We also have (XXII) en I dn sin coam ( cos coam A coam (^"'l) = ik' k cos am (m, k) Next put iu in the place of u and observing that the complementary 1 ik' modulus of - is -- , it is seen that A; k (XXIII) and (XXIV) (ik'\ ku, — j= cos coam {u, k'), cniku, — )= sin coam (m, k'), dnlku, ^ ) = — — ; \ k / a am (w, A; ) (ik'\ ku, -7-)= cos am (it, k'), (ik'\ ku, -7-)= sin am(zt, A;'), . /, ik'\ A am (it, k') . A coam Iku, -7- 1 = 7 ' (ik'\ ku, — j = cot am (u, k'). * See also Hermite, CEuvres, t. II, p. 267. 250 THEORY OF ELLIPTIC FUNCTIONS. and Art. 224. It follows from Art. 204 that H(w; K,iK') ■ H(tt; K',iK) @{iu; K,iK') Hi(m; K',iK) ei(0; K,iK') _ ei(0; K'AK) Hi(0; K,iK') 0(0; K',iK) We have at once (cf. also Art. 220) (XVI) sn{iu; K, iK') = i cn{iu; K, iK') = dn{iu; K, iK') = , sn{u\K',iK) cn{u; K', iK) 1 Art. 225. Quadratic transformations. t cn{u;K',iK) dn{u;K',iK) cn(u; K', iK) If we write (1 + k)z 1 + /CZ2 ' we have where Writing dz Mdt \/(l - Z2)(l - A:2z2) V(l - <2)(1 _ ^2^2) 1 + A; \ +k Jo \/(l - it follows that (1 + k)u = / — =^ Jo Vd - dz 22)(1 A;2z2) and consequently (XXV) V(l - <2)(i _ ;2^2) [< cn\ (1 i-l,, 2 Vfc "I _ cn(M, k)dn{ u, k) 4(l+fc)u,2^1=l^lAi!ii(3^ L 1 + A;J 1 + ksrfi{u,k) (u, k)dn(u, k) I + k sn^{u,k) — k sn^ (u, k) + k sn^{u, k) In a similar manner write and we have where t = dz (1 + k')z VT^ Vl - k^z^ Mdt \/( 1 - Z2)( 1 - /c2z2) \/( 1 - /2)( 1 _ ^2^2) 1 -k' 1 +k' and M = ■ 1 1 + /b'' THE FUNCTIONS snu, cnu, dnu. It follows at once that \r,[(^ 4- h'\i, 1 ~^n - (1 + k')sn{u, k)cn{u. k) (XXVI) c.[(l + k')u, 1^;] = ^-(i+k')snHu,k) dn{u, k) (1 - k')sn^(u,k) dn{u, k) 251 In formulas (XXVI) change k to \/k and u to wA; and observe formulas (XXI). It is seen that (XXVII) sn (A; + ik')u, k — ik'~\ _ (k + ik')sn{u, k)dn{u, k) -1 k + ik' \ cn{u, k) cn\{k-\- ik')u k ~ ik k + ik ^1 = 1 ik'j ■ {k + ik')k sn'^ju, k) cn{u, k) { J f/; , -7^ k — ik'~\ I —{k — ik')k sn^(u, k) dn\ {k + ik')u, \ = i^ , , . ■ L k -\- zfc'J cn{u, k) The formulas just written are the very celebrated formulas due to John Landen {Phil. Trans., LXV, p. 283, 1775; or Mathematical Memoirs, I, p. 32, London, 1780) and may be derived as follows: Write sin (2^ — ^i) = fci sin^i, (1) where Since it is evident that fci 1 ~k' I +k'' (/c2 + A;'2 = 1). ki < k, sin {2(f) — (})i)< sin^i, {2-4>i)i, Solving (1) for (j), we have sin2 2 ^ = (1 + k{)^ sin2 ^i fl - 4fci (1 + A;i)2 sm- i\, or, smce ^ = (1 + k')^ and ^, ^^,' ,, = k^, (1+A;i)2 (1+A;i)2 it is seen that We further have • - /I , 7,xsin^cos^ sm^i =(1 +k') — ^ ^ ■ Vl-fci2sin2<^i Vl - A;2 sin2 ^ 252 THEORY OF ELLIPTIC FUNCTIONS. Art. 226. Development in powers of u. — If we develop by Maclaurin's Theorem the three functions sn u, en u, dn u, we obtain the following series : sn u = M - (1 + A;2) ^ + (1 + 14 fc2 + A;4)^ _ . . . , cnu=l -|^+(1+4A;2)|*- dnu = \~~^+{k* + 4: k^) |-* where the coefficient of any t^rm, say — — - or — — - , is an integral (2n + 1)! (2 np. function of k^ with integral coefficients. Following Hermite* we wish to determine these coefficients. From the formulas derived above snlku,-\= k sniu,k), cn[ few, — )= dn{u, k), it is seen that the coefficients of sn{u, k) are reciprocal polynomials in k and that those of dn{u, k) may be derived immediately from those of cn{u, k). Gudermann f has shown that the coefficients of en u are 1 + 4 F, 1 + 44 A;2 + 16 A;4, 1 + 408 A;2 + 912 fc* + 64 k^, 1 + 3688 A;2 + 30768 /b^ + 15808 k^ + 256 k», We note that if we put A; = cos 6 and introduce the multiple arcs instead of the powers of the cosines, the above coefficients when multiplied by k may be written A: + 4 fcs = 4 cos i9 + cos 3 (9, k -V 44 k^ + 16 A:5 = 44 cos 5 + 16 cos 3 (9 + cos 5 6, k + 408 P + 912 k^ + 64 A:7 = 912 cos d + 408 cos 3 (? + 64 cos 5 (9 + cos 7 d, In these equalities it is seen that the powers of k and the cosines of the multiples of 6 have precisely the same coefficients. * Cf, Hermite, Comptes rmdus, t. LVII, 1863 (II), p. 613; or CEuvres, \,. II, p. 264. t Gudermann, Crelle, Bd. XIX, p. 80. THE FUNCTIONS snu, cnu, diiu. 253 In general, if we denote the coefficient of (2 n + 2) ! by 1=71 -•lo + Aik^ + A2k* + ■ • ■ +Ank^''=^A^k^i = cn»2«+2) (0, fc), i = we will have the relation S.Ucos2' + i6' = 2.4. cos (2 n + 1 - Ai)0, which may be demonstrated as follows: From formulas (XXVI) we have en[(fc + il^')-' ^] = - (A; + ik')ksn^(u.k) cn{u, k) and changing i to —i it follows that {k — ik')k sv?{u, k) \ II ■i'\ k + ik'l 1 cn^k - ^k )u, -^— ^J = - cn{u, k) From these two formulas it follows at once that {k + ik')cr\(k - ik')u, ^-ti^l + (fc _ ik')cn\ {k + ik')u, ^ ~ ^^' 1 [_ fc — ifc J L A; + ik'} = 2k cn{u, k). In this formula write k = cos 0, k' = sin 6, and we have e'*cn(e~**M, e^'') + e~'"cn{e'-^u, e"^*') = 2 cos (9 cw(m, fc). Noting that cn(2»+2)(0)= 1 + .-liP + .42^)4 + ■ • + AnP", it is seen by equating the coefficients of — on either side of this (2 n + 2) ! equation, when expanded by Maclaurin's Theorem, that 2.4iCos2^'+ifl = S.4, cos (2n + 1 - 4i)(9. From this formula the quantities ^o = l^ -■ii- ^2, • , may be determined at once. For example, let n = 4 and for brevity put Ai = 4}ai. If the multiple arcs are replaced by the powers of the cosine, we have cos (? + 4 ai cos3(9 + 16 02 cos^^ + 64 a^ cos'O + 256 a^ cos^^ = cos ^ + ai (cos 3 (9 + 3 cos 8) + a2(cos 5 5 + 5 cos 3 (? + 10 cos d) + 03(008 7 (9 + 7 cos 5 fl + 21 cos 3 (9 + 35 cos 0) + aiicos 9 ^ + 9 cos 7 ^ + 36 cos 5 (9 + 84 cos 3 (9 + 126 cos 0) = cos 9 (9 + 4 ffli cos 5 ^ + 16 02 cos i9 + 64 03 cos 3 (9 + 256 04 cos 7 0. 254 THEORY OF ELLIPTIC FUNCTIONS. We thus have among the a's the five equations 1 = a4, 4 ai = 02 + 7 03 + 36 a^, 16 aa = 1 + 3 oi + 10 ag + 35 03 + 126 04, 64 03 = ai + 5 02 + 21 as + 84 a^, 256 04 = tta + 9 04. Since the sum of these equations leads to an identity, we may omit any one of them, say the third; and from the other four we have ai= 922, 02= 1923, 03= 247, 04= 1, which agree with the above results of Gudermann. Since dnlu, -]= cnl-, k] {"■D^'^Kf" the coefficients of dn{u, k) are at once deduced from those of cn{u,k); while those of sn{u, k) may be obtained from the formula sn' {u, k)= en {u, k) dn (w, k). [See Table of Formulas, LVII.] Development of the Elliptic Functions in Simple Series of Sines and Cosines. First Method. Art. 227. In Art. 206 we saw that 0/'2_^\= 4(l_25cos2M + g2)(l-2g3cos2w + 56)(i_255cos2w + gio)-- • . Noting that A=ao t>- \og{\-t)=-X\ /l=l ;=<» X iog(i + 0= -X(-i) )• and that 1 - 2gcos2M + g2 = (1 _ ge2*») (1 -gg-^*"), it is seen that 1 Q COS 4 ii — K log (1 — 2 g cos 2 M + gr2) = q, cos 2 M + - — q^ cos 6 M o* cos 8 u + 7i + -A + • • • . THE FUNCTIONS snu, cnu, dnu. 256 We therefore have 1 log /?-^^ = const. - cos 2 m (g +^ + gS +...) or COS 6 w , o , Q , , ^ , , — g — iq^ + q^ + gi5 + ■ • • ) ^—^ (g4 + gl2 + 520 + . . . ) q^ cos 4 w 1 , ^ /2 KmN ^ Q cos 2 M - Iog0^-__j = const. - \zr^ - 2(1 -g4) q^ cos 6 w g* cos 8 u ~ 3(1 - g6) - 4(1 _ g8) - ■ ■ • • The logarithms of the other Theta-functions may be expressed in a similar manner. Art. 228. Hermite (CEuvres, t. II, p. 216) gives the following method for the expressions of sn u, en u, dn u in terms of the sines and the cosines. We have the formulas , d log (dn u — k cnu) k snu = s_!^ 1 du d loe (dri cnu = ., d log (dn u + ik sn u) ik cnu = 2-^^ — - — ■ — — . du ■ , d log (en u + i snu) idnu = — 2-^ ■ — — ^ • du We shall next derive the formulas /, N J 2 Ku , 2 Ku (1) ,dn— — ■ — ken Tt t: _ _ 1 — 2 Vg cos u + q 1—2 Vg3 cos u + q^ 1 — 2 \/(f' cos u -\- (f 1+2 \^q cos u + q 1+2 Vg^ cos m + g^ 1 + 2 Vg^ cos u -\- q^ (2) dn h iA; sn 1 — 2 \/ — g sin m — g 1— 2\/ — g^ sin u — q^ \—2\/ — {u)= 1 — e ^ ; the expression which we wish to demonstrate equal to cnu + i snu will take the form ^ <^{-u + JK') 4>{u + ^ JK') 4>{- u +oiK'). . . (u + 3iK'){- w + 5iK') . . . , where ^ is a constant, and putting Triu (w)= Ae^4,2{- u + iK')2{u + 2.iK')^{- u + biK') . . . , we have to demonstrate the formula cnu + I snu = ^ ■ &{u) We further note that ^(u + 2K) = - 4>(m), ^{u + 3iK') or ^{u + ^iK')= e ^ («)= CH(m)+ CiHi(m). Divide this expression by (u) , and we have ^?^ cp{~ u + iK'){u + 3iK')cj>{- u + 5iK') . . . ^ CHju) + C iRiJu) 4>{u + iK'){u + 5iK') . . . e(M) = D cnu + iB sn u. Writing m = and u = K respectively in this formula we have Z) = 1 and 5 = 1, which we wished to demonstrate. From the formulas (1), (2) and (3) we have (see Jacobi, Werke, II, p. 296) _ _ _ kK 2 Ku y/q sin u a/o^ sin 3 u Vo^ sin 5 u 27r n 1 — 9 \ — q^ \ — (f kK 2 Ku _ Vg cos u Vq^ cos 3 u Vg^ cos 5 u 2~T^''^r~ 1 +9 + 1 +g3 + 1 +g5 + • • • ' K 2 Ku _ 1_ g cos 2 M g2 ^Qg 4 ^^ gS cos 6 m 2^"*""^ ~ 4 + 1 +g2 + 1 +g4 + 1 +g6 + THE FUNCTIONS snu, cnu, dnu. 257 Second Method. Art. 229. Suppose with Briot and Bouquet {Fonct. EUipt., p. 286) that /(m) is a doubly periodic function of the 2 nth order with periods 4 K and 2 iK' such that ^(^ + 2K)^- f{u) and further suppose that f{u) has n infinities ah within (see Art. 91) the period-parallelogram ABDC, where A is an arbitrary point Uq, while B and C are the two points uo+2K and uo+2iK': Form the parallelogram EFGH whose vertices E and H are the points mq — 2 m'iK' and Mq + 2 m'iK', while /^ and G are the points Uq + 2 K — 2 m'iK' and Mo + 2 /f + 2 m'iK'. The infinities of /(m) situated within the parallelogram EFGH may be represented by a = ah+ 2 miK' , where m varies from —m' to m'-l. Let t be any point situated within this parallelo- gram. The function fiv.) sin— (u 2K^ t) Fig. 69. has the period 2 i^; its poles are the point t and the points a = ah+ 2 miK'. It follows from Cauchy's Theorem that the definite integral i± 2: -1, r f^ du, Jsm— (..-0 where the integration is taken over the sides of the parallelogram EFGH, is 'equal to the sum of the residues relative to the poles that are situated within this parallelogram. The two sides FG and HE give values that are equal and of opposite sign, while on the sides EF and GH the function 1 f{u) has a finite value and mod, m' becomes verjr large. tends towards zero * when sm (u 2K t) Thus when m' becomes very large the definite integral tends towards zero and consequently the sum of the residues is zero. 2 K The residue relative to t being f{t), we have the equation m- ^VRes .K-^ (a) f{u) 2K sm (t 2K^ u) * In write u sin u X + iy and note that 1 2 2 Jx^V = for y — oo . 258 THEORY OF ELLIPTIC FUNCTIONS. If fiu) has only simple infinities, which case alone is necessary for our investigation, the above equation becomes m= +00 h^n . ^ ^m^-^oo A= 1 sin^^ {t-aK-2 miK') 2K where Ah is the residue of f{u) relative to an. The series is convergent in both directions. This equality is thus demonstrated for all points t situated within two indefinitely long parallel hues EH and FG. Since both sides of this equation change signs when t is replaced by t + 2 K, the equality is true for all values of t; and consequently we have for the finite portion of the w-plane 7n= +00 h = n . /(w)=2^X X n m=-x /i=i sin (u — ah — 2 miK') 2K^ Art. 230. Consider next a doubly periodic function f{u) with periods 2 K and 2 iK' and having n infinities uh within the parallelogram ABDC of the preceding Article. The function f(u) tan^(M -0 admits the period 2 K, and the definite integral J>>!1 i« LJ- tan— (m -t) relative to the contour of the parallelogram EFGH is equal to the sum of the residues with respect to the poles situated within the parallelogram, that is, for the point t and the points a = ah + 2 miK', where m varies from — m' to m'— 1. The sides FG and HE give equal results with contrary sign. If we represent by m a point on the line AB, the congruent points on HG and EF are u + 2 m'iK' and u — 2 miK', and the parts of the integral relative to these two sides are 1 1 lizi I Jal tan -^ (m - « - 2 m'iK') tan -% (m - i! + 2 m'iK') 2 K 2 K f{u)du. /AB When m' becomes very large the first tangent tends towards — i (see Art. 25) and the second tangent towards i, so that the integral just written tends towards a limit equal to the rectilinear integral 1 pu„ + 2K M=- I f{u)du THE FUNCTIONS sn u, en u, dn u. 259 along the line AB. The residue of the function relative to the point t 2 K being fit), we have, as in the preceding Article, tan— (<-w) and consequently if the function has only simple infinities m= +00 A=n ■kM , r. -^ -^ Ah /(0 = if + 5%_2.X— : m= -co h=ii,an~{t ~ ah -2 miK') 2 K where t is any point in the finite portion of the w-plane, and Ah is the residue of f{u) relative to ah- Art. 231. To make apphcation of the results of the two preceding Articles consider the ratios of the four Theta-functions. Of these twelve ratios eight satisfy the relation f{u + 2 K) = — f{u) and four the relation /(m + 2 K) = f(u) . Take the two functions ^^^ and %1^ . Form a paral- H(m) H(w) lelogram EFGH with the origin as center and vertices ± K ±{2m' + l)iK'. The infinities of these two functions are the zeros of H(w). Those infinities within the parallelogram are represented by the formula a = 2 miK', m varying from — m' to + m' ; all these infinities are simple. The residue of ^^, relative to the infinity 2 miK' is:^;^ ; that of ^i^ H(m) H'(0) H(m) is (_ 1^01(0). ''^ ^' H'(0) We therefore have n^ Q(") _ ^ Q(0) "V" 1 ^^ H(.) 2KH'(0)_^„^i„ . (^_2^.^,; 2 K ^'^ H(.) 2KH'(0)„f^^^.^ . (^_2mK') 2 iv Replacing in these two formulas u by the quantities u + K, u + iK', 1^ + 7^-1- iK' we have six additional formulas including HM = ^ 0(0) "y" ! , ^^^ @iu) 2KWm^^_^^^j^^^ _ (2^ _ ,).^,j 2K HiM = JEL QiM^y " (-1)" ^ ^ 0(«) 2K H'(0)„f^„3j^^f„ _ (2^ _ i).^,j 2 if 260 THEORY OF ELLIPTIC FUNCTIONS. To develop the function ^ . , say, which admits the period 2K, we apply the method of the preceding Article. We note that for congruent points on the sides EF and GH of the parallelogram EFGH, the difference of the values u being equal to (2 m' + 1)2 iK', the function f{u) takes equal values with contrary signs; and the values of the tangent on these fwo sides being T i, the definite integral relative to these two sides is zero. We therefore have (5^ Qi(^) ^ jEL Hi (pr y" (- ir e(u) 2KW{0) -^ 'n r ,o ...j^n ^ ' ^ 'm=-m tan — — [u — {2 m — l)iK] 2 K Further, since 1 H(m) ./FHi(m) , wr;©i(M) snu = — = — ^^^ , cnu =\/ ^-^^-' , dnu =\' k ^^ ' , Vk@{u) ^ k &{u) @{u) we have by differentiating sn u with regard to u, and then writing m = ®M.= 1 ;andsinceHi(0).JA, H'(0) Vfc 0(0) >^ k' Hi(0) = -i= and similarly ©iW _ ^ H'(0) Vk' H'(0) Vkk' It follows immediately from (3), (4) and (5) that 771= +00 (^) ^"" = p-f^X 1 (7) . 777= +CC en M = 2kK "sin^[M-(2w- (- D- Dt/i:'] "sin^[w- {2m- Dtif'] (8) dn M = -^ ^ 2 -^^,,7= -co tan -^[m - (2 to - 1)1^'] 2 /t If we group the terms two and two the equations (6) and (7) become 2-\/q . nu""^ o'"-i(l + 02™-!) ''^ ^^TTi'Ti 1-2927^-1 cos 2i^ + g4— 2 2 K 27rV'g nu""^ (- D^-itf^-iCl - oS"--!) ^''^ ^"" = -^ -^2-x X / ,3,7. 2.u' J; 771=1 1 — 252m-l gQg. _|_g4m-2 2 if THE FUNCTIONS sn u, en u, dn u. 261 The series (8) is not convergent in both directions; but if from dn we subtract dn u, we have the convergent series =00 (-1)'»-V J l_+j2"'-i (11) 1 -d7iw = -^sm2-^ y 1 Observing that (1 + q) sin t ^ 2K ^ = sin ^ + g sin 3 < + 5^ sin 5 < + 1 - 2 5 COS 2 < + <2 it is evident that (9) and (10) may be written (12) «„,== 3 2:T4^2;;^i«^^(2m-i) , m=l ^ 27!:V'7'"^" g™-i ,„ ,, -u (13) ^«^ = ^Z- X l+%2^-^ '^°^(^^-^^2K m=l These values are the same as those given at the end of Art. 229, where the corresponding value of dnu is found. By considering the quotient } ' as given in equation (1) and also the H(m) quotient — ^ , we may derive in a similar manner Hi(m) 2 A' 1 1 /-^ q2m-i _ ^^ sm— .=, (15) = + 4 y(-ir ^ o^_i cos (2 m - 1) ;-^- r cnu nu ^ 1 + g^"" ' 2, K cos m=l ^ 2K [See Jacobi, Werke, I, p. 157.] 1. Prove that sn(iu + K) ■■ EXAMPLES cn{iu) dniiu) 2. Show that " ' sin am u 3in am ( IKU, 77 J = — V k'l c IDS amIifc'M, 77) = Aam(i.'.,n = -i \ k / cos a: cos am u A am M ) cos am u am u 262 THEORY OF ELLIPTIC FUNCTIONS. 3. Show that A am 4. Prove that (tk\ ik sin am u iku, —- 1 = — k / A am u am u cos am u A am u + 1 sn^(iu, k) sn^iu, k) = 1. 6. Derive the formulas T/, . ,x • 2 Vk' "1 i(l + fc') sn(u, k) cn(u, k) snia + k')^u, ^,\ = ,_(, ^,,;^.(,^;:/ . V ^ '1 + fc'J 1 -(1 +k')snHu,k) L^ ' '1 + k'^ 1-(1 +fc')s«'Kfc) Suggestion : apply formulas (XVI) to formulas (XXV). 6. Show that sn\{k'+ik)u,^^^'] = (k'+ik)sn(u,k)dn(u,k) L ' ' fc' + ifc J 1 - (fc - ik')k sn2(u, fc) r,,, J ..^ 2\/iA;A;'1 cn(u, fc) ere (fc'+iA;)if, — = i-LJ. L ' 'k' + ikA 1 - (fc - a') fc sn^u, k) dn\ik'+ik)u,^-^^] = l-(fc + ^fcO fe^'fa, k) L ' 'ft' +ifc J i-(fc- ifc' 7. Show that 8. Show that 9. Show that [ik-ik,u.'^ (fc — ik')u, (fc - ik')u, fcjMfc'' ' fc - ik\ k + ik' ') fc sn'{u, fc) (fc - ik') sn(u, k)dn {u, fc) cn{u, k) 1 - (fc - ^^fc')fc sn^(M, fc) cn{u, fc) 1 -(fc + t:fcOfc6w'(M, fc) cn{u, fc) fc - ifc' ^'{u) =Vk cnu dnu^iu) +\/k snu&(u). ©(iq=®(0) Vfc' H(-K)=\/^,0(O), V^ = H(g) e(js:)' THE FUNCTIONS sn u, en u, dn u. 10. Prove the following relations : &{iu, k) V k' eiKit Riu + K', k') 0(0, A:) \ k'" 6(0, fc') H(m,fc) ../k^^,n(u,k;) 0(0, /c) 'Vfc' 0(0, A;')' miu + K,k) /k ^, @{u,k') 0(0, A;) \k'^ 0(0, fc')' @{iu + K,k) /k ^ @{u + K',k' 0(0, k) V fc' ^ 0(0, fc') 11. Show that and that d /en du\ ^^^^]= dn'iu + iK')-dn^u snu / d /sn u dn u\ du\ cnu I dn^u + dn^{iu, fc')— 1. 13. Prove that sn u dn"u — sn"u dnu — snudnu; and that {sn uy, sn u sn'u, (sn'uY (en uf, en u en'u, (cn'u)^ (dn uf, dn u dn'u, {dn'uf 263 13. Show that 2kK 2 Ku 4 Vo sin u 4 Vq^ sin 3 u 4 Vq^ sm 5 u cos COam = ;; — ; 5 1- — = ^ T 1+g 1+3 1+?^ = k' snuen u dn u. (G. B. Mathews.) 4V'9=i ' 14. Show that 2k'K 1 dn 2Ku Aq A(f = 1 - — -^ cos 2 M + , ^ ■ 1 + 5^ 1+q* cos 4 u cos 6 M + CHAPTER XII DOUBLY PERIODIC FUNCTIONS OF THE SECOND SORT Article 232. From the formulas (X) and (XII) of the preceding Chapter it follows that dn u has the period 2 K and sn u the period 2 iK', although 2 K is not a period of sn u and 2 iK' is not a period of dnu. There is consequently an irregularity in this respect. In order fully to understand this, it is well to consider the doubly periodic functions of the second sort which were introduced by Hermite.* The Germans use the word "Art" for the word "esp^ce" which I translate by " sort " (see Art. 84 where the doubly periodic functione of the third sort were treated under the name " Hermite' s intermediary functions "). In this connection see Jordan, Cours d' Analyse, t. II, No. 401, and Halphen, Traitc des fonctions elliptiques, t. I, pp. 325-338, 411-426, 438-442, 463. Art. 233. A doubly periodic function of the second sort with the primitive periods 2 K and 2 iK' is defined through the functional equa- tions f(u + 2K)=vf(u), fiu + 2 iK') = y7(u), where v and v' are constants called factors or midtipliers and are inde- pendent of u. When n = I = v', we have the doubly periodic functions properly so called, which belong to the category of doubly periodic functions of the first sort. In the case before us of the preceding Article sn u, en u, dn u belong to the class of functions of the second sort, as appears from the formulas (X) and (XII). For the function snu we have v =— 1, v' = 1; for en u we have 1^ = - 1, v' = - \, while y = + 1, i/'=— 1 for dnu. We may now consider more closely these doubly periodic functions of the second sort. * Hermite, Comptes Rendus, t. 53, pp. 214-228, and t. 55, pp. 11-18 and pp. 85-91; Hermite, Note sur la iheorie des fonctions elliptiqiies, in Lacroix's Calcul, t. 2 (6th ed.), pp. 484-491; see also Cours de M. Hermite redige en 1882, par M. Andoyer, p. 206; Appell, Acta Math., Bd. 13, 1890; Picard, Comptes Rendus, t. 90, pp. 128-131 and 293-295; Picard, Crelle, Bd. 90, pp. 281-302; and in particular Forsyth, Theory of Functions, pp, 273-281, where references are made among others to Frobenius, Crelle, Bd. 93, pp. 53-68; Brioschi, Comptes Rendus, t. 92, pp. 323-328. 264 DOUBLY PERIODIC FUNCTIONS (SECOND SORT). 265 Art. 234. Formation of the dovbly 'periodic functions of the second sort which have prescribed factors v and v'. — In the following Article it is shown that it is always possible to form a fundamental doubly periodic function of the second sort f{u) with factors y and v', which function is infinite of the first order at only one point within the parallelogram with sides 2 K and 2 iK'. The infinity of this fundamental function is denoted by w = c. This admitted for the moment, let F{u) be an arbitrary doubly periodic function of the second sort which has the periods 2 K and 2 iK' and has the same factors v and v' as f{u). Further we shall assume that F{u) is determinate at every point of the period-parallelogram. Suppose that the function F{u) is infinite of the ^t order at the points tti (i = 1, 2, . . . , n), where the points ai, a2, . ■ . , an all lie within the period-parallelogram. We shall show that F{u) may be expressed in terms of /(m). For simplicity suppose that the parallelogram is so situated (Art. 91) that F{u) does not become infinite upon its sides. Consider next the function no=F{of{u-$), where u is any point within the period-parallelogram, while f is to be regarded as the independent variable. Instead of f write f + 2 K. It follows that Vr(f + 2K)= F{$ + 2 K)f{u -e-2K). But we have j-^^ +2K)= uf{u) , /(M - e + 2 K) = i/(w - ?)• If we put $ + 2 KioT $ in this last formula, the result is /(M-f-2X)=i/(w-0. V Also, since Fi$ + 2K)= vF{^), it follows that ^{^ + 2K)=F{^)f{u-^), or t (f + 2 K) = t(f), and similarly It is thus seen that '>^(f) is a doubly periodic function of the first sort. For such a function we have proved that 2Rest(O=0, where the summation is to be taken over all the infinities within the period-parallelogram. 266 THEORY OF ELLIPTIC FUNCTIONS. But i/r(f) becomes infinite on the points where F{$) is infinite and besides on the point w - f = c, where f{u - f) is infinite. The points ai, a2, ■ ■ ■ , an must be distinct from the point u — c = $. The expansion of f{u — f ) in the neighborhood of the point c is of the form /(w-0= ? + Ao + 4l(M-f -C)+ A2(u-f -C)2+. . M — f — C "^ +Ao-Ai[^-{u-c)]+- ■ ■ . f - (M - C) (In the sequel we shall choose a fundamental function f{u) such that the quantity C is unity.) Next if we develop F{^) in the neighborhood of u — c by Taylor's Theorem, we have F{^) = F{u - c) + F'iu - c)[f - (u - c)]+ ■ • • , and since f{0=F{e)fiu-o, we have Res ■yjr{$) = — CF{u — c). f = u — c In the neighborhood of the infinity at, the expansion of F{^) is of the form (cf. Art. 98) (f - a*)''' (f - at)^' 1 (? - akV while the expansion oi f{u — $) in the neighborhood of this point is fin - 0-f(u - a,)-^^^^^(i - .,) + r(u^i^ _ «,)2_ /'^--'>(^-«. ) ± (4-1)! ^^ "*'' + Through the multiplication of these series it is seen that Rest(e)= ^,,i/(M-afc)- Al2 ^,(^ _ ^^) + ii^y„(^ _ ^^)_ Since ^Res y}r{$) = 0, we have = - CF{u - c)+ 2 r4ft,i/(w - a,) - 4^/'(u - aft) + ft=lL 1! ^a Ift-l)! J CF{u) = y I Ak.ifiu + c-ak)- ^f'{u +c-ak) DOUBLY PERIODIC FUNCTIONS (SECOND SOET). 267 II' next we write m + c in the place of u, it follows that k = n + • • • ± TT^^^ /<''-!'(" + c - ak)\ {^k — 1; '• J which is the expression of F{u) in terms of the fundamental function f(u). Art. 235. Formation of the fundamental function f{u) which has prescribed factors {or multipliers) v and v', where v and v' are any constants different from zero. We had the formulas H(u + 2K) = -Il{u), H(m + 2iK')=- //H(m), where // = /i(w) = e If we write iu)=B.{u+^), it follows that {u + 2K)=B.{u+l3 + 2K) = -B.{u + ^); or, cj}iu + 2K) = -iu), and similarly nip (f>{u + 2iK') = - u.e~^4>{u). Consider next the function ^y,._ B.{u+P) _4>{u) ^ ' H(m) H(m) We have immediately ^(w + 2 K) = -¥{u), ■^{u + 2iK')=^{u)e ^. The function •*■(«) is therefore a doubly periodic function of the second _ '^ sort having as factors +1 and e ^ . Suppose that v and v' are the prescribed factors. To form a function having them, write f{u)= e««^(M); so that f{u + 2K) = e«(»+27i:) ^(^ + 2 K) = e-^^^fiu) and _ j^ f{u + 2iK')= e«(»+2*^')^(w + 2iii:')=e"'^'^e */(w). Hence /(w) is a doubly periodic function of the second sort with the factors e"^-^ and e 268 THEORY OF ELLIPTIC FUNCTIONS. The arbitrary constants a and ^ may be so chosen that (1) e«2^ = V, (2) e ^ = v'. From (1) it follows that a=^logv; and from (2) or _ K'logv + Kilogv' The quantities a and /? being thus determined we have f{u + 2K) = vf{u), f{u + 2iK')=v'f{u), The function f{u) is infinite of the first order for w = (see Art. 203) and for no other point in the period-parallelogram, since the other vertices of the parallelograms are counted as belonging to the following parallelograms. Art. 236. There is one case * in which we cannot determine f{u) in the above manner, viz., when the multipliers or factors v and v' have been so chosen that j3 = 2mK + 2 niK', where m and n are integers. We would then have y(,,) = e«" H(u + 2mK + 2niK') H(w) = (■_ 1-)77.car. H(M + 2niK') H(w) Further, since (cf. Art. 91) - % (TIM +nHE') H(w + «2iK')=(-l)"e ^ H(m), it follows that - ^ (MM +'nHK') TT (.,\ /(«)=( -l)™+"e ^ e««^, H(w) so that f{u) is an exponential function and no longer a doubly periodic function of the second sort. * See Forsyth, Theory of Functions, p. 279. DOUBLY PERIODIC FUNCTIONS (SECOND SORT). 269 Art. 237. We must proceed differently for this exceptional case. We had by hypothesis P = 2mK +2 niK', and consequently 2 mKK + 2 nK'in = K' log v + Ki log v'. Further, since log v = 2 Ka, it follows that Ki log v' = 2 mKn + 2 nK'tTC - 2 KK'a, or log v' = — 2 mm + 2 n -— tt + 2 K'ai. K We thus have v' = e -^ =e ^ •*^'', and If we put nni « - ^ = r, the above expressions become We have the exceptional case* when v and v' have this form. The quantity y is arbitrary; but if the factors v and v' are given, then y is known. We now write H(w) where H'(m) is the derivative of H(m). • From the formulas H(w + 2K)= - H(m), H(u + 2zA:') = - e"-^^""''^'H(M), we have at once H'(m + 2K)=-H'(w), H'(M + 2iii:') = e ^ [|H(m)- H'(«)J- It follows that /(M + 2K) = /(M)e2^V = v/(M). We further have B.'{u +2iK') ^_m E.'(%i) B.{u + 2iK') K H(w)' so that f{u + 2iK')=v'f{u)- i^'^ev". * First noted by Mittag-Leffler, Comptes Rendris, t. 90, p. 178; see also Halphen, Fonct. EUipt., t. I, p. 232. 270 THEORY OF ELLIPTIC FUNCTIONS. The function f{u) is therefore not a doubly periodic function of the second sort. It will nevertheless serve for the formation of a doubly- periodic function of the second sort with the factors e^-^'i' and e'^^'^'', which function becomes infinite on an arbitrary number of points within the period-parallelogram. Let F{u) be the function required, so that F{u+2K)=^vF{u), v = e2^> and F{u + 2 %K') = v'F{u), v' = e^^''^. We shall express F{u) in terms of /(w) = ; ' e^"- -ti(M) The period-parallelogram is to be chosen so that F{u) does not become infinite on its sides. We again form the function We shall see that '«|r(f) is here not a doubly periodic function of the first sort as was the case in Art. 234. From the formulas /(w + 2K)=v/(,0, /(M + 2iK')=v'j/(u)-ger«j, it follows that i/r(f + 2 K) = -<|^(f), and further that ^^^ + 2 ^K') = f (O -f e^C" " e)F(f ) g • is. We again note that 2 iK' is not a period of V^C^). We compute next 2 Res ■V^(if) for the interior of the parallelogram whose sides are 2 K and 2 iK' . It is seen that f = m is an infinity of i|f($); for H(0)= 0, and as H(u) is an odd function, its expansion is H(m)= m(co+ CiM^-t- . . .), so that TT,/ N 1 Itt - - + P(«), H(m) w where P(m) is a power series in positive integral powers of u. Similarly we have H(m - f ) M — t Further, since e^" = 1 + ^ -f • • • , we have , ,H'C?y — P\ 1 /(^ - = e^("-^' "Ir I = - r^— + 'PiC" - f), H(m - f ) f - M where Pi(m — f) denotes a power series in positive integral powers of DOUBLY PERIODIC FUNCTIONS (SECOND SORT). 271 The expansion of F(6) in the neighborhood of f = w is We therefore have As in Art. 234, F{^)=F{u)+ F'{u)^^+. ■ ■ 'Resy}r{$)=~F{u). Res ^(f) = Ak, ifiu - ak) - ^f'{u - ak) A It follows that (4 - 1)! 2) Res t(f) = -F{u) + X \Ak.if{u-ak) - ^T^'^" " "*^ (4-1)! We cannot put S Res "^i^) = 0, as in Art. 234; but after Cauchy's Theorem ^ 2niJ where the integration is to be taken over the four sides of the parallel- ogram in the figure. p + 2iK' p + 2K+2iK' We have as in Art. 92 P + 2K 2 7ri ^Resi/-(0 Jnp+iK pp+2K+2iK' rp + 2iK' fp ' ■iri?)d$ + / t(f)rf? + / fiOd$ + / fiOde, p ^p+2K 'Jp + 2K-¥2iK' 'Jp-¥2iK' or bv Art. 92, or = 2K rVcP + 2Kt)dt + 2iK' P f(p + 2 K + 2iK't)dt Jo /(^-i) (m - afc) j . (h— 1)! ) On the other hand if we put u + 2 iK' for u in the expression above, we have F{u + 2iK')=v'Aey- + i^'X\Ak.if(u-ak)-- ■ ■ ± ,f ^'^"'. /(u-a/c) l k = i' (4-1)1 ) ■ k = n, 1! du "^ (4- 1)! dw^'-i ^' Comparing the two results just derived, it is seen that ^rj \ ■ 1! '^^ 2! ^ =^ (4-1)'/ )y This condition must be satisfied by the A's in the formation of the function F{u). Since ;- is an arbitrary quantity, it may be made equal to zero. We then have 2^A.i = 0. But Ak,i is the residue of F(f) for f = ak. We therefore have ^Ak,i= llUesFiO; and consequently S Res FiO = 0, when 7- = 0. DOUBLY PERIODIC FUNCTIONS (SECOND SOET). 273 But if r = 0, then F{u + 2K)= F(m) and F(u + 2iK')= F{u), so that F{u) is a doubly periodic function of the first sort. We thus have another proof of the theorem* (see Art. 99) that for a doubly periodic function of the first sort the sum of the residues with respect to all its infinities vnthin a period-parallelogram is equal to zero. Art. 238. A preliminary formula of addition.^ — By means of the above results, and as an illustration of them, we may compute the addition-theorem for sn u. In the function sn{u + v) we consider v as constant and u as the vari- able. This function becomes infinite on the points where @{u + v) is zero, viz., u + v = 2 mK + {2n + l)iK' It is seen that ©(u + v) vanishes on the point u + v = iK' or m = iK' — V and on all congruent points (modd. 2 K, 2 iK'}. It is quite possible, when we consider the parallelogram of periods, that the point iK' — v does not lie within it. There is, however, some congruent point which does lie within it, and we shall simply denote this point by iK' — v. Consider the product sn(u + v)[snu — sn {iK' — v)\. If u = iK — V, the expression within the braces becomes zero of the first order, while sn{u + v) is infinite of the first order. The product therefore remains finite for u = iK' — v. We form next the function ' G(u)= sniu + v)\snu - sn{iK' - v)]\snu - sn {iK' + 2K - v)}. This product remains finite for u = iK' - v and ior u = iK' + 2 K - v and for all points congruent to these two points (modd. 2 K, 2 iK'). We have sn{iK' — v)= r = — ; it sn{ — V) k snv It follows that -i- > KoiLu,— - k snv) ( K = sn{u + v)\ sn^u G{u) = sn{u + v)\snu + i \snu - [ ^ ■' ^ ' ) ksnv) ( ksnv) 1 } k^sn^v ) or G{u)k^sn^v = sn{u + v) { k^sn^u snH - 1 } = ^^^^' ^^y- * See Forsyth, Theory of Functions, p. 280. t Hermite's " Cours " (Qxmtneme edition, p. 242) ; see also Appell et Lacour, Fonctions Elliptiques, p. 129. 274 THEORY OF ELLIPTIC FUNCTIONS. It follows at once that F{u + 2K)=- F{u), so that v =- I and F{u + 2iK')= F{u), or v' = L We note that F{u) is a doubly periodic function of the second sort with the periods 2 K and 2 iK'. Consider the parallelogram with the sides 2 K and 2 iK' in which the point iK' lies. The function F{u) becomes infinite on this point but on no other point of the parallelogram. To determine the order of the infinity of F{u) for the point u = iK', it is seen that snh = h -\- c^h^ + cji^ + • • ; and consequently if we put u = iK' + h or h =^ u — iK' , we have 111 sn{iK'+h) = ksnh kh 1 + egh^ + egh* + ■^{1 +e2h^ + e4h* + - ■ ■ }. kh It follows at once that and consequentl}' k^sn^u sri^v — 1 = sn^v + 2 e-isn^v + ■ ■ • — 1 (m — iK'Y d STi 1) Noting that — - — = cnvdn v, it is seen that the expansion of sn{u + v) in the neighborhood oi u = h + iK' is 1 sn{u + v) = sn{v + iK' + h) = k sn{v + h) which by Taylor's Theorem 1 T cnvdn V , k snv k sn^v _ 1 _ cnvdnv k snv k sn^v We therefore have {u - iK') + Writing VfnA- 1 swt; cnvdnv 1 , _/ r^,^ ^^^^ -kiu- iK'Y - —ir- v^r^ + ^(" - ^^ >• cnv dnv , snv . k "' k we have F{u) = -^^ + \ + Piu - iK'). u — iK (u — iK')^ DOUBLY PEEIODIC FUNCTIONS (SECOND SORT). 275 We shall next express F(u) through a fundamental function f{u). The function f{u) must be a doubly periodic function of the second sort with the factors + 1 and - 1 and with the periods 2 F and 2 iK'. We may consequently choose —for this fundamental function. We have ®^^ = — h positive powers of u. snu u Consequently we have Res/(M) = 1 = C (of Art. 234). Hence (see the formula at the end of Art. 234) it follows that F{u)= Aof{u - iK')- Aif'iu - iK'). We have further ., -r^,. 1 f{u - iK') = — — - = k sn u, sn{u — iK) and also /' (m — iK') = kcnudnu, so that p/, N cnv dnv , snv , , p (u) = ; k snu — kcnu an u. k k Equating the two values of F(u), it is seen that sn(u + v) [k^sn^u snH — I] = — snucnv dnv — snv en u dnu, or finally , -. _ snucnv dnv -\- sn vcnu dnu 1 — k'^sn^u sn^v which IS the addition-theorem for the modular sine. When k = 0, we have sn u = sin u, en u = cos u, dn u = 1, and consequently sin {u + v)= sin u cos v + cos u sin v. The above addition-theorem may also be written in the form d snv , d snu sn u \- snv , , \ dv du sn{u + v) = 1 — k^sn^u sn^v As an exercise the student may derive the addition-theorems for cn{u + v) and dn(u + v) and compare the result with those given in Chapter XVI. Art. 239. As a further application of the doubly periodic functions of the second sort we may develop in series of sines and cosines such expressions as 0(it -I- o) H(m -I- g) ©i(M_±_o), Hi(m + g) 0(m) ' 0(m) ' 0(m) ' ©(m) which appear in Jacobi's investigations relative to the rotation of a body which is not subjected to an accelerating force.* * Jacobi, Werke, II, pp. 292 et seq. 276 THEORY OF ELLIPTIC FUNCTIONS. Consider with Hermite * the series e^ sin — — (m + 2 niK') 2 K where n takes all values from — oo to + oo , a being a constant which will be represented by a + ia' . We shall first show that this series is convergent, whatever be the value of u, provided that a' is less in absolute value than 2 K'. Writing the general term in the form xma it is seen that we may neglect the first or the second exponential term in the denominator according as n becomes positively or negatively indefinitely large. We thus have either -2te-*^ ^^ or 2te^ ^^. If we write — n in the place of n in the second of these quantities and take the limit for n indefinitely large of the nth root of the moduli, we have after a has been replaced by a + ia' -^A0L' + 2K') -J.(a'-2Ji7) either e or e If for the first a' + 2 K' > and for the second a' — 2 K < 0, the two limits are less than unity and the series in question is convergent. Consider next the function Tzina ^{u) = y '— : sin ~(u + 2 niK') 2 it and noting that, since n varies from — oo to + <» , we may change n into n + 1, we have (n+ l)ma ^na K !^ '~K' («)=2; =e*'x sin -IL-{u + 2{n+ l)iK'] sin -ZL. [„ + 2 iK' + 2 niK'] Z ti. 2 K It follows at once that ina 4>(u) = e"^$(M + 2iK'), ina or *(m +2iK') = e~^4)(M). * Hermite, Ann. de I'Ecole Norm. Super., 3^ s6rie, t. II (1885); see also Hermite, Sur quelques applications des fonctions elliptiques, p. 35. DOUBLY PEEIODIC FUNCTIONS (SECOND SORT). 277 On the other hand we have immediately so that 4>(m) is a doubly periodic function of the second sort with the lira multipliers — 1 and e ^ . The poles are obtained by writing sin -^ (m + 2 niK') = 0, 2 K from which we have M = 2 mK - 2 niK', where m is an arbitrary integer. We therefore see that on the interior of the rectangle of periods 2 K and 2 iK' there is only one pole w = 0, the corresponding residue being 2 K We further note that the quantity 2K H' (0)8 (it + g) Tt H(M)0(a) has the same multipliers, the same pole, and the same residue. We may therefore write (see Art. 83) S- 2K H^(O)0(M + g) _ 7. H(w)0(o) ^sin^(u + 2mK') 2 K If a and u are permuted in this equation, we have jtinu 2K H'(O)0(M + a) _y ^^^ n H(a)0(t.) ^si,^(« + 2niK') 2 K We may deduce the others as follows: If we change a into o + iK' , we have itinu riu x e -X— ^ 0(w)0(a) -^sin ^^[o + (2 n + l)iK'] 2 K or 2Kir(0)Ii(^i+-«l=V ±- ^^ , 2K B.'mii{u + a) _^ ^^ n 0(w)0(a) ^sin^[a+(2n + l)iK'] 2 K 278 THEORY OF ELLIPTIC FUNCTIONS. If further a + K is written for a in (1) and (2), these formulas become „ A' ,o^ 2K H'(O)0i(m + o) _y . e(.)H,(a) ^,os^[a + 2niK'] 2 K (2n + l)?rtu 2K H^(0)Hi(m + a) _v e ^^' 2 K. If w + if is written for u in the four formulas above, we have the four following formulas, in which @i{u) is found in the denominators: ,.. 2K H'(0)9i(u + a) ..y ± n 0i(w)H(a) ^ ■ it IK H'(0)9i(m + a) _y ( - l)"e ^ 0i(^^)H(a) ^sin^(a + 2mK')' 2K^ (2n+l)rtM 2gH'(0)Hi(u + a) _^:^ t^" + ^e ^-^ n 0i(M)0(a) ^ ■ K r , ,„ , -,\irn '^^ ' ^ ' sm— — [a + (2n + l)iK] 2 K Ttinu .-,. 2g H'(O)0(m +a) y ( - l)"e^^ ;: 0i(m)Hi(o) ■^ TT / , o -i^/x ^ ^ -^ ' ^ -^ cos (a + 2 ni/t ) 2X (2n + l)7rtu 2_K H'(0)H(u + a) _ y j-iy^+U ^^ n 0i(M)0i(a) •^ ;r r , .„ , is-i^/i ^ -^ ^ ^ cos-— [a + (2w + l)t/t J 2 K. (8) Art. 240. Hermite next formed a series entirely different from the one of the preceding Article which is represented as follows: izina cot ^ + S '^ P* 2% ^" + ""'^'^ + "] ' where n takes all even integral values from — oo to + <» , while the quan- tity £ must be supposed zero for n = and equal to unity positive or negative according as n is positive or negative. If we allow n to take only the positive integers n = 2, 4, 6, • • ■ , the series above may be decomposed into the two partial series JTMWI cot II + cot II + 5) e^ [cot H (^ + niK') + i] DOUBLY PERIODIC FUNCTIONS (SECOND SORT). 279 which by aa easy transformation becomes cos -— {u + mK') 2K e ^^ COS (u — niK') 2K To prove the convergence of this series, note that for large values of n the two denominators cos — — (u + niK') and cos — -; (w — niK') may be replaced by ie ^^ and ie^^ , the general terms becoming ^{na + 2niIC+2u) - ^(.na-2niK' +2u) If we put a = a + ia', we have for the limit of the nth root of their moduli as n becomes very large, the quantities -Ji-(,a' + 2IC') '^a--2K') e ^^ and e^^ and consequently the conditions a'+2K'>0, a'-2K'<0. It follows that the series in this Article is, as the one in the preceding Article, convergent when the coefficient of i in the constant a is in abso- lute value less than 2 K'. This series also defines a doubly periodic function of the second sort; For writing Trt'na — I ^(^,) = cot II + 5; e^ [cot ^ (^ + niK') + eij , we have the relations ^(m + 2X)=^(m), S^{u + 2iK')= e'^'iriu). The second of these relations is evident from the expression of the Tzia product e^'^iu + 2iK'), viz., e^^iu + 2iK') = e''cot^ + e''%e'''\cot^{u + 2iK'+niK') + ei^. 280 THEOEY OF ELLIPTIC FUNCTIONS. We have na A' . na ^ na , ■[ K , ^ ] ^ 2K "° 2K ^ '' and if we change, as is permissible, n to n — 2 in the general term, it becomes :^a I Trig \ irina p -, e^'i'{u + 2iK') =cot-^+i[e^ + l) + ^6^" ^cot~^{u + niK') + dj, where now there is a modification regarding s. The quantity e must be = 1 for n = 4, 6, 8, ... , while £ = for n = 2 and e = — 1 for n = 0, — 2, — 4, • ■ . We note that in adding jtia to the terms corresponding to n = 2 and n. = on the one hand ie ^ and ( ivia \ e'*- + 1/ to enter the summation, we find for e precisely the significance which was accorded it in the function '^(m). We further note that within the rectangle of periods there exists the one pole m = 0, to which corresponds the residue - — . We may therefore represent the function '^(m) by 2K H'(0)H(m + a) K H(w)H(o) If we interchange u and a we have finally where n represents all even integers and the unity s must be taken positive when n is positive and negative when n is negative. «« Next changing a to a + iK' , we have, after having multiplied by e^^, the formula IrtU TrtTTlU . where m denotes the odd integer w + 1. Since e^^ cot ^^ = _i— + iP^, 2K . Ttu sm — — ; iriu 2 JK. we have, if the term ie^^'is introduced under the summation sign, (10) ' ' -' , , hXe^'' cot— -(a + miK')+ a , n H(M)e(a) . nu -^ \_ 2K J where m represents all odd integers and e must be taken + 1 or — 1 according as m is positive or negative. Changing a to a + K we have DOUBLY PERIODIC FUNCTIONS (SECOND SORT). 281 the formulas (11) and (12) below, and by replacing u by u + K in the formulas (9), (10), (11), (12) we have the formulas below, (13), (14), (15), (16). ninu ,TT, 2g H'(0)Hi(M + a) ^ OTi V axr n , , -j^,. .1 ^'^^~ H(.)H,(a) -''°^2g -^' [tan— (a + mK')-r,J, . (12) 2KirMM2ii^ =eoseci^-X«^rtan-^(a + mi/v")-«-l, n H(w)0i(a) 2K ^ I 2K J ^ H-(0)H,(.+a) ^_^^^^^^(_,)l^wr ^ (a + ™X') + .r, ;r Hi(M)H(a) 2a ^ L 2a J ,, .. 2K;H'(O)0i(M+a) TTW ,V-mT^r ^~ , , ■L-t\ , -1 (14) ^ ' /' — -^ =sec — - + y.i'^e \ cot —— ia + miK) + SI \, ^ ' n Hi(M)0(a) 2K ^ I 2K^ ' \ n TTinu jrt'mu (16)2^^(«J^ =secil| -2x-e-^rtan-^(a + m^7.")-c-.| TT Hi(M)0i(a) 2 a; "^ L 2 A J the quantities m, n, and £ being defined as above. EXAMPLES 1. If TO = 1, 3, 5, . . . ; n = 2, 4, 6, . . , show that 2K Bmmii±A=^osec^+4Xq^sin^(ma-nu). Tt 0(w)H(a) 2K ^^ IK Z, Further if m'= 1, 3 5, . . . , prove that 2X Wmmu+a) ^ ^y '^ ^^ jr. ^^ _ ^,^) ;r 0(u)0(a) ^^ 2i!C' 3. Show that m — 1 mn 2K H'(O)0i(M+a) _ ggj, iE£L + 4V ( - l)"2~g 2 cos _?L (^„ _ „w). TT 0(w)Hi(a) 2ii: ^ 2e: 4. Prove that m — 1 mm' 2K H'(0)Hi(M + a) ^ 4^ (_ ^"^ g-2- eos -^ (ma - m'u). n @{u)@i(a) ^' 2K [Kronecker.] CHAPTER XIII ELUPTIC INTEGRALS OF THE SECOND KIND Article 241. From the investigations relative to the integrals of the first kind in Legendre's normal form (see Chapter VII) it is seen that the elliptic integral of the second kind Z2dz / \/(l - 22) (1 - k^T?) is finite and continuous on the finite portion of the Riemann surface. In the neighborhood of the point z = qo, we have V(l - z2)(l - A;2z2) ^ J_ Oi 02 7? 2.'^ Z6 so that /: ■2?d-z = z +^ + k + \/(l - Z2)(l - /c2z2) z z 3 where the a's and h's are constants. It follows that the elliptic integral of the second kind is algebraically infinite of the first order for the value z = oo in both the upper and the lower leaves. In the Weierstrassian normal form /: VS{t) the expansion in the neighborhood of the point t = oo, which is a branch- point, is the limits of integration being so chosen that no constant term appears in this development. The question naturally arises whether it is possible to form a one-valued function of position on the Riemann surface which is algebraically infinite at only one point. To investigate this question, consider the integral /( Cdt {t - a)2 where C is a constant. 282 ELLIPTIC INTEGEALS OF THE SECOND KIND. 283 This is the faimplest integral which is algebraically infinite of the first order at the two points* a,VS{a) and a, -VS{a). We note also that the integral J it- a)WS(i) where A and B are constants, becomes infinite in the same manner at the same two points as the integral above. Neither of these integrals is infinite for i = oo. We shall so choose the constant s A a nd B that the latter integral be- comes infinite on the point a, — \^S{a) in the same manner as does the first integral. By Taylor's Theorem we have in the neighborhood of the point t = a A\/S{a)-l (Aa + B) S'(a) M + B^Aa+J^ I v;^W(, -«)+.. .. VSit) VSia) S{a) It follows, if we put (1) AVsi^) -liAa+ B) -^1^ = A = 0, that r (At + B)dt ^ _ Aa + B ^ ^ log (t - a) ^ p^^ _ ^^ J (t- a)^VS{t) it - a) VSicc) S{a) will not contain a logarithmic term in the expansion according to ascend- ing powers of i — a. Further, since " Cdt C h {t - a)^ t - a it is seen that the two integrals become infinite alike on the point a,-VS(^), if (2) -C^^^. It follows from equations (1) and (2) that 2 -s/Sia) L y/Si^ J * The following results are true not only when S(t) is of the third degree in t, but also when this degree is n, where n is any positive integer. 284 THEORY OF ELLIPTIC FUNCTIONS, and consequently that the integral /( At + B =.\dt {t-aY {t~a)2VS{t)/ ^(j I 2VS{a) dt {t - «)^ VS{t} is an integral of the sec ond kind, which is infinite of the first order* at only the one position (a, v'S(a)). Write C = ^ and put ri S'_{a)_ ^^ _ ^^ +VS(0 + VS(^) 2{t-af VS{t) + P{t-a). We may regard this integral as the fundamental integral of the second kind. Art. 242 . We next raise the question: Is there another integral Ei{t,VS{t)) of the second kind which becomes algebraically infinite of the first order on the point a, \/S{a)? If such an integral exists, its development in the neighborhood of i = a is of the form EiO, Vm) = -^^ + Pi(t - a), t — a Writing J E 1 (<, VSit)) = E {t, VSit) ) , it is seen that E{t,vs(t))-^oit,^m) does not become infinite for any point on the Riem ann surface. It is therefore an integral of the first kind, = F{t,\^S{t)), say. It follows that E(i, VS(i)) = EoO, ^m) + Fit, vsiT)). Hence, if we add to an integral of the second kind an integral of the first kind, we h ave an integral of the second kind which is infinite only at the point {a, \/R{a)) provided the original integral of the second kind is infinite only at this point. There are consequently an infinite number of inte- grals of the second kin d whi ch are algebraically infinite of the first order on the one point (a, \/R{a)). * Cf . Koenigsberger, Elliptische Functionen, p. 250, ELLIPTIC INTEGRALS OF THE SECOND KIND. 286 Art. 243. If we put ^(0 =: - ''^'^ {t - a) + V5(0 + V^ 2\/5(a) and ^(,)^ Vyx/p , 2 (< — «) then ^-±^= ^ffl ■ aa 2 (< - a)2 We further write (see Art. 2S7) tdl :{t,vsit))= f^t >S(o which integral, as we saw above, becomes algebraically infinite at infinity. It is then evident that the expression remains finite and continuous in both the finite and the infinite portion of the Riemann surface. It is therefore an integral of the first kind. Similar results hold when mutatis mutandis S{t) is of the fourth degree in t. It is thus seen that the elliptic integral of the second kind, which becomes algebraically infinite at the point infinity, may be replaced by one which is algebraically infinite at only one position on the Rie- mann surface, the latter position being a definitely prescribed one. Art. 244. If in the integral of the first kind — == 0.1 V(l-; z2)(l-A;2z2) we put z = sin <^, we have in Legendre's notation F(6,k) = r '^ — = P4^' where A Vl — k'^ sin2<^ The complete integrals of the first kind are therefore d4> \2 I JoVl-k^sm2(f> = F, # = F{k')= F'. Vl-A;'2sin2 In Legendre's notation {Fonct. Elliptiques, t. I, p. 15) the integral of the second kind is E{k,4>)= PVl - k^sm2. Jo Jo 286 THEOEY OF ELLIPTIC FUNCTIONS The complete integrals are (see also Art. 249) : TZ fi/|,fc^= rVl - k'^ sm.^4>d = E, It E(~,k'\= rVl ~k'2 sin^ d = E' = E{k'); or E = I gz: E = I — , dz. Jo \/i _ z2 Jo y/l _ z2 If we put d(f> = dnu du, A(f> — dnu, we have E{k,4>)= E{u)= f"dn^udu = ^(1 - k^sn'^u) du. Jo Jo (Jacobi, Werke, I, p. 299.) Art. 245. To study the integral of the second kind z^dz 1/0,1 '0,1 V(l - z2)(l - A;2z2) as a function of u, where dz = r^ Jo.i \/(l V(l - z2)(l - A;2z2) we may with Hermite * multiply this integral by k^ and put /2(w)= / k^sn^udu. Jo We note that the function sn^u has the periods 2 K and 2 iK' ; and from the developments above it is seen that l2{u) is a one-valued function of z. But z considered as a function of I2 is not one-valued, and con- sequently the problem of inversion for these integrals, which is effected with difficulty, does not lead to unique results (see Casorati, Acta Math., Bd. 8). Art. 246. We saw (Art. 217) that snu became infinite on the points 2mK +{2n + V)iK' = a, say. Writing u — a = h ot u = a + h,y{& must develop sn'^u = sn^ [2 mK + {2n + l)iK' + h] in powers of h. Since sn^u has the periods 2 K and 2 iK', we have 1 sn2 [2mK +{2n + l)iK' + h]== sn^ (iK' + h) ■■ k^sn^h * Hermite, Serret's Calcul, t. II, p. 828; CEuvres, II, p. 195; Crelle's Journ., Bd. 84. This integral Hermite denotes by Z{u). We shall, however, reserve this symbol for the integral employed by Jacobi (Art. 250) . ELLIPTIC INTEGRALS OF THE SECOND KIND. 287 so that It follows that k^sn^u = — 4- eo+ ei(M — a)^ + • • • ; [u — a)'' and consequently, since the integrand does not contain the term (w— a)~i, the integral Iziu) = I k^sn^u du Jo is a one-valued function of u. Art. 247. The analytic expression for /aCw). — The function k^sn^u is doubly periodic of the first sort, having the periods 2 K and 2 iK'. The only infinity within the period-parallelogram having the sides 2 K and 2 iK' is iK'. We may, however, consider k^sn^u as a doubly periodic function of the second sort with the factors v = 1 and v'= 1; or i^ = e^^^, v' = e^^'^'^, where y = 0. We have here the exceptional case of Art. 237 where k=n r- A -I F(w)=(7ev«+2 \Ak.ifiu-a)--^f'{u-a)+- ■ • , the function F{u) being k'^sn^u and /(m) = ^^e^ being /(m) = ^^, smce ;- = 0. The development of k^sn^u in the neighborhood of the infinity iK' is k^sn^u = - K7^^+eo+ei{u-iK')^+- •. Hence in the formula above, Ak,i, the coefficient of (m — iK')-'^, is zero: and 4^ 2, the coefficient of — (m — iK')~^, is — 1. du We consequently have kHn^u = C -f'{u-iK'). It follows that "^"k^sn^u du =[Cu - fiu - iK')]l, r Jo ,., „ W{u~iK') , W(-iK') or hiu) =Cu- jj(^ _ .^,) + H( _ iK') ■ It is thus seen again that l2{n) is a one- valued function of u. 288 THEORY OF ELLIPTIC FUNCTIONS. Since we have 7: K' Tciu H(m - iK') = - ie*^ ""^^QCm) --^(-2u + tJir')^ = -ie "^ e(M). It follows that 2 /t We therefore have H'(m - ^gO _ jyi_ ^ e'(M) H(M-iK') 2K e(M) and H^(- tigQ _ jri_ , e'(0) _ _7ri_ H(-iii:') 2K 9(0) 2K' since 0'(O) = 0. It has thus been shown that* /2(m)= Cu -~r^- 8(m) To determine C, we have from above du (m) Equating powers of u on either side of this equation, we have ^ e"(0) It follows that l2{u)du = hGu^- log0(M) + C, where C is the constant of integration. From this it is seen that log0(M) = C" + * Cm2- riziu) du, Jo or r\, X C'-t-iCu'-f''l,iu)du Finally we may write f 0(w)=C"e*''"^--^o"'=<">''", where C"=0(O). * Hermite, Serret's CoIquI, t. 2, p. 829. t Jacobi {CreUe, Bd. 26, pp. 86-88; Werke.II, pp. 161-170) defines the G-function by this formula and therefrom derives directly the series through which this tran- scendent may be expressed and its other characteristic properties. ELLIPTIC ESTTEGRALS OF THE SECOXD KIND. 289 Akt. 248. We may next consider the integral of the second kind /2(z,s)= p k^zHz \/(l - Z2)(l - ifc222) regarded as a function of z, s on its associated Riemann surface. In the simply connected Riemann surface T', we saw that u{z, s) was a one- valued function of z, s. If z, s are given, then m(z, s) is uniquely determined, and if u is known, then also l2{u) is known. Hence in T' not only the elliptic integral of the first kind but also the elliptic integral of the second kind is a one- valued function of z, s. Since 72 (z, s), that is, the elliptic integral of the second kind in T', is a one-valued function of z, s, it is independent of the path of integration. This, however, is not true of /2(z, s), that is, of the integral of the second kind in the Riemann surface T which does not contain the canals a and 6. For the elliptic integral of the first kind u(z, s) we had iu(X)— u{p) = A{k) = 2 iK' on the canal a, ( u{p) — u(X) = B{k)= 4 K on the canal b. In a corresponding manner we shall represent the constant differences of the integral of the second kind at opposite points of the banks as follows: * ( l2i^) — l2{p) = 2 iJ' on the canal a, ' hip) — l2{^)= 4 J on the canal b. We had (Art. 193) K • ^^ \/(l - Z2)(l - fc2z2) Jo Vn - z2) (1 - /fc'2z2) Jl 1 * dt V{1 - 7?) (1 - /fc'2z2) Jl V{fi - 1)(1 - /C2f2) In a corresponding manner we may write with Weierstrass (Werke, I, pp. 117, 118) Jo Vn - z2)(l - A:2z2)' Jl 1 * kH^dt \/(l - z2)(l - A:2z2) Jl Vifi- 1) (1 - A;2<2) We note that J' is not deduced from J by changing k to k'. From these definitions of J and J', it is seen in the remark at the end of Art. 249 that the formulas (2) above follow. * Hermite, loc. cil., p. 828; Fuchs, CTelle, Bd. 83, pp. 13-38. 290 THEORY OF ELLIPTIC FUNCTIONS. Art. 249. We had above /a u)= Cu-—^- ©(w) If in this formula we write m = K, we have From the formulas ©(m + K)=0i(w), 0'(m + K) = 0i'(u), it is seen that for w = 0'(K) = 0i'(O)=O, and consequently l2iK)= CK. To compute hiK) we put u = K m z = snu, and if Zq is the value of z that corresponds to m = i^, we have i,iK)^r-=j:m=^j. Zo= sriK = 1 (Art. 218). It follows that V{1 - Z2) (1 - fc2z2) We therefore have J = CK, or C = -^; and finally j e'(^i) /2(m)= t;M- KT-f- We may next compute the constant C in a different manner. If in the equation J , . ^ e'{u) 72(w)= Cu - -f-f, 0(w) we write K + iK' for u, it becomes h{K + iK') = C{K + iK') - f^f^^' hiK + iK') = C{K + iK') + ^- Z K To compute li{K + iK') we put w = K + iK' in sn m. If Zi is the corresponding value of z, we have Zi = — . Further, since h 1 1 -'I Vd - Z2)(l - Fz2) Jo.l tV -/■ t/0.1 \/(l - Z2) (1 - /e2z2) Jo.l V(l - Z2)(l - fc2z2) fc2z2dz V(l - z2)(l -A;2z2)' ELLIPTIC INTEGRALS OF THE SECOND KIND. 291 we have iJ'=l2iK+iK')-J, or l2{K+iK')=J + iJ'; and consequently J + iJ'=C{K+iK') + ^. Eliminating C from this formula and the formula CK = J, it is seen that J'K - K'J = -• 2 We note that K-J=f ^1 -^^^ ^ dz = E (Legendre); «/0 VI — z2 and making the transformation t = -Vl - k'^w^' k it is seen that It follows that KE' + K'E-KK'=^~, which is the celebrated formula of Legendre (Fonct. Ellipt., I, p. 60). Remark. — The characteristic properties of /2(m) are expressed through the formulas l2(u + 2K)= I2{u)+2J, Uiu + 2iK') = hiu) + 2iJ' . These formulas follow at once, when we note that Q{u + 2iK') = -e '^ 0(m). Change w to m + 2 X and w + 2 iK' respectively in the equation and use the relation ^ji _ j^/ ^ ^. Art. 250. We note that k^sn^udu ■= _; / - ©(w) or e'(M) i^-.t' \ K) Jo 0(m) 292 THEOKY OF ELLIPTIC FUNCTIONS. With Jacobi (Werke, I, p. 189) we define the zeto-function by the relation Z(w) = ^1 - |] M - rk^sn^u du, which is Jacobi's elliptic integral of the second kind. It follows * also that e(w)= 0(0)6-^"^'"''", where 6(0)=!/^^ (Art. 341). The ©-function may thus be considered as originating from the function Z(m) [see Cayley, Elliptic Functions, p. 143]. From the formula Z(w)= / ldn^u — ---\du w E we have dn^u = -r: + Z'(m) and consequently Z'(0)= 1 — — • A. ii- It follows at once that fc2sn2M= Z'(0)- Z'(m), and k^cn^u = fc2 - Z' (0) + Z' (m) ; Z' {K) = Z' (0) - k^. It is further seen, since &{u) that 7(,, 4. j^.&^u±jQ _ e/(M) ^^^ + ^^-e(u + K)-0;(^- As 0i(m) is an even function, its derivative is odd, so that Z{K) = 0. Akt. 251. With Jacobi (Fund. Nova, § 56; Werke, I, p. 214) we shall derive other properties of the Z-function and at the same time we may note the connection with the ©-function. We emphasize the following results because the properties of the ©-function are again derived inde- pendently and at the same time we have an d priori insight into the Weierstrassian functions. In Art. 220 we made the imaginary substitution sin.^ = itant, |^ = ^i^, Fi)=^ iFi.lr,k'). It follows at once that A.^ d^ = ia+lcH^n^f)df ^ iM±J^d^ ^ ^ A{t,k') cos2i/r ^ * Jacobi, Werke, I, pp. 198, 224, 226, 231. or ELLIPTIC INTEGRALS OF THE SECOND KIND. 293 This expression, when integrated, becomes r (1) E{)=i{ta.n^Mylr,k')+Fif,k')- E(f,k')}. It follows that (2) ^^^^^^^^^^ = Ftan./rA(^,fc')-{^E(-f,'fc') + (E-f)i^(V^,A;')}- From the formula (Art. 249) FEik') + F{k')E - FFik') = ^ we have at once FE{ir, k') + {E- F)F{y}r, k') = ^ [F{k')E{y}r, k') - E{k')F{^, k')] 2 F{k') Equation (2) becomes through this substitution (3) FE{)- EF( = am iu, yjr = am {u, k'), F{)-EFi<^) ^ p ^ " F{k')E{^, k')- E{k')F{ylr,k') _ ^ ^,.. F{k') and consequently from (3) we have (4) iZ {iu, k)=-tn {u, k') dn (u, k') + ^^^ + Z {u, k') . Multiplying (4) by du and integrating, this equation becomes riZ{iu, k)du = log cniu, k') + -^^ + r"z(w, k')du. Jo 4il/l Jo Further, since £ziu)du^l0,^J^y it follows that (5) f^-^^^^^^^-^'^ljl} i^i-^^^.20^). 294 THEORY OF ELLIPTIC FUNCTIONS. Formulas (4) and (5) reduce the functions Z(w) and @{iu) to real argu- ments. If in (5) we change u into u + 2 K', that formula becomes — --e cn{u,k)^j^^~ e ^^^^ In this formula change iu to u and we have rr(K' - iu) (6) 0(w + 2iK')=- e ^ e(u) (of. Art. 202). Again write u + K' for u in (5) and note that cn{u + K',k')=-k f^'^'^'\ an{u, k) e(w + K', fc') = ^^^^^^e(w, fc'). Vk It follows that ®(iy'+^K') ^_i^'^^Vksniu,k') ®}'''^'^ 6(0) ^e{0,k') jr(2 u + K') — e *^ ^ktn{u,k')^^. 9(0) Write iu for u in this formula and it is seen that ir{K' -2iu) _ (7) Q{u + iK') = ie *^ Vk snu@{u), which is a verification of formulas (V), Art. 202, and (VIII), Art. 217. By taking the logarithmic derivatives of (6) and (7), we have (8) Z(M + 2iX')=-^+Z(w), K (9) Z(m + iK') =-^ + cotnudnu + Z(m). 2 K. Write M = in formulas (6), (7), (8), (9) and we have 0(2 iK') = - e ^ 6(0), &iiK') = (of. Art. 203), Z(2iK')=-'§, ZiiK')=oo. K ELLIPTIC INTEGRALS OF THE SECOND KIND. Art. 252. In Art. 227 we saw that 1 log© (11^] = const. - l^^^lJi _ 5icoslM 2 ^ V ^ / 1 - 9^ 2 (1 - g4) 295 From the relation it follows that (1) We also have Z(m) = ) _ q^ cos 6 u _ q* cos 8 u 3 (1 - g6) 4 (1 - 58) @iu) _ ■ mTzu m=ixi g^sin 2 7r -o K z«=.f 2- [Jacobi, Werke, I, p. 187.] 5sin^*-2g4sin2|« + 3g9sin^ ^ -^ 1 — 2gcos — + 25*cos— — — 2g" cos— ;:7-+ ■ K K K To be noted is the equality of the right-hand sides of (1) and (2). We further note that k^K^ , 2 Ku JK ■sn'^ K F q cos 2 u , 2 g^ cos 4 u , 3 g^ cos 6 u , 1 27r2 "■" ff 2 Aet. 253. Thomae * introduced the notation Zoo(w)=f logei(u), du du Zio(w)=;flogHi(w), du Zn(M)=|-logH(M). Differentiate logarithmically Vk'^lM^dnu, @{u) and we have r, , x r? / n A^sn m en w ZooW- Zoi(m) = - Similarly we have Zn(w)-Zoi(M) Zio(w)- Zoi(m)=- dnw [Jacobi, Werke, I, p. 188, formula (6).] cnudnu snu snudnu en u ♦ Thomae, Functionen einer complexen Veranderlichen, pp. 123 et seq.; Sammlung von Formeln, etc., p. 15. 296 THEORY OF ELLIPTIC FUNCTIONS. Art. 254. The derivatives of the Z-functions are one-valued doubly periodic functions; for differentiating J &(u) Jr>u J k^sn^u du = —u K it is seen that -^log 0(m) = -i - k^sn^u = ^ - fc2z2. du^ K K Further, since it follows that -^log Gi(m) = ^ - k^snHu + K) = -^ - A;2 ^ du^ K K dn^u K 1 - A2z2 Similar results may be derived for H(m) and Hi(m). The functions 6(m), 0i(m), etc., when for u is written the integral of the first kind m(z, s), are functions of z, s, but not one-valued, since m(z, s)'is not one-valued in z, s. But from the formulas just written it is seen that the second logarithmic derivatives of these functions are rational, and consequently one-valued in z alone {i.e., the s does not appear). This is fundamental in the derivation of the Weierstrassian theory, which we shall consider in the next Chapter. EXAMPLES 1. Show that E(k,'^=E= j dri'iu, k) du, E'= C^ dn^{u,k')du. Jo 2, Through the definitions of the zeta-functions of Art. 253 derive independently the formulas given in Chapter X for©j(M), Hj(w) and H(u). 3. Prove that iZg^iiu, k) = Z^(u, k') + ^^ and *Zoi(m, k) = Zig(M, V) + ,=00 (-g)"* sin 2KK' nu 2KK' rrmu 4. Prove that Z^=— V K -^ 1 — q 2 TT y ^ "' K m= 1 5sinf+25^sin2^+35»sin3^+. 1 + 2 o cos h 2 g' cos h 2 g"cos 1- • • K ^ K K Derive similar expressions for Zjj(m) and Zji(u). (Thomae, Sammlung, etc., Tp. 16.) ELLIPTIC INTEGRALS OF THE SECOND KIND. 5. Verify the results indicated in the table : 297 iK' ^»^'I- QC ^-r. oc Zx„+ '" Zu+ '" " 2 A z«, z„, z,„ 00 Zu oc 6. Show that / K los sin ^ , , . I- , 1 ;: „, 7. Prove that Roberts {Liouville's Journ. (1), Vol. 19), Wangerin (Schlomilch's Zeit., Bd. 34, p. 119). Z(iA)=i(l-fc'), Z(iiA')=ii(l +k)-j^' iK Z(iK +i iK') = Hfc + ik')~ ^. 4 A 8. Complete the table of Ex. 5 by letting u take values ^ A, K + \ iK', i iK', i A + § lA', I A + I iK', etc. CHAPTER XIV INTRODUCTION TO WEIERSTRASS'S THEORY Ahticle 255. In the previous study we have followed the historical order of the development of the elliptic functions and have made funda- mental Legendre's normal form. We may just as well use the one adopted by Weierstrass, J^ V4(t-ei){t-e2){t~e3) where Vi (t - ej) (i - eg) {t - €3) = \/S{t) (see Chapter VIII). We have taken infinity as the lower limit, because this value of t, as we shall later see, corresponds to the value w = 0. We saw, Art. 185, that this integral could be transformed by a simple substitution into the normal form of Legendre. Consequently in the derivation of the new formulas we need not always return to the consideration of the Riemann surface, but in this respect we may rely upon our former developments. Art. 256. If in the above integral we write (see Art. 195) t = pu, it follows immediately that - \^m- ~ = iP'u. du In Art. 185 we saw that the transformation of Weierstrass's normal form to that of Legendre is effected through the substitution t = €3 -\ ) where £Z ei - 63 We therefore have , 1 j?w = 63 + ■ Since snf—= J is a one-valued function, the function ^ must also be one- valued; and since sv?{u Vei- eg) has the periods 2 VIk and 2 VeiK', these are also the periods of fiw. We put (Art. 196) 2 VIk = 2 w, 2 V~EiK' = 2 io', so that the function gw has the periods 2 id and 2 w'. We further note that sn'^u being an even function, the same is true also of gw. 298 INTRODUCTION TO WEIERSTRASS'S THEORY. 299 Art. 257. As we have introduced the new function pu in the place of sn u, following Weierstrass we shall introduce new functions for the 0- functions, which new functions are, however, closely connected with the ©-functions. If in the formula of Art. 254 k'^sn^u = — — — -log0(M) K du^ we put u + iK' in the place of u, we have ^ i-:^logH(M). Since ^{vVs.^=ez^ —< it follows that |»(t> V £ ) = 63+ - — r„ e K e dv' Noting the identity or 1 J\_ d2 S^^„ .-li'^^y^" — |e3-^ it is clear that Writing vV i = u, this formula becomes \ J \u- _^ = ^|.-^(--)^H(^|| which is a one-valued function of z (see Art. 254). We thus have or, if we put where /? is a constant, then is _^ = _log,«.. The arbitrary constant /? we may so choose that in the development of au, the coefficient of the first power of u is unity. 300 THEORY OF ELLIPTIC FUNCTIONS. By Maclaurin's Theorem this development is au = (7(0)+ uo'{0)+ ■ ■ . Since H(0)= 0, we also have (v)dv + 7,, CfA)' = — (w') = )?' = - / '^{v)dv + )j. If we put (see Arts. 195 and 256) jfw = /, dv = _^ , ^w = ei, gw"=e2, ■"(^) It follows that aiv) ( £ K) Ve h/ ^ ^ a{aj) ( £ k) Vs H(K) From the formulas H(m + K)= Hi(w) and H'(m + K)=Hi'(m) we have at once H(K) = Hi(0) and H'(K:)= Hi'(0)=0. It is seen that acD \ e K/ Further, since J = K — E and m = VTK, we may write * f 61-63 ) Further, since ■VeiK'= 10', we have ,'= — = - /'e3+ 1 l]o^'+ J- 2^0 . ooj' \ £ kI Ve UiiK') ' or, since (see Art. 247) WjiK') m H(rK') 2K' we have 7Zl (2) ,.= -(,+ l|y_ or. (2') ,' = 2ai t «i-e3 * See Schwarz, he. cit., p. 34. INTRODUCTION TO WEIERSTEASS'S THEORY. 303 It follows at once from (2') and (1') that g= ,— iv + eiw), Vei- ez E'= /- (V+e3a>0. Vei- 63 From the formulas w"= w + oi' or w"= VI(K + iK') we have 1 H'(iv + %K') Further, since (Art. 247) H^(g + t/JCQ m H(i!C + iX') 2/i:' it follows that From the formulas (1), (2) and (3) it is evident that '? + '?'= i'- It is seen from the preceding article that and since 1?' = C'^ tdt B '' "> A VS{t) 2 , B A we fiorther have A '' = ¥ and ,,_I + 5 2 the congruences being taken with respect to the moduli of periodicity of the integral of the second kind. We also have the relation corresponding to Legendre's formula of Art. 249, ■qoj'- i]^ = Y We may note that ,;;(« + 2 w) = Cw + 2 Tj, ,^(w + 2w')= Cw + 2ij'; for ^ being an exien function, its integral ty is odd, and writing u =- m and - w' respectively in the two formulas just written, we establish theii existence. 304 THEORY OF ELLIPTIC FUNCTIONS. Art. 260. We have already derived the formulas and If we put u = 2 (jjv, then is au = j3e^i'^' lti{2 Kv), from which it is seen that au is an odd function, the function H being odd. It follows immediately that a{u + w) = ^e2i'"(''+»' H(2 Kv + K) The following new notation is suggested: <72M =/?2e2'"«''0i(2Kv), asu = Pie^^-^'(d{2Kv), where fii, /?2 and ^3 are constants.* It is seen that u-e3), d- 63 sn^2Kv we have t _ i _ cw^^ Kv ^ " m^2Kv so that /£i^Y ^ ^,(f _ 1) = „' ^1^LJI_^, \, ^ ^^ ~sn22Xi; or /£2!i"\ = C2(^ - 62), where C2 is a constant. \aul 306 THEORY OF ELLIPTIC FUNCTIONS. We have accordingly , (Jim V jfm — e\= dx — ^) "S/jpM — 62 = ^2-^' Vgm - 63 = ds ^2!!£, where di, d2, and ds are constants. To determine the constants we note that au may be developed in the form cm = If + 63^3 -I- 6gu5+ . ■ • , and also that aku=l + b2.ku^+ &4.fc M*+ • ■ ■ (A; = 1, 2, 3), where the b's are definite constants. We therefore have okU 1 1+62 ku^+ • • • 1 , J , J 1 , au u 1 + 03U^+ ■ ■ ■ u In the neighborhood of the point m = we also have snv = V + ezv^ + ■ ■ ■ , so that and 1 e . , / uY , Since ci — ea jfW - 63= -1 i, 9 M sn'' — - it foUows that &u - 63= — + ho+ h2U^+ • • • , j?M - 62= -J + ''0+ 63- 62+ ^2W2 + gW - 61= — + ho+ 63- 61+ ^2^2 + On the other hand we had \2 It follows that dk^= 1 or dj = ± 1, and consequently (TM INTEODUCTION TO WEIEESTEASS'S THEORY. 307 Since the quotient ^^ is a one-valued function, we may take the positive sign (see Schwarz, he. cit, p. 21). We further have Similarly it is seen that Wei V^ - 63 Ve (^au en m pu — ei _ pU - 63 (7lU\ CT3M Kau/ or and also that Ve <^3U Art. 263. It follows from the formulas that dn^iV, ei - 63 . m) = 1 - Further, since CTsM = e"''" we have 62- 63 j?M - 63 (7(m + gi') d^ d^ — log (73M = — log a{u + cd') = - p{u +0)') = - p{u - io'). Admitting the relation (see Art. 316) p(u - c,') =63+ (^i-«3)(e2-e3) , pu - 63 we have f(^ + e,u)= e,- 63- (iLZ-falfenial =(,^_ e3)dnKV^^s -u). au\ CT3W / pu — 63 Since E{u)= / dn^u du, Jo it is seen that * * See Schwarz, loc. cU., p. 52, 308 THEORY OF ELLIPTIC FUNCTIONS. Further, since (Art. 259) E = JL±^ and K = V^T^ • w, it follows from Z(m) = E(u)-w| that Z(v^7^.w)= -^L^/^ + e,u)- -^=(^ + eA Vei - 63 V <^3W / V ei - 63 \<^ / _ 1 f as'u _ Tju\ Vei- esWsw w/ The last formula may be written * Z{Vei-e3.u)= ^— [cCm + cuO-^m- i?']. EXAMPLES 1, Jacobi, Werke, I, p. 527, wrote dlog^jx) _^'ix) dx ^(x) If

) - E{4>)]. - f(x) = KE{4>) - EF{<}>). srru 3 „, , 1 , \-k^+k* J , 2-3P-3fc^+2/c» , , P(w) = 1 w^ H w* + ■ • ■ . M^ 15 189 4. If F^k"^) is the coefficient of m2»»-2 Jj^ ^j^g preceding example, show that fc"i^/^] = F(fc2) and that F(l - fc^) = ( - l)«F(fc2). 5. Prove that the function P{u) of Example 3 satisfies the relation P'iuf = 4 P(«)' - J (1 - F + ¥) P{u) -4j{l+k^){l-2 k^) (2 - fc2) ; or P'(u)^=4P(M)'-gjP(w)-ff3. (Hermite, Serret's Calcul, t. II, p. 856.) * See Enneper, EUiptische Punctionen, p. 221. show that 2. Prove that 3. Let Show that CHAPTER XV THE WEIEBSTRASSIAN FUNCTIONS pu, ^u, {u) _ 1 , P{u) . b u^ h The constant term that occurs in the power series P{u) is put on the left- hand side of the equation, and the function which we thus have was called by Weierstrass the Pe-funcHon and denoted by p{u) or more simply jpu. This function is of the form ^ = \+ * +((m)). The " star " indicates that no constant term appears on the right-hand side of the equation, since it has been put on the left-hand side, and the symbol ((w)) denotes that all the following terms are infinitesimally small when M is taken infinitesimally small and are of the first or higher orders. If the point at which the function becomes infinite is not the origin but the point v, we may transform the origin to this point and consequently have to write everywhere u in the place of m — v. 309 310 THEORY OF ELLIPTIC FUNCTIONS. We may show as follows that the constant a is zero: We had C^{U) =A+£+c + CiU + C2m2+ C3m3+ .... u^ u Consider also the function ^(- m). It is doubly periodic, having the same pair of primitive periods as has (f>{u), and consequently like ^(m) is infinite of the second order on all points congruent to the origin. It may be written i _ S(- u)=-^--+C- CiU + C2U^- • • ■ . u^ u We therefore have ^(w)-^(-m)=2- + 2ciw + • • • . It follows also that {u)— 4>{— u) is a doubly periodic function with the same pair of primitive periods as {u) and <^{— u) become infinite and therefore only on the points congruent to the origin. But, as seen from the last equation, ^(u) — ^(— v) becomes infinite at the origin only of the first order. We thus have a doubly periodic function which becomes infinite at only one point within the period-parallelogram and at this point of the first order. We have seen in Art. 101 that there does not exist such a function. It follows that a = 0; and we further conclude that (j){u)— (f){— w)= Constant, otherwise we would have a doubly periodic function which is an inte- gral transcendent contrary to Art. 83. As there appeared no constant term on the right-hand side in the development in series of the function (j)(u)— (— u), we conclude that <^(m)-<^(-m)=0, or u, since the series V is not convergent. For if we give to u the value zero, we have ^ (u — w)^ 2—-, which is not convergent (see next Article). But if we form the series l+yj 1 -J-j and impose the condition that the minuend and the subtrahend which appear in the difference under the summation sign cannot be separated, then this series is absolutely convergent (Art. 266). If we put an accent on the summation sign to indicate that the value u> = is excluded from the summati6n, we may write ^ u^'^^ \{u-w)2 w^l Art. 266. We must show that the series X 1 1 (u — w)^ is absolutely conver- gent. Let the shortest dis- tance from the origin to any point on the periph- ery of the parallelogram passing through the points 2 CO, — 2 w, 2 w', — 2 w' be di, and let ^2 be the longest distance from the origin to any point on the periphery Fig. 70. On the periphery of this parallelogram there lie 8 = 3^ — 1^ period- points. For these points we have di=\w\=d2. 312 THEORY OF ELLIPTIC FUNCTIONS. On the second parallelogram passing through the points 4 w, - 4 w, 4 ai', - 4 oJ there are 5^ - 3^ = 8 • 2 period-points, and for these we have 2 di S I w I S 2 dg. On the third parallelogram indicated in the figure there are 7^ — 5^ = 8 • 3 period-points, and for them there exists the inequality 3 di ~ \w\ = Sdz, and for the n + 1st parallelogram there are (2 n + 3)2 — (2 n + 1)^ = 8(^-1-1) period-points, and for them we have (n + l)di S I w I S (n + 1)^2- In the first parallelogram we have in the second parallelogram we have in the third parallelogram we have w^ (2 d2)2 1 (3^2)^' It follows that and consequently X' 8-1 8-2 (2d2) ■ 8-3 " (3d2)2 for the first parallelogram, for the second parallelogram, for the third parallelogram, A 1+2.+ 3_ + d^2 ^2 22 ^ 32 The series on the right is the well-known divergent harmonic series. We have further V I — I S — ^ for the first parallelogram, I w IP di 8-2 X— = — '—— for the second parallelogram, Iwl (2di)3 ^ ^ .r-* I 1 |3 and consequently 8-3 (3di)3 for the third parallelogram, /|1 P-= 8 \ 1 A \w\ di3n^ 23 33 which is absolutely convergent.* * Eisenstein, Genaue Untersuchung, etc., CreUe, Bd. 35, p. 156; Vivanti-Gutzmer, Eindeutige Analytische Funclionen, pp. 168 et seq.; Osgood, Lehrbuch der Funktionen- theoric, p. 444. THE WEIERSTRASSIAN FUNCTIONS m, Ku, 2R. It is clear that The series Sir^ [u — w)^ w^ ) is composed of a finite number of terms and has a finite value if u does not coincide with any of the values w. It is seen that this series has the character of an integral rational func- tion and is continuous for all points except u — w' which are situated within the circle with radius 2 R. We consider next the series V^_J 1-] and limit u to the interior of the circle with radius R about the origin as center. We then have (, We also have ^ 1 (w"-m)2 ^ ^ <\ and since u w' the expression may be developed in the series < 1, (w' '-«)2 W"^-1 ^t«" U"/ W'/ (w" - m)2 w"- u>"^ ( w" W 314 THEORY OF ELLIPTIC FUNCTIONS. By reducing all the terms to their absolute values we have 1 {W"- U)2 W < , ^_ _ j 2 + I + 4^ +5^+ • w ^r The expression in the braces converges towards a definite limit, 6, say. It follows that 1 1 (w"- uY w"^ <^«X7^' which we saw above is an absolutely convergent series. It follows that {u - w"Y w is a finite quantity, and since 2' 1 (u - w'Y I w'2 is a finite quantity, it is seen that 21 1 {u — wY 1 I is absolutely convergent within any finite interval that is free from period- points. The series is also seen to be uniformly convergent within (Art. 7) the same interval. We have thus shown that the function 1+V'S— J J- iY „ f (m — wY vY /w = 2/^w + 2/iV;^,j = 0, ±1, ±2, . . . ; u; = excluded 1 has only at the points m = lu (including u; = 0) the character of a rational (fractional) function; at all other points it has the character of an integral (rational) function. At the points u = w the function becomes infinite of the second order. Aet. 268. In order to show that the function corresponds completely with the function pu defined in Art. 264 we must first show that it is doubly periodic. THE WEIERSTRASSIAN FUNCTIONS pu, ^u, )2 ~ (2w)2 i "^ ^ i (m - w - 2 a>)2 ~ (w + 2w)2 ) ' Adding these two expressions and dividing by 2, we have (I) «^ = -5 + ) 7 7r^2 ~ i^r^ [ + u^ {{u — 2 (dY (2 (dY ) 2^ \{u + wY {u-w-2(xjY w2 (w + 2u)YS' * See Osgood, toe. cit., p. 444; Humbert, Cours d' Analyse, t. II, p. 194. THE WEIERSTRASSIAN FUNCTIONS m, Ku, = 2p'aj + 2 q'uj', where pq' - gp' ~ 1 {p, q, p', q' being integers). It is clear (Art. 80) that the totality of values w remains unaltered by this transformation and consequently we have p{u; oj,u)')= p(u;oJ,ai'). It is thus seen that pu remains unchanged by a transition to an equivalent pair of primitive periods. 318 THEORY OF ELLIPTIC FUNCTIONS. The Sigma-Function. Art. 272. By integrating twice the ^-function we may derive another important function. It is clear that - fpu du = - +y\'\ — — + ^ [ + Constant. J u ^ (u — w w^) or The sum of the terms on the right-hand side is not convergent, but it may be made convergent by a proper choice of the arbitrary constant. For writing /pudu^ - + y\'] 1 ^- "T ( ' u ^^ (u — w w w^) we shall show that this expression is absolutely convergent and becomes infinite of the first order only at the points u = and u = w. It is seen that 1 1 ( 1-- w w [ w \w/ \w/ 1 , 1 , u u^ \^ , u , u^ , or 1"- + ^ = 5ll + - + -5 + - • u — w w w^ w^ [ w w^ As in Art. 268, it may be shown that the series is convergent, so that the above development of — I ^u du is convergent. It is also seen that, the above series is infinite only of the first degree at the origin and its congruent points. It follows that — i pu du cannot be doubly periodic. Integrating again the above expression we have - fdu fffudu = log,. + X' Slogfl - ^)+ ^ + J 4!' where we have introduced the constant of integration under the logarithm which comes after the summation sign. We shall next show that this expression is also absolutely convergent if u does not coincide with one of the periods of pu. To do this we limit u to the interior of a circle with radius R, where R is arbitrarily large but finite. THE WEIERSTRASSIAN FUNCTIONS pu, ^u, au. 319 The quantities w we again, Art. 267, distribute into two groups, so that U' S 2 B, w >2R, — <-• \w"\ 2 We then have 2 ty2 '^ ( \ wj w 2w)M t^(V w/ w 2 ^. ( \ w/ w 2 w^) where the first summation on the right consists of a finite number of terms, and is consequently finite so long as none of the logarithmic terms which appear is infinite, that is, so long as u does not coincide with one of the quantities w'. Noting that ^\ w"l w" 2\w"l 2,\w"} ■ ■ ■ ' it is seen that which is an absolutely convergent series (Art. 268). It follows that — I du I pudu is absolutely convergent for all values of u other than m = and u = w. . Since the logarithmic function is many-valued, the above integral func- tion is many-valued. To avoid this difficulty we no longer consider this function but the one-valued function -fdufpu rfu XT' U 1 w\ „S + 2 ^" This sigma-function is therefore expressed as a product of an infinite number of factors. As shown in a following Article this product is abso- lutely convergent if the two factors that occur under the product sign are not separated. The agreement of this function with the function defined in Art. 257 follows in the sequel. The function au is one-valued and becomes zero at the origin and at the points congruent to the origin. The accent on the product sign denotes that the factor which corresponds to u) = is excluded. The sign o is chosen on account of the similarity of this function with the sine-function. 320 THEORY OF ELLIPTIC FUNCTIONS. It is seen at once that

-,-T~7i.)- where A and A' are quantities such that while pi and ju' take all values ± 1, ± 3, ± 5, ■ • ; and on page 287 he formed the products '^'-1aTYa)'^'-JaTTi) a^')= ±2, ±4, ±6, • • • , // = ± 1, ± 3, ± 5, On page 288 Eisensteia says that the quotient of any two such products gives rise to the doubly periodic functions and he closes the article with the remark: " Die hier angestellte Untersuchung ist iibrigens so elementar Natur, doss sie sich wohl eignen mochte, den Anfanger in die Theorie der elliptischen Functionen einzufiihren." In Crelle's Journal, Bd. 30, p. 184, Jacobi called attention to the fact that Eisenstein had formed defective ©-functions owing to the fact that the above products are not absolutely convergent. Jacobi at the end of this article claims that the "exact formulas" are given (by Jacobi) in Crelle's Journal, Bd. 4, p. 382; Werke, Bd. I, p. 297 (see also Werke, Bd. I, p. 372). Cayley (Elliptic Functions, p. 101) remarks that such products as the above "in the absence of further definition as to the limits are wholly meaningless; " but Cayley, loc. cit, pp. 301-303, fixed these limits (see also Cayley, Can^. and Dvhlin Math. J own.. Vol. IV (1845), pp. 257-277, and Liouville's Journal, t. X (1845), pp. 385-420), and illustrated them by means of a " bounding curve." THE WEIEESTEASSIAN FUNCTIONS m, ^u, -function. Art. 274. We saw in Chapter I that the infinite product y = 00 V = 00 JJ (1 + a„) is absolutely convergent if ^ |a„| y = 1 "=1 is absolutely convergent. To prove the absolute convergence of the infinite product through which the sigma-f unction is expressed let \u\ < R, \ w' \ S 2 R, \ w" \ > 2 R as above. We omit from the infinite product all those factors which correspond to the quantities to'. Such factors being finite in number exercise no influence upon the question of convergence. 322 THEOEY OF ELLIPTIC FUNCTIONS. The factors remaining in the product are of the form ('-#■ , 1 "' ' 2 w'"- . >(-^)^^ U , 1 V? 2u)"' Since u w" < i , we may develop the logarithm in a power series and have or finally u 1 m' 1 m' " to" 2 m"2 3 w" l_u^ ( ,,3_u_3jf^ "3u/" I 4 to" 5m"^ 1 «2 Since w I 1 • ■ — < -, this expression is w" I 2 and consequently < e' < e 5'l(. 1 I _M_ 3 w" -1^^- 1 + + 1.2 + It is thus seen that the quantities in the sigma-function corresponding to a„ above are such that a„ < < '^S + 1. 1 + 1.1 + 2! 8 3! 82 u w" ^ + 16 + 16-2+ -p l^'l 77T; 16 or finallv It follows that 1.161 u VI 1^161 l^vl 1 which we saw above was absolutely convergent. To the S |a„| we must add the quantities |a„| which correspond to the quantities w'; but the convergence is unchanged by the addition of these terms. It foUows that the -product through which the sigma-function is expressed is absolutely con- vergent. Since an absolutely convergent infinite product is only zero when at least one of its factors becomes zero, it is seen that au vanishes only at the points m = and u = w and at these points au is zero of the first order. THE WEIERSTEASSIAN FUNCTIONS m, ^u, ^|. the star denoting that of every pair of values w and —w only one value is to be taken. It follows that au --IT If u is chosen smaller than any of the values w, we may write au = u n * . 2 w< 3 io« ' n*)l 1 M* 1 U^ ) 9. ini .-? ,„6 2 W* 3 W' and consequently au ^* w^ or au = u ^ w w * Cf. Daniels, Amer. Joum. Math., Vol. 6, p. 178. 324 THEORY OF ELLIPTIC FUNCTIONS. We may write o? o c V 1 w 22. 5. 7 V' -1 = ^3, where, as will be evident from the sequel, the quantities 32, ffs are the invariants introduced in Art. 184. It is also evident that 32 and g^ remain unaltered when we pass from one pair of equivalent primitive periods to another pair. It is seen that au = u + * -^ w* 22 li'— • • • , 2* • 3 . 5 23 . 3 ■ 5 • 7 the star indicating that the term with m^ is wanting. The function au is an integral function that is regular in the whole plane and may be expressed through a series that is everywhere convergent (Art. 13). The fw-FuNCTiON. Art. 277. From the formula just written it follows that g2 -.,.5 .93 7 . . .1 log au = log \ u -— u — --= u- — ■ ■ ■ ( ^ * 2* • 3 . 5 23 • 3 • 5 • 7 ) log M + log j 1 - .'72 .^4 g3 24 . 3 • 5 23 • 3 . 5 • 7 = logM 22 y4 g3 6_ .... ^ 2* • 3 . 5 23 • 3 • 5 . 7 It is evident from the consideration of the product through which au is defined that this series is convergent within a circle with the origin as center and a radius that passes through the nearest period-point. If this expression is differentiated with respect to u, it follows that — = — h * ^^ — w** ^^ u°— ■ • • . au u 22 . 3 • 5 22 . 5 • 7 The quotient 2-SHl jg often denoted by tu (Art. 258, see Halphen, Fond, au Elliptiques, t. I, Chap. V). Differentiating this expression again and multiplying by —1, we have rf2 , d { a'(u) ) 1 , , C9 o Oi ^ ^U = -_l0g..=--j-l-i5=-+* + ^^«2+_^,4+... M"* 10 7 The series through which pu, p'u and (^u are expressed are convergent within a circle which has the origin as center and which does not contain any period-point. THE WEIERSTRASSIAN FUNCTIONS PU, Ku, Art. 256, we have ■ du or , r ds J V 4 s3 - gfgs - Qa agreeing with the results of Chapters VIII and XIV. No confusion can arise from the fact that here we have written s for the variable t before used. The double sign is accounted for by means of the Riemann surface of Art. 143. Since s = oo for w = 0, we may write this integral in the form ds u = 2 si y/l - -22- - -23 J, 2s'V 4s2 4s3/ 4s2 4s3 * See for example, Humbert, loc. cit., p. 204. 326 THEORY OF ELLIPTIC FUNCTIONS. If we consider values of s lying in the neighborhood of infinity so that 1 — S2 S3_ y Q^ ^g jjia^y expand the integrand in a power series and then integrate term by term. We thus have u 1 Vs follows that «-.©■ All the coefficients of this power series are clearly functions of §2 and 33 with rational numerical coefficients. When this series is reverted, it is seen that - may in the neighborhood s* of the origin be expanded in powers of u; and it is also evident that s = pu may be expanded in the neighborhood of the origin in a power- series whose coefficients are integral functions of ^2 and ga with rational numerical coefficients. The functions ru = -^^ and loge au have the a{u) same properties, and by passing from the logarithm to the exponential function, it is found that the same is also true of the function au, so that the development of au in the neighborhood of the origin is such that all the coefficients are integral functions of 92 and ga with rational numerical coefficients. The sigma-function is therefore a function of u, g2, gs- A method of determining the coefficients of au by means of a partial differ- ential equation is found in Art. 336. Art. 280. It follows from the equation above that "-H'+iM^((?))l uH 20 s2 28 s3 \\s*jj ) Hence as an approximation (up to terms of the order u^) we have 1 2 s If then on the right-hand side of the last equation we write — for s, we have , 2 " u^ 20 28 Writing gnt = _ + * + C2u'^ + c^u* + c^u^ + . . . + c^^2>-2^. . . . ^ jt follows that "" C2=^92 and c,= i^g,. We shall express the other constants C4, C5, . . . through these two quan- tities. THE WEIERSTEASSIAN FUNCTIONS m, ^U, au. 327 From the relation (S?'m)2= 4jp3„_ ^2Sm- ff3 we have through dififerentiation 2 ^'u ^'u = 12 f^u p'u — g2&'u, or, if we give to u such values that p'u ^ 0, f?"w = 6 j?2w _ .^ . (Eisenstein, Crelle, Bd. 35, p. 195.) Multiplying through by u* we have (A) u*p"u = 6 u^p^u - i g2U*. From the equation S>U = -^ + * + C2U^+ C3U*+ ■ ■ ■ + CxU^^-2 + • • • it follows that (P'm = - -^ + * + 2 C2M + 4 C3m3+ • • • + (2 >< - 2)c;w2^-3+ .... S'"u = -^+* + 2c2+3-4c3m2+ . . .+(2;- 2)(2/i-3)Qu2^-4+ • . . , or mV'w = 6 + * + 2 C2m4 + 3 • 4 C3U6+ • . • + (2 -^ - 2)(2 /i - 3)ciM2i + • • ■ . We also have V?^ = 1 + * + C2M*+ C3M6+ • - • + c^w2i + . . . ^ or W^jj^ = . . . -I- CjM^^ -f Cj_im2^-2 -^- . . . _f- Cj-^U^'-Z" + . . . + 1; and consequently v=X-2 „_2 where we have written down only the terms that contain u^*-. Writing these values in the equation (A) above and equating the coefficients of w^*, we have * v = l-2 12c,+ 6 2 C„C;_„ = (2>l-3)(2>i-2)C;, 3 (2A + l)(yi-3) This is a recursion formula by means of which each of the coefficients Ci in the development of jfnt may be expressed through coefficients with smaller indices. * Cf. Schwarz, Formeln und Lehrsdtze, etc., p. 11; the Berlin lectures of Prof. Schwarz have been freely used in the preparation of this Chapter. 328 THEORY OF ELLIPTIC EUNCTIONS We have, for example, 3 2 or, since C2 = 5*5 92, it follows that 1 .2- and similarly 3 g2g3 24-5 -7 • 11' cs = '^'"2^71^ It + 2.3.53/ c,= ^[£23 , etc. ^ 25.3.52.7.11 We may therefore write »'«--2+*+22.5''+22.7 +2*.3.52'^ + ^M = i + * 22— m3 _ J3 ,,5 gli^ 7_ ^ It 22.3.5 22.5-7 2*. 3. 52. 7 cm = w + * — —^ — „ " u' — ■ • • . 24.3.5 23.3.5.7 Art. 281. We saw in Art. 268 that «^" == ^ + X' ) T U 2 ( ■ If we make the condition that | m | < w, we may write w ~ u w w^ u i2 i/'^ ?/t^ + *■ This equation differentiated with respect to v. becomes It follows at once that We note that all terms in which w appears with an odd exponent vanish, since a value — w belongs to every value + w. THE WEIERSTEASSIAN FUNCTIOXS pu, ^u, au. 329 If then we write n— 1 = 2^-2, or n = 2^-l, and compare the above expression with ^ = i- + * + c2m2 + . . . + c;w2»-2 + . . . it is seen that ,_i, i c,= (2^-l)V'4-. It follows from the results of the preceding Article that 5^'— r- may be ^"f vr'- integrally expressed in terms of g^ and ^3. This is a very remarkable fact (cf. Halphen, Fonct. Ellip., t. I, p. 366). In Art. 272 we saw that

We have thus shown that the function gjt) L 2w J 2w corresponds in its initial terms with the development of pu, so that it differs from gw only in quantities which become infinitesimally small of the first order when u becomes indefinitely small. 332 THEOEY OF ELLIPTIC FUNCTIONS. Art. 283. We had The quantities w may be distributed into two groups. The first group contains all values w for which /t' = 0, so that w = 2 (i(x>. The second group contains those w's for which i^ S 0, so that w = 2 ^ w + 2 /i'w'. If then we let w! become infinite, the values w of the second group become infinite, and we have j?(M, w, oo) = — + X' ] 7 7, Ti ~ 7^ \2 ( ■ u^ ^ ( (m — 2 /iw)^ (2 pico)^ ) It is seen from Art. 22 that this expression is none other than the function {t - 1)2 If then the period 2 cu' becomes infinite, the function pu is represented by git) {t - 1)2 UJ 1 1 . 9 UTT 3 sm^ — 2w Art. 284. We shall next write (cf. Eisenstein, loc. cit., p. 216) F{t) = - ^ ^ a>2 (< - 1)2 and we shall seek to express * gm through t even when the second period Jrtu 2 w' is finite. Fif) being a rational function of 1 = 6" remains un- changed when u is increased by m + 2w; but when u is increased by 2a»' , , o ,\ art' 2 w'ffi then e ^ = e " t. Weierstrass used the letter h to denote the quan- , / tity e " , which Jacobi denoted by q. In Art. 86 we wrote — = a + i3, where /? > 0. From this it is seen that and consequently r ^ i ^ g-^si Noting Art. 81, it is evident that we may always choose a pair of primitive periods so that I fe I <^ 1 Since t becomes hH when u is increased by 2 w', it follows that when u becomes u + 2w' p^^^ ^e^^^^^^ ^^^2^^^ FQiH) becomes F{hH), FQiH) becomes FQiH), * See also Halphen, FoTict. Ellip., t. I, Chap. XIII. THE WEIEESTRASSIAN FUNCTIONS m, ^U, au. 333 If we consider the infinite series (SO F{t)+ F{hH)+ F{hH)+ . . . +F{h^H)+ • ■ ■ , then, if u is increased by 2 w', each term becomes the following term. Hence the series + F{h2t)+ F{h*t)+ ■ ■ ■ + F{h^H)+ ■ ■ ■ F{t) + F{h-^t)+ F{h-H) + . . . + F{h-^H)+ . . . is a doubly periodic function having the two periods 2 w, 2 w'. At the point u = and all its congruent points this function becomes infinite of the second order; for then t equals unity or some even power of h. Art. 285. We shall next show that this series is absolutely conver- gent for all points except the origin and the points congruent to it. We limit m to a region in which \ u\ < R, where R may be arbitrarily large, but finite. The quantity t has everywhere within this region the nature of an integral function and is different from zero. Further, since lint it is seen that \t\ = el', so that 1 1 1 becomes a maximum with fi, that is, with R|— J- If we put u = (1)', then is \ "* / R (=)=-,,. If 'M is the greatest value that R(— j can take for values of u within the region in question and m the smallest, it is always possible to find an integer no, say, such that — no^TT < m and M I -\h^\. We therefore diminish the denominator of the terms in question if instead of (1 — h^'-tY we write {\ — \h^ \Y, and consequently we increase .the value of the term FiJfiH). The numerators of the terms which have been separated from the first no terms are h^^t, h^'^+^t, A2»o+4<, • • • , which is a geometrical series whose common ratio is less than unity. It follows that the series (S') is absolutely convergent for the region in question. It follows also (see Osgood's Lehrbuch der Funktionentheorie, pp. 72, 259) that this series is uniformly convergent and represents an analytic function. The terms Fit)+ F{hH)+ F{hH)+ ■ ■ ■ , which also belong to the series (S') but which were not taken into con- sideration above, do not affect the question of convergence, since they constitute a finite number of finite terms. We shall next establish the convergence of the series (S") Fit)+ F{h-2t)+ F{h-H)+ . • ■ . We may write F{t)=- \ , ^2(1 - fe2i-l)2' By separating a finite number of these terms from the series (S") it may be shown as above that the remaining terms are less than the corre- sponding terms of a decreasing geometrical series. THE WEIERSTRASSIAN FUNCTIONS mi, ^u, au. 335 It follows that the series + F{h2t)+ F{hH)+ ■ • • (S) F{t) + F{h-H)+ F(h-H)+ ■ ■ ■ is absolutely and uniformly convergent in any interval that is free from the points u = 0, u = w. This series therefore represents a one-valued doubly periodic function of u which for all finite values of u has the character of an integral or (fractional) rational function. At the points u = and the congruent points this func- tion becomes infinite of the second order. Art. 286. We note that F(0)= F(oo)= 0. It is also seen that the series (S) has the same periods and becomes infinite of the same order at the same points as the function pu. Two doubly periodic functions which in the finite portion of the plane have everywhere the character of an integral or (fractional) rational function and which become infinite of the same order at the same points can differ from each other only by a constant (Art. 83). Hence the above series can differ from pu only by a constant, which constant it will appear later is — •2- w Further, put z^ for t, retaining the notation of Weierstrass, as no confu- sion can arise between the z used here and the z formerly employed. It follows,* since z = e^", that 71 = 00 71 = 00 ^ CO OJ^liz- Z-l)2 "^^-f^ (1 - /l2"2-2)2 "^^f/ (1 - /l2«s2)2 where h = e "" = q. In order to determine the constant rj, it follows, when we expand z = e^ and z-^ = e~ ^'■', that z-z-i= — ]1 + «U~H" ■ ■ ■(' and consequently {Z-Z-1)2 ^2y2\ 3.4 ^2 \ We note that (jj^\z-z-^) u^ 3-4 ^2 * See Schwarz, Formeln und Lehrsdtze, etc., p. 10. 336 THEORY OF ELLIPTIC FUNCTIONS. If we write this value in the above expression for gm, we have It follows that * The above expression for ^w is not unique, since the period 2w may be chosen in an indefinitely large number of ways. Art. 287. Since the series derived in the last Article is uniformly convergent, we may integrate term by term. If in this integration we make a suitable choice of the constants, we again have a convergent series. Multiplying the series by — Au, it follows through integration that ^U = £l (m) = i + * + ((m2)) a u X""" S 2h^^z^ _ 2/t2" |-| „=i n - h^^z^ 1 -/i2"ij' where the choice of constants has been such that the constant terms occurring in the expressions under the summation signs, when expanded in ascending powers of u, are zero, this being already the case on the left-hand side of the equation. The above formula simplified may be written f ^w = — (m) = 2- M 2a)\z-z-^ ^^l -h^^z-^ ^^1 - h^"z'- If with Eisenstein {loc. cit., p. 215) we note that 2h2"z-2 _ 2fe2«z-2 _ z/i-" + z-ife» * Schwarz, toe. «'<., p. 8. t Schwarz, loc. cit., p. 10. THE WEIERSTRASSIAN FUNCTIONS m, Ku, a J J VS where mt = s and du = — ■ The coBstant of integration on the right-hand side is so chosen that for sufficiently large values of s the series on the right-hand side is (cf. Art. 279) J VS L 24 s2 40 s3 J ujri Art. 288. If u is increased by 2 0. \uji) Art. 289. Following a method given by Forsyth {Theory of Functions, p. 257) we offer another method of proving the formula last written. Consider the period-paraUelogram with vertices 0, 2 a), 2 w', 2w" = By shding this parallelogram parallel with itself, it may be caused to take Mo+au)'=u3 / xto+ am'^iu a position such that for all points on its boundary and within the interior (except the point u = 0) the function Cu has the character of an integral ui «^+2ii)^ui function, being of the form Fig- 71. ^ w 60'' 140'' It follows that j l^udu = 2 m, where the integration has been taken over a small circle about m = 0. Since this integral is the same as that taken over the parallelogram U0U1U2U3, we have 2m= I \udu+ i l^udu+ j t^udu + j (^udu; Juo t'«i •-'"2 'J^a 27zi= f\u - t:iu + 2co')]du + jlZiu + 26j)- l^uldu = f"'- 2 -q'du + I ""2 7) dw = - 4 rj'a> + 4 -qd)' . Art. 290. If we multiply by du the expression ^{u + 2w)=-(w)+2ij, a a we have through integration log a{u + 2 ai) = log cm + 2 1JM + c, or a{u + 2 w) = aue^i^e". or 340 THEORY OF ELLIPTIC FUNCTIONS. If the value — w is given to u, it is seen that We consequently have a{u + 2co) = -e2i("+'">(7(M). If — u is written for u in this formula, we have ct(m - 2 w) = - e-2»(«-<")crw. Combining these formulas into one formula, we may write (A) ct(m ± 2 w) = - e±2»(«±<") (j(m). In a similar manner it may be shown that (B) a{u ±2u)')=- e±2V(«±<"')(7(M). Further, if 2 5J = 2 pw + 2 qo)', where p and q are positive or negative integers (including zero), it is seen that a{u + 25)= a{u + 2pa> + 2 qui') = Ce^Pi^e^ii'^^au. Writing 2pTi + 2qTj'=2^, it follows that a{u + 2 oi) = Ce2'«CTM. To determine the constant C, write u = — w + v, where d is a very small quantity. It follows that a{aJ + v)=— Ce-25S+25«£,(S} _ ^y If we develop by Taylor's Theorem, it is seen that (C) , a{a> + v}= o{w) + va'(a>)+ ■ ■ =- Ce-2'"+25''(T(5 - v). Two cases are possible: (1) either | a{a>) | > 0, or (2) I c7(S) I = 0. In the first case we have by writing -y = 0, a{aJ)= - Ce-2;s^(2i). It follows that C = - e25s and consequently f7(M + 2 5) = — e2'("+")J„=o or log It follows that * and u +(fa5)) :7^+((w^)) 2(jj = C. u=0 ^2" „ 1 _ h^n where w = 2 ojv. . ./ ■ Writing — = t, it is seen that a> 1 — /t^"g-^ ^ sin[(-t> - Tri)7r] ^-i_ sin[(Tn — 'v)Tt] ^_i 1 _ /j,2n fe-"— A" sinriTTT with a similar formula for 1 — ft''" It follows that t (2) ^ = ,,,^'2J±^i^^JlSm[(J^^_^,JMe-v^ j-|-sin [(nr + .k] ^, "^ ' TT „ sm WTTT „ sm nrn or n = oo /l- -n h^ 2i 1 - - h^ "g2 au = e^'°™' ^^ — sin ffv 3T -pr /^ _ _sirAr^y -*- , V sm27:nr/ n = \ ^ ' * Compare this function with Eisenstein's x-function, loc. ciX., p. 216. t Schwarz, loc. cit., p. 8. Formulas (2) and (3) are precisely the same as those derived by Jacobi for H(«) [Werke, I, pp. 141-142]. 342 THEORY OF ELLIPTIC FUNCTIONS. The formula (2) may be written ,Ox 2„™,' 2 W . -1-1-1-2 A.2WC0S 2 VTT + h*"* (3) au = c2'"" — sm tw Ji t:, ,,„,o Since 2 oj may be chosen in an infinite number of ways, it is seen that au may be expressed in an indefinite number of ways in the form of a simply infinite product. Through logarithmic differentiation of formula (3) it follows that ^^^ Cu = — cot wr + 2 TO H V — • ^ 2a; ' (^ „tl 1 - 2/i2»cos2wr +/i4" Noting that ^ = 1 + M + u^+ ■ ■ ■ + u^+ • ■ ■ ( I M I < 1), 1 - u it is evident, if m = r(cos ^ + t sin 6), that 2 r sin 1 — 2 ?■ cos 6 + r^ = V 2 r"" sin m^. an identity which is true for complex as well as for real values of r. If we put r = /j2", we have 7n = CC 71= CO r'l 1 -/i2 "cos (9 4-/1*" ^, '^, n=co n= 1 and consequently C" = r— cot wr + -t_ H >, -— sm 2 niw. 2 w CO a; -^ 1 — /l2»t If we differentiate with regard to u, we have . n=oo (A) ^„ = -^ cosec2 vn~^-^V T^^cos 2 ni;;:. 4a;^ CO oj^ ^ 1 — h^" 71=1 The right-hand side of this equation is while the expansion of cosec^ t is cosec2 i = 1 + i H 1—^2 .^ 8 — ^4 I . . . i2 3 3 • 20 27 . 28 By equating like powers of u on either side of (A), we have * f-y,, = JL + 2o"y -i^!^, (^]%,- 1 7 "v n5/t2n Uy ^' 12 ^ ^t'l 1 - /^2" U/ ^' 216 3 „% 1 - A2"- * Harkness and Morley, Theory of Functions, p. 321; Halphen, Fonct. EUipt., t. I, Chap. 13 THE WEIERSTRASSIAN FUNCTIONS 9U, ^U, au. 343 Art. 292. Homogeneity. — Write the functions au, l^u, pu in the forms a'l = ct(m; w, w') = a{u; 32, gs), ^u = C(w;w,w')= C(u; 92,93), 8fm = p{u; 0), o') = piu; 92, 93). It follows at once from the infinite product through which the function au is defined (Art. 272) that a{Xu; Xo), hij')= Xaiu) u),w'), where X is any quantity real or imaginary. We also have a'{Xu; koj, k(i)')= a'{u;aj,(D'), and consequently Hiiku; koj, k(jj')= ic(w; "J,w'), &{Xu; ku), ;.w')= -^^{u; to,oj'). In the formulas 32=22.3.5y'i- 33=22.5.7V'-1-, when tx) and w' are replaced by Aw and/^(";92,fif3), The above formulas are particularly useful when in Volume II we make a distinction between the real and imaginary values of the argument. Art. 293. Degeneracy. — When w' = rxi, v/ei saw in Art. 283 that ^" ^ feJ 1 ■ n -KU 3 sin'' — 2w We further have 02 = 22. 3.5 V' i — , 03= 22.5- 7 V' — ^- From Chapter I we have ■^ m4 32 .5' -^ ot6 33.5.7' 344 THEOEY OF ELLIPTIC FUNCTIONS. It follows that and consequently A = 92^- 27g^^= 0. The discriminant being zero, the roots of the polynomial ■4s3- g2S - gz= = 4(s - ei)(s - e2)(s - 63) are equal. Further, since 61+62+63=0 and ei>e2>e3, the quantity e\ must be positive and 63 negative. Two cases are possible: eitherr 62 coincides with 63, or €2 coincides with ei- In the ^rsf case: 62= 63= — i 61; 9^2= 361^, 93= ei^, 33 > 0; ' ' 2^2 V2W 2gi' 61-63 2 We also have ^^ = e:H = jLcot^+i/^Yw, 2 7?W = — , l/>ro\! fi\2^) 2(0 ■ TZU au= e"^-""' — sin — - • n 2 (11 In the second case: 63 = ei= — ^63; 33 < 0, e"— e~" 32= 3 632, ff3= 638; fc = 1, S71W= — -^, K =00, w= 00. g,„ = 3^ _ ? 23/£!+£l!y, where 1; = iu ./^, ru = —^ u ■ 1 where t = -, ^ 2^2 2w'e-'''-6"« 2w' ' 12 ^ w 6^-6-^ .-i(^r XK 2 When the roots of the polynomial 4s3- gigs - 93= are equal, it may be shown directly that the values of s = g>u derived from the integral n\ C" ds (1) w = I — Js ^4*3- g^S- 33 agree with the results above. THE WEIERSTRASSIAN FUNCTIONS pu, ?m, tru. 345 When both periods are infinite, then g2 = = g^ and ei = = 62 = 63. The integral (1) becomes ds 1 > or s = — r u = \ —~~' or s = — = $>w, 1 ru = - , cm = M. u EXAMPLES 1. By making a>' = 00 in the formula "2 to2 w l\ W derive the results of Arts. 283 el seq. (Halphen, loc. cU., Chap. 13). 2. If /= r ^ ■ t/o ^4 s' — ^jS show that 3. If F(0 is any rational function of t = e " , such that F(0) = = ^"(00), show that n=QO n = oo 4>{u) = i^(0 + ^ Fith^"") + 2) F{th-^'') + C n = l is a one-valued doubly period function of u. CHAPTER XVI THE ADDITION-THEOREMS Article 294. It is the purport of this treatise to consider as far as possible the ultimate meaning of the functions which have been intro- duced. The simplest funct'onal elements have been found in the Jacobi Theta-functions which are made the foundation of the theory. It is therefore natural first to develop the addition -theorems from this stand- point. We have seen in Art. 90 that there exists a linear homogeneous equation with coefficients that are independent of the variable among any n + 1 intermediary functions (w) of the nth order, which have the same periods. We may next make an application of this theorem for the case n = 2. If in Art. 87 we write a = 2K, 6 = 2 iK', n = 2, it follows that (4>(w + 2K)= ^{u), (I) \ 2m, .„,. [^{u + 2iK')= e ^ 4>(m). Among any three functions of the second order which satisfy these func- tional equations there must exist a linear homogeneous equation with coefficients that are independent of the variable.* Three such functions are 02 (m), H2(m) and 9(14 - v)@{u + v), where v is an arbitrary parameter. It follows that C0(m + v)@{u - v)+ Cie2(it)+ C2H2(«)= 0, where the C's, are quantities independent of u. The C's may, however, be functions of v. None of these quantities can be zero; if, for example, C = 0, we would have H(m) which is not true. ^ ^ Constant, * See Hermite in Serret's Calcul, t. II, p. 797; and Koenigsberger, EUiptische Functionen, p. 368. 346 THE ADDITION-THEOREMS. 347 Writing we have &(U + V)@{U-V) = f{v) ©2(m) + giv) H2(W). If we coiisider/(v)G2(M)+g,(v)H2(M) as a function of v, say ^(u), we have It follows that Sp-(i; + 2K)= ■*•(?;) and from which it is seen that ^{v) satisfies the functional equations (I). If we write v + 2 K m the equation (II) 0(m + v)@{u - v)== f{v)®^{u) + g{v)B?{u), we have @{u + v)@{u~ v) = f{v + 2K)Q^(u)+g{v + 2K)H2(u); and consequently through subtraction it follows that U{v + 2K)-f{v)]®^{u)+{g{v + 2K)- g{v)]B.Hu)= 0. As this relation is true for all values of u, we must have f(v + 2K) = f(v), g{v + 2K)= giv). On, the other hand, if in the equation (II) we write v + 2iK' for v, we have in a similar manner /(v + 2iK')=e-^ f{v), g{v + 2iK')= e^ g{v). It follows that f{v) and g{v) satisfy the functional equations (I) that were satisfied by02(w) and H2(m). We thus have the following relations: f{v)=aYP{v)+pQ^(v), g{v)= rH2(i;)+ d^^iv), where a, /?, y, d are constants. When these relations are written in the equation above, we have (1) @{u + v)@{u - v)=[aH2(«)+/?G2(2;)]e2(M)+[^H2(v)4-Je2(y)]H2(u). 348 THEOEY OF ELLIPTIC PUNCTIONS. To determine the constants a, /9, ;-, d, write v = 0. We then have 02(m)= ^02(0) 02(m)+ ^02(0) H2(w), °^ 02(u)[l - /?02(O)]= 502(0) H2(w), a relation which can exist only if 1-^02(O)=O and 502(0)= 0. We thus have 3 = -— !— and d = 0. ^ 02(0) If next we write m = in the above equation, we have a = 0. To deter- mine y, we write the values of a, /3, d just found, in (1), then write u = V + iK' and note that @{iK') = 0. It follows that r = - 02(0) These values of a, /?, 7-, d when written in the equation (1) give us the formula 02(0) 0(m + v)@{u - '!;)=02(v)02(m)- H2(t;)H2(M), which is fundamental in the Jacobi theory (see Jacobi, Werke, I, p. 227. formula 20). Aet. 295. We introduced in Art. 208 the following notation: @(2Ku)=Mu), «?o(0)=t?o, B.{2 Ku)=Mu), ??i(0)=??i, Hi(2Xw)=!?2(w), ^92(0)=t?2, We also saw in Art. 215 that and in Art. 217 that 1 01 (0) MO) ^3 Vk Hi(0) ^2(0) ^2' Vk' - ®(o> »?o(0) ^0. 01(0) ^3(0) M sn2 Ku : _^3^i{u) en 2 Ku ■■ ^0 «?2(m) ^2 ^o(w)' dn2Ku ^3 t?o(M) The addition formula above for the function may be written (1) ^o^oiu + v) ^oiu - v) = ^oHu) ^oHv) - «?i2(m) ^i^{v), THE ADDITION-THEOREMS. 349 if in the original formula we write 2 Ku for u and 2 Kv instead of v. In a similar manner we may derive (2) &2^3l^l{u + V)MU - V)= ■»i{u)MuW2iv)Mv) + MU)MU)MV)-»0{V), (3) !9o«?2^2(m + v)Mu - v)= Mu)Mu)M^)Mv)- ^i{u)Mu)Mv)Mv), (4) t?ot?3t?3(w + v)§oiu - v)= do{u)^3iu)^o{v)&3{v)- Mu)^iiu)Mv)^i{v). All four of the above formulas were also derived in the table (C) of Art. 211. Abt. 296. If we divide equation (2) above by (1) we have T?2»?3 ^iju + v)^n{u - v) ^i{u)Mu)Mv)Mv) + Mu)^3(u)^i(v)^o{v) that is, ^sn2Ku'^cn2Kv'^dn2Kv + '^cn2Ku'^dn2Ku^ sn2Kv ^3 ^0 ^0 ^0 ^0 t?3 l-'^sn^2Ku'^sn^2Kv or rn z^f 1 \i sn2 Ku en 2 Kv dn 2 Kv + cn2 Ku dn 2 Ku sn 2 Kv sn\2 K(u+v)\ = ■ . ^ ^ ^ '^ 1 -k^m^2Kusn^2Kv If we divide the equation (3) by (1) we have ro I'/ , M cn2 Kucn2 Kv — sn2 Kusn2 Kvdn2Kudn2Kv cn[2K(u + v)]= l-k^^^2Kusn^2Kv ' and similarly when (4) is divided by (1) we have J rnz^/ , M dn2Kudn2Kv — k^sn^2 Kusn2Kvcn2Kucn2 Kv dn[2K{u + v)]= l-k^sn^2Kusn^2Kv If we write u and v for 2 Ku and 2 Kv, we have snucnvdnv + cnudnusnv sn(u + v) = k^srfiu snH Further, since ^ — snu = cnudnu, du it follows that dsnv , dsnu sn u 1 snv — - — , , s dv du sn{u + v) = 1 — kHn^u sn^v We have thus shown that sn{u + v) is a rationed function of snu, snv and the first derivatives of these functions (see Art. 158). Remark. — If for brevity in the formula above we put snu = s, snv = s'; cnu = c, cnv = c'; dnu = d, dnv = d', it becomes , , ^ sc'd'+ s'cd ^(" + ^)=l_fc2,2,.2- 350 THEORY OF ELLIPTIC FUNCTIONS. We further have 2/ , N 1 2/ _L ^ (1 -FsV2)2- (sc'd' + s'cd)^ cnHu + v)=l- sri^iu + v)= i (i _ fcW2)2 ' (cc'- ss'dd')^ (1 -/C2s2s'2)2' so that cc' —ss'dd' cn(u + v) = ± 1 - /c2s2s'2 Writing v = 0, and consequently s' = and c' = 1 in this formula it fol- lows that en u = ± c, so that the positive sign must be taken. We may derive the formula for dn{u + v) in a similar manner. Art. 297. Addition-theorem for the elliptic integrals of the second kind. — From the formula 02(0) e(M + v) e{u -v)= e2(w) 02(1;) - rhu) n^iv) we have at once 02(O)0(. + .)0(.-.) ^ _ ^,^^,^^2,. 02(u)©2(t;) This formula differentiated logarithmically with respect to u and v respec- tively becomes @'(u + v) . @'{u — v) _ ,-y Q'(m) _ _ 2 fc2sw ucnudnu sn^v @{u + v) @{u — v) 0(m) 1 — k^sn^u sn^v 0'(m + v) _ ) = - e2 ,(«+<„) (j,^^ j7(y + 2w') = - e^ ■''("+'"') cm, it follows that v=—-^+ ((m2)), It ds further seen that the function is doubly periodic and becomes infinite in the same manner and at the same points as ^u — pv. Other developments are a{v + u)= av + uc/v + —■ o"v + • • • , <7(— V -\- u)= — av + ua'v — c/'v + • • • , , , -,_ a^v + \ava"v - {a'v)^]u^+ iju*)) 'P^^'''~ 'v)p'u and 2 {pu - pv)^ (F) piu±v) = ^^lj-\^^^^^^l 2, dv\_ gnt — pv } * See Schwarz, he. cit., p. 13. t Schwarz, loc. cit., p. 13. t Daniels, Amer. Joum. Math., Vol. VI, p. 268. THE ADDITION-THEOREMS. 353 It follows, since (^'-u)2 = 4 jp3„ _ g^ ^^^ _ g^ and p"u = 6p^u - h g2, that the formula (E) becomes 2(jpU - fJ)D)2 while formula (F) may be written (F') S>(m ± v) = gw + (g'^ - g^) (6 j?^»- i 92) + 4 (p^t) - g^y^) - ffa T »'u t?'v 2{u — pv ) that the partial differential quotient which appears on the right-hand side must be infinite for the value u = — v. To observe the nature of this infinity, write u =~ V + h. It iollows that f?'^ - ^'^ = 2 p'v - hp"v + ■ • ■ = 2 ^ ^ ^(^^)) pu — ^ hp'v — i h^j>"v + • • • h and that , >, a S yu-p'v l 2 ^ _ . . _ du\ pu — pv S h"^ Noting these results we may obtain another formula for p{u ± v) aa follows: The function / I ^ 1 ( p'u - p'v \^ 4\pu-pv/ is one-valued and doubly periodic. It is also finite at the point u = — v and -the congruent points. We further note that this function remains finite at the point u = +v. At the point w = the function becomes infinite as -• If then we add to the above expression the function ipu, we have a doubly periodic function which remains finite everywhere 354 THEOEY OF ELLIPTIC FUNCTIONS. in the finite portion of the plane and is therefore (see Art. 83) a constant. It is easily shown that this constant is — s>v. We may consequently write 4 L f?w - gw J Art. 301. If in the formula just written we put u + v ioT u and —v for V, we have pu + p(u + v)+s)V= - 21^ — -^ — SI- . • 4L p{u + v)- pv } It follows that fp'u — p'v _ , p'(u + v)+ fp'v gwt - gw p{u + v)- ^ If both sides of this equation are developed in powers of u, it is seen that the negative sign must be used. In determinant form this formula may be written 1, ffU, ^'u 1, pr, p'v = 0. 1, ff{u-^v), —p'iu + v) By differentiating with regard to v the formula . , ^ Id a'u T %>'v 2 3m ^u — gw we have &{u ±v)=-- — — - S^ S^ 2 du av pu — pv i'ii^2 ifp'v) t?"v {pv - pu)^ 2{pv - pu) \pu±] "-^ > (ipu- P' P"u {pu - pv)^ 2{pu p'v. Remark. — If in the formula (F') of Art. 299 we write tt» in the place of V and observe that 4 p^oj - g2poj - g3= 4: ipo) - e,){pui - e2)(g>w - 63) = 0, it is seen that f 1 \ 6 p^w — A 02 p{u ±oj)- ei= -^ ^, 2{pu - ei) piu ± w) - ei = - — '- ^-^• 2 pu — e\ From the relation ,, „ „ p u = 6s^u - ig2 it follows that p"cu = 6 ei^ — ig2, and consequently that p{u±co)-e^=l~^-^. 2 pu — ei THE ADDITION-THEOREMS. 355 Further, since (8?'w)2= 4(jpM - e{){^u - 62) {pu - 63), and therefore also S>"u = 2[(g>u - ei)(jpw - e2) + (»nt - ei)(gJM - e3) + ipu - e2)(g«^-e3)J, it follows that p"(o = 2(ei- 62) (ei- 63;. We consequently have f < \ (^1— 62) (ei— 63) ^M - ei and similarly ^^^ ^ ^„^ _ ^^ _ (eg- eQfe- 63) ^ pu - 62 / 1 )\ (es~ Ci)(e3— 62) pu - 63 Art. 302. The reciprocal of formula (G), Art. 300, is 1 2(g?M - fpv)^ g>{u ±v) 2(ffupv - \ gz) (pu + pv) - 93 T p'u p'v ^ 2ifpu - pv)^ { (2 pufpv - i (72) ifpu + fpv) - ffs ± p'u sp'v \ _ [2iipupv - ig2)is>u + pv)- gsf- p''^up'H Noting that (j?'m)2= 4 p^u - g2pu - gz and {p'vY = ipH - g2pv - g^, it is seen that [2ipupv - i gi) {pu + pv)- g^p- [4 pSu - gzpu - gs] [4 p^v - gzpv - gfg] = 4{pu - pv)^[p^upH+ ig2pupv + -i'ff?2^+ gaipu + pv)]; and consequently 1 ^ 2ipupv - ig2)(^ + pv)- g3± p'usp'v p(u ± v) 2{pupv + i g2V+ 2 gaipu + pv) If we write u = v, we have (2^)=(£!?L+l22)!dl2^3g«. Ap^u- g2pu - gs It also follows that -3p*u + ig2P'u + Sg3fim + ^g2' "^ 4p^u- g2S>u - ffs From the formula just written we have pu- - — log p'ujdu. 356 THEORY OF ELLIPTIC FUNCTIONS. Inteerating we have „/ ^ i w a a 2 du a 2 pu Developing both sides of this expression in ascending powers of u, it is seen that the constant C = 0. We therefore have — (2m)= 2— (m) + - '^, — a a 2 pu This formula multipUed by 2 du and integrated becomes log (t(2 m) = 4 log au + log (ff'ii + log c, so tnat /r* \ / \-i / (t(2 ii)= c( Co Co 16^2_ (1 -co^)(F-co^) Writing these values in equation (I') we have 1 A;2 fV+ Co(e2+ ,,2)=_ ^V(l - Co2)(/c2- Co2)?JJ, Co Co Co or [1 + ^$2,^2+ Co2(f2+ ,2)p= 4[fc2_ (1 + fc2)co2+ co4]e2rj2. Arranged in powers of — , this equation is Co (1 - fc2^2,2)2 2(1 + fc2f2^2)(^2 + ^2) _ 4(1 4. fe2)^2^2 ^ ^^ ^^^^ ^^ Co* or 1 _^ '? Vl - 7)2 \/l - fc2,;2+ ,; Vl - f2 Vl _ Ji.2^2 Co 1 - A2f2,2 which is the algebraic integral of (III')- After deriving the transcen- dental integral Euler proceeded to the addition-theorem in practically the same manner as is given in the next Article. Art. 308. Professor Darboux * proceeded to the above algebraic integral as follows: He assumed that 41 - ./^?7K „„ „ _ A.^^(f) df \/Z(0 or u= p du Jo.i v'Z(f) where Z(0 (1 - f2)(l - A;2f2), and required that $ be determined as a function of u. He further introduced an auxiliary variable v, such that (i) J^ = VZ(^ or V = r'-^^"'^5_ (ijs dv ^-^W J„_j ^2(,)- ^"^ * Darboux, Am. de I'Ecole Norm., IV, p. 85 (1867). THE ADDITION-THEOREMS. 363 We therefore have from (III') du + dv = 0, or u + V = c, V = —u + c, where c is a constant. It follows that i„ so that f and ij are functions of u, both being integrals of the equation We next form ^ ^ ^ ^^^ ^ ^ j ^ du^ 2v/Z(0 <^w 2 = - (1 + fc2)f + 2 /fc2f3 and .2„ |-i = _(l +A:2)ij + 2A;V. We have immediately ''^(£r~ ^'O'^ '^~ ^^^ fc2f2,2(f2_ ,2). Through division it follows that rf^_^rf^ or du^ du^ ^ 2 k^$Tj - ,2/^V_f2/d5\2~ -l+fc2f2,2' dw dw This expression, when integrated, becomes d$ __ ^dju du du _ ^ 1 - A2f2,2 - ' where C is a constant. Further, since ^ = V(l -f2)(l _fc2f2), ^ = V(l -,2)(l_fc2,2)^ rfw aw we have at once ^- V V(l - f2)(l - fc2^2) + f V(l - ,;2)(1 _ fc2,;2) (J^) 1 - A;2f2,2 which is the algebraic integral of (III')- 364 THEOEY OF ELLIPTIC FUNCTIONS. The addition-theorem may be derived as follows: If in the relation U + V = c we write for u and v their values from (i) and (ii), we have (N) py^) d? _^ A.V^) dri _ ^ Ai \/Z(f) Jo.i VZ(7}) This is also an integral of (III') but in transcendental form. Suppose next that 7) becomes tjq for the values $ = 0, \/Z(f) = 1. It follows from (M) that '?o= C, and from (N) that Ai \/Z(j)) If we write $ = snu, 7) = snv, v^l — ^= cnu, Vl ~ Tj^= cnv, then from (P) we have )jo = sn c. But since c = m + w and also ijo = C, the equation (M) may be written snvcnudnu + snucnvd nv 1 — k^sn^u sn^v Write D = 1 — k^sn^u sn^v = sn{u + v). and note, since 1 = sn^u + cn^u, that D = cn^u + sn^u dv?v = Di, say, and D = cn^v + srfiv drfiu = D2; and also that It follows that cn^{u + v)= 1 — sn^{u + v) ^ D^— {snucnvdnv + snv cnu dnu)2 or (cf. Art. 296) cniu + v)= '^^ ^^^ ~ ^^^ ^^^ '•^^^ ^'"■''^ . 1 — k'^srfiu sv?v THE ADDITION-THEOREMS. 365 Similarly, if we note that D = dri^u + k^sn^u cn^v = D3 and that D = dn^v + k^snH cn^u = D4, we may derive from drfi{u + v) = 1 — k'^sn'^{u + v) the formula J , , s dnudnv ~ k^snucnu snv cnv dn{u + t)) = -— — 1 — k'^sn^u sw'v Art. 309. A direct process for finding the algebraic integral was given by Lagrange as follows: For brevity write X = a + bx + cx^+ dx^ + ex*, Y = a + hy + cy^ + dy^ + e/y*. The differential equation to be integrated is of the form (I) _rf^ + ^ = 0. vx Vy Considering x and y as functions of u, we have as in Art. 308 ^=VX and ^ = -VY. du du It follows * that 2^=& + 2cx + 3da;2-h4 ex^, du^ 2^ = & + 2cy + 3dj/2+4e2/3. du^ If next we introduce two new variables defined by J) = X + y and q = x — y, we have df£=^ + ^^ = 6 + cp + f d(p2 +q^)+i c(p3+ 3 pq2), du^ dv? dv? ^.^ = X-Y = bq + cpq + i qd{3 p^+ q^) + i epq{p2+ q^). du du It is seen at once that du^ du du or 2.rf!Erf£_2/d£Y^ = (d + 26p)^. 52 du"^ du g3 \du) du du * See Cayley, he. cit., p. 337. 366 THEOEY OF ELLIPTIC FUNCTIONS. The integral of this expression is where C is the constant of integration. Writing for q, ^' p their values, we see that the general integral of (I) is du (II) / VZ -Vf V^ C + dix + y)+ e{x + y)2. \ x~ y / Cayley {Elliptic Functions, p. 338) gives several interesting forms of this algebraic integral and of the addition-theorem. Art. 310. The formula (II) above suggests at once a form for the inte- gral of the corresponding differential equation in the Weierstrassian theory. Write (a, b, c,d,e) = {- gs, - g2, 0, 4, 0) and consider the integral (I') '^^ + ^' = 0. Vi s3 - g2S - 93 Va t^ - g2t - gs The algebraic integral is seen at once to be (IF) | ^^^4.^- 92S - 9s-^^t'- 92t - g^ J _ 4 (, + ^) = c. Writing du= '^^ =^. dv '^^ V4 63- g2S - gz V4. t^- gzt - gz the transcendental integral is (T) u + V = c, where s = pu, t = pv = p{c — w)= p{u — c). When these values are substituted in the algebraic integral, it becomes Ipu- g>{c - m) J From (A) it follows (Art. 300) that C = 4 p{c), and from (T) we have ^(„ + ,)=ir£:^i^^?_^„_^, 4 L jpw - gw J or g>{u + v)= 2 (pufpv - i g2) (pu + pv) - ff3 - p'up'v _ 2 {pu - pv)'^ THE ADDITIOK-THEOEEMS 367 Art. 311. Equate to zero the determinant* 1, JpM, g>'u 1, fpv, p'v = p'w(pv - pu) + p'v{gm - pw) + p'u{pw - pv)= 0. 1, pw, p'w Squaring we have {p'w)2{pu - pvy~ {p'v{s>u - pw)- p'uipv ~ pw)]^= 0, or ipu - gw)2[4j?% _ g^^ ~ ga]- [p'vpu - p'upv - pw{p'v - p'u)f= 0, an equation which is satisfied for w = u, v and also (Art. 301) for — ty = u + v; that is for pw = pu, pv, p{u + v). The equation (pu — pv)^ {s — pu\\s — pv}{s — p{u + v)] = has the same zeros, viz., s = pu, pv, p{u + v); and since the coefficients of ipw)^ and s^ are the same in both equations, the two equations, since they can differ only by a multiplicative constant (Art. 83), must have all their coefficients the same. The coefficients of (pw)^ and s^ give immediately -{p'u- p'vY= A{pu - pv)^{-pu - pv - piu + v)\, or p{u + v)=\ y^^-tp'^ C- g>u - ^. 'H pu - pv ) Art- 312. In Art. 193 we derived the formulas li{+\)=-ZK or sn{-ZK)=\, K4) = ^K-iK' or sn{-ZK-iK')=\, k u{oo,ao) = — iK' or sn{—iK')=cc, u{0,1) =0 or sn (0)=0, u(-l)=-K or sn{-K) = -l, 1 =(4) = K - iK' or sni~ K - iK') = - u{'x>,-oo) = -2K-iK' or sn{- 2 K - iK') = x, u{0,-l)=~2K or m{-2K)=0. By means of these formulas and the addition-theorems we may verify the formulas IX-XV of Chapter XI. * See Daniels, Am. Journ. of Math., Vol. VI, p. 269. 368 THEORY OF ELLIPTIC FUNCTIONS. Art. 313. Duplication. — In the addition-theorems above if we write V = M, we deduce the following formulas : sn2u= ^^^<'^^dnu ^ ^^^^^ Z) = 1 - k^sn^u, cn^u — sn^u dn^u cn2u dn2u = D dn^u — k^sn^u crfiu D Writing snu = s, en u = c, dn u = d, we have l-cn2u = -^ = __, 9r2 1 +cn2u = ^, 1 - - dn2u = 2 k^s^c^ D 1 +dn2u = ^- Art. 314. Dimidiation. — From the above formulas we deduce at once , 1— en 2m ,1 1 — cnu sn^'u = or sn-' -u = — > 1 + dn2u 2 1 + dnu , dn2u + cn2u , , k'^ + dn2u + k^cn 2 u cn^u = ■ ) dn^u = ; l+an2w 1+ an 2w Changing uto^u we have formulas* for the determination of sn{^K), sn{\ iK'), sn (§ K), etc.; for example K J l - cnK J 1 iK en ilC _ / dnJK'+cnJK' _ / - iki - il ^ j \ + k 2 V 1 + dn ^"K' y I - ikI V fc ' [Table of Formulas, No. XVII.] In a similar manner we have 'g-4-)=v/H^(m^> 5^, - . , -.. . - .-+k+ ik'\__ Vk + ik' Vk where we have written k = Vl+k'Vl-k', l=Vk + ik' Vk - ik', and noted that Vk - ik' + V/c + ik' = Vl - fc' + \/l + fc', * See Table of Formulas, No. XVII. THE ADDITION-THEOREMS. 369 Art. 315. To determine the value of the g>-function for the quarter- periods, we note that i-c^] ei-g3 . r^^ 2 (ei -e3)^cri(\/ei -es-iddnjVej -63 • m) ^ sn^{y/ei — e3-u) sn^(\/ei — e3'u) V ei— 63 V ei - 62 We have for example (i)=''"3(|)=''"<^'-'''"^''' sn = ei+ V{ei- 63) (ei- 62); en = - 2i{ei- e3)ikk'ik - ik') ^'(^-^\=-2{ei- e^W 62-63- 2i{e2- esWe^- 62, a formula which is incorrectly derived and given by Halphen, Fonct. EllipL, t. I, p. 54. Art. 316. We also find that or sn^{v + K) (ei- e3)dnH _ (ei- e-i){pu - 62) cn^v pu — ei jptf — 6] It follows, if we write 5(/) = 4(< - ei)(< - e2)(< - 63), that p{u + (o)= ei+- ^^-> i pu — ei and similarly , ^ ,„ ,1 ^'(62) £>(W + W") =62+7 ^^' 4 ^W - 62 4 jpM - 63 370 THEOEY OF ELLIPTIC FUNCTIONS. If further we let P),(u) = pu — ex (A = 1, 2, 3), we may derive at once the formulas * P2{u + w) = (ei— 62) Psiu + w) = (ei- 63) Pi(m + w") = (62-61) Pzju) Piiu)' P2M Pi(u)' P2{U)' P2{U + W") - 1 S'{e2) 4 Pziu)' P3(m+w") = (62- 63) Pi(m + w') = (63-61) P2{U)' P2iu) Psiu)' P2{u + co') = {es~e2)p^> P3(. + .')4f^- 4 Psiu) 1. Show that 2. Show that 3. Prove that EXAMPLES sn(u + v) = cn'^u — cn^v snv cnu dnu — snu cnv dnv 1 2 k'^cnu cnv cn{u + v) cn{u — v) dn'u dn?v — k'^ cn(u + v) cn{u — v) _ 2 snu cnu dnv sn{u + v) sn{u — v) sn'u — sn^v A -D 4.v,i r , \ I ^13,l+ k snu snv 4. Prove that sn{u + v) — sn(u — v)= log fc 3m 1 — fc snu snv 5. Prove that tan am u + V _ snu dnv + snv dnu 2 cnu + cnv 6. Verify the formulas given in the Table of Formulas, No. LXIII. 7. Derive the addition-theorem for the g)-function from that of the sn-function. 8. Show that , 2 ©2(0) R{v - u) B.{v + u) sn^v - sn'u = — —^ — ^-— — '-—i ■ k e'(M) %\v) * See also Art. 327. THE ADDITION-THEOREMS. 371 9. If am a = a, am 6 = /?, am(a + b)= a, show that (1) sin a sin /? A (T + cos a = cos a cos /?, (2) cos ^ cos a + Aa sin ^ sin ct = cos a, (3) Act + F sin a sin /9 cos o- = Aa A/3. 10. Show that the algebraic integral of dx Vx dy where X = a^* + 4 a,i' + 6 02^:^ + 4 OjX + a^, I^ = a^y* + 4 Oii/^ + 6 ay + 4 Ojt/ + a^, may be expressed in the form of the s)rmmetric determinant X + y 0, 1, x+ y 2 ' xy, 1, da. xy «1, a^+c, 2c a^- 2 c, a,, = 0, (Lagrange.) where c is an arbitrary constant (Richelot, Crelle, Bd. 44, p. 277; Stieltjes, Bull, des Sciences Math., t. XII, pp. 222-227). CHAPTER XVII THE SIGMA-FUNCTIONS Article 317. In Chapter XIV we derived the function au from a certain theta-function and we then proceeded to the other sigma-functions. In Chapter XV the function au was defined through an infinite product which followed from the definition of the g)-function and the character- istic properties of the sigma-function were thus established. We shall now prescribe these characteristic properties of the sigma- functions and derive therefrom directly the functions themselves.* In Art. 298 it was shown that a(u + v) a(u — v) We write u = 5, where 2 5 = 2 pw + 2 qu>'. The quantities p and q are integers, and here one of them at least is taken odd, so that ib is different from a period. Since 5 is a half period, we may write fpa> = ex {^ = 1, 2, 3). The formula above becomes a(u+ a>) a(u — co) pu- ei = ^^ o 2- In .'^rt. 290 we derived the formula (m)= ffMe^+fi"+C"=, and we shall so determine the constants A, B, C that {u) has the period 2 o^. This function {u + 2a;)= 4>{u) we have, since , „ . , , , , ct(m + 2 W) = - e2'>("+"') CTM, the formula _ g2,fu+<,j)+A+B(M+2iu) + C(u+2iu)2^jj ^ g,4+Su+Cu2j^ It follows that 2T)iu+w)+2Bw + 4: CvM + 4 Cw2= (2 fc + l)7ri, where k is an integer; and consequently 2rj + iCoj = 0, or C=-l2; 2 w and 2 Tyw + 2Bw + 4 Cw2= {2k + 1)ot; or, if A: = 0, ^ ^ m^^ 2a)' The remaining constant A being arbitrary, may be taken equal to zero. THE SIGMA-FUNCTIONS. 375 We then have (j){u)= aue ^^ 2«. _ We further write u = 2u)v and put Since <^(w + 2w)= ^(u), it follows that 4>[2a){v + 1)] = i'(«+""') au, and consequently 2i)'(u+<,/)-^(u+2(u + 2w')=-e 2. 2. ^^. or 2ii'(u+{u). Writing — = t, we have it) cj){2ajv + 2w') = - e-'^'^^'-fiv), or Since it=+oo we therefore have ■^ (>, g2*m(i'+T) = g-m'(2?)+l) '^ (J g2k!!iv 3 A:= —00 fc=— 00 A;=— GO 376 THEOEY OF ELLIPTIC FUNCTIONS. If the coefficients of e^*^" on either side of this equation are equated, we have _ q^^ c^_^^2{x-i)ri.^ which is a transcendental equation of differences. In the formula q^ ^ Cx-i e2(/i-i)Mt+m change ^ to ^ + 1 and write log Cx = Sa- We then have Bx+\ = Bx+2hzir+m. Suppose that ;^(A)= Ao+ AiX + A^i)? and consequently that l{l + 1)- ;i:(A)= 4i+ 2 AgA + ^2= Ihzix + m. It follows that ^2= Tzix and A\= izi{\ — r). As Ao remains arbitrary, we choose it equal to zero. These values sub- stituted in 7 (A) give j;(A)=ot(1 - t)A + OT7/12. Let us further write Bx — xi^) = ^>-- We then have Ex+i= Bx+i- zi^ + 1)= Bx + 2XmT + m-[x{X)+2h:iz + 7n] = Bx-xi^)=Ex. We note that Ex+i= Ex= • • • = So- Further, since Bx = xi^) + ^o, or logCx= xW+ Eo, we have Cx = e^"e^''". Writing e^»= C, it follows that Cx = Ce"'!"''^"^"'^", and consequently A:= —00 Further, since jl-u^+iLu it is seen that fc=+QO (TM = a{2u)v)= e^'"^-'^'"C ^ ga(,l-x)k+^zk^g2k7dv k——oo _ g2iiaiv-(J X^ giri(l-r)fc+rtt*;2g(2fc-l)mt. k=—oo _: g2iOT2(^ '^ gmiig'^'l 2 / g(2J:-l)!rt»g 4 fc = — « MI fc = QO . / 2k— \ \- = g2iiM)2g 4 (-r '^(_i)ig""\ 2 /g(2*;-l)ri»_ k= —00 THE SIGMA-FUNCTIONS. 377 jrtT Letting — e * C = c and substituting fc + 1 for fc, we have J;=+QO . / 2<:+l\ ' fc = — M We note that au is an odd function and we shall assume that the constant c is such that the coefficient in the first term in the expansion of au is unity, that is, au = 1 • u + • • • . The sigma-function is thus completely determined. Akt. 320. If we write w = co, cu", w', we have directly from Art. 317 the formulas „^^ = e.u q(co^:J!) ^ ^_,„ aiu + u^) ^ au) aiD am aw ,,„ ajw' - u) _ g_,,„ a{u + m') _ aau = e aw uLu where w" = w + w' and rj" = rj + i)'. The argument 2 6li(?; + i) corresponds to the argument w + w. We may consequently write *:= -00 SO that 25ot;2+2ii<.w+-5- /. --\ — (2fc + l)* ,„,,,, . ^ •, t "S- ^^ = g-2,<»»e' 2jC_ V(-l)'e* e^2k+l)nv(^gr,^kg2^ or riuj id fc=+00 If we write crtt> then is *=+" aiM = /?ie2i«'' ^ gi(2fc+i)=«re(2fc+i)«v^ and similarly t.+oo 2A:friv (73U = /Jge^"^' ^(-i)*e*"""'e' J:- -00 378 THEORY OF ELLIPTIC FUNCTIONS. If with Weierstrass we write gtiti = ji and e"" = z, we have i=+m (2A+1)' k=-x k= + x CT2M = /Jae^'"™' ^ ;i«^22fc, fc=+oo J:=-a> ffl — — JT -rr Using the notation of Jacobi: h = e " = e ^ = q, and writing with him J:- +00 (2t + l)'' = 2 gi^ sin nv — 2qi sinSTtv + 2q'^ sin Snv — ■ ■ • , k= + ao (2«:+l)' «?2(f)= X 5 * ^'*^' A:= —00 = 2 gi cos KV + 2q^ cos 3 tt?; + 2 9^ cos 5 ttv + • • • , J;-+oo A:= —00 = 1+25 COS 2 TT?; + 2 g^ cos 4 ttv + 2 5^ cos 6 ttv + ■ • • , i-+oo fc= —00 = 1 — 2 g cos 2 ;r« + 2 g* cos 4 ^rv — 2 g^ cos 6 TTV + • • • , we have au = l^e^i'^'^iiv) [P = ci], Art. 32L By differentiating both sides of the formula above for au and then writing m = 0, it is seen that au du ^-^ '^I'W THE SIGMA-FUNCTIONS. 379 Further since (7i(0)= 1 = 02(0)= (73(0), it follows that MO) "^ MO) ^' ^o(O) In Art. 340 it is shown that n = oo «9l'(0)=2;r;iiJJ(l -;i2n)3. n=l When this value is substituted in the formula above for au, we have In a similar manner if we write for /Sj, ^2, fis, their values we have a:(u)= eWcos./nl + 2/^=^"cos2.. + ;.4n n = oo n-1 n = M aou = e^'"^'' TT ^ -"--^ (l-;i2»-i)2 n= 1 ^ ' Take the logarithmic derivatives of aiu, a2U, a^u and equate the coeffi- cients of u on either side of the resulting expressions. We then have 2 ,^ = - 2 e2c.2+ .2 2; ^^ ^\'.7-V' o 9 " ."v 4fe2»-i 2 ,a> = - 2 63^- ^2 X (,_^2n-l)2 ^\^l (1+/^-)^ 1 + 2;(.2n-i(.Qg2iw + h*"-- -2 (1 + fe2n-l)2 1 - 2fc2n-i(;os2wr + ;i*" -2 -12 Since ei + C2 + 63 = 0, it follows from Art. 286 that 3 f 2 „t'l (1 + /l2")2 + A (1 + /,2.-l)2 ^^ (1 _ /j2n-l)2 = 2,.=| -32 71=00 tMI oV 4?i2" 3^2 „t'i (1 - ^'")' ) We note that au is an orfd function, while aiU, a^u and asu are even func- tions. The zeros of these four functions are given in the Table of Formulas, No. XXXI. 380 THEOEY OF ELLIPTIC FUNCTIONS. Art. 322. If the formulas \(m) \au / \au/ are multiplied together, we have in virtue of the equation (j?'u)2= 4(83M - ei)(j?M - e2){s>u - 63), the formula ^,^^2y/p^^g^' To determine the sign to be used before the root, write m = and it is seen that the negative sign must be employed. We thus have (1) p'M=-2^i^^^2^^-^33f. In Art. 302 it was seen that (2) "k"--^"- It follows from (1) that u - ex u\ p'u _ xuj {pu - ej)2 c oxU a^u a„u au au au \au} or Since the equation may be written and further since we have d au _ OiM Oyii du oiu axU axU a^u — a^u +(e^— e^a^'u = a?u 1 , s a^u / d au \^_ / a^u '^fa^u^ \du axu) \axuj \axu/ \du axu) \_ \axiij J L \oxuJ J In the same way it may be shown that / d omY Ti aj^ul r , , ■. a„hi] \du a^u/ L o(m + 2a),) = ~ $m(u) ) <^h (^m ^^ for ^' <^"' ^'• f^„(w + 2t0 (see Art. 288). Hence among the six quantities above there exist the relations -i\/en- 63, Ves - ■' — i Vei— 63, V62— ei= —i Vex- es, Ves— 62= — ^ V62- 63, ve3— ef or iV'63- 62= Ve2— 63, tVes- 61= Vei— 63, ■iv'e2— 61 = ^61— 62. We hav e thus re duced t he six roo ts without any ambiguity to the three roots v^ei — 62, Vej— 63, "^62— 63, which three roots are real and posi- tive if the discriminant of 4 s^ — g2S — J3 = is positive. * Schwarz, loc. dt., Art. 21. THE SIGMA-FUNCTIONS. 385 Remark. For the sake of a greater symmetry some recent writers on this theory have written wi, W2, W3 for the quantities which at the outset with Weierstrass we denoted by w, co", o)'. When such formulas that result are compared with those given by Weierstrass, much confusion, in particular with regard to sign, arises; for example with these writers v'eg— 62= -i v'e2— 63, Ves— ei= — iVei— 63, Ve2— ei= i Vei— 62. The explanation they give to — a»2 is not entirely satisfactory, especially if these quantities are defined on the Riemann Surface with reference to K and iK'. Aet. 329. From the equations (2) above it follows that (A) ao) Vei— 63 "Vei— €2 ye2— ez Vej— €2 it follows that 4ai and 2w' are periods of az{u + 2 a) ) a^u ■■ A closer investigation shows that 4 a» and 2 w' are a -primitive pair of (T3M periods of this function ; for in the period-parallelogram with the sides 4 w and 2 w' the function a^u becomes zero only on the points m and 2 ai + w' , being zero of the first order. Hence becomes infinite of the first order OzU on these points. Since only two infinities lie within the period-parallelo- gram with the sides 4 (n and 2 oj' , and since the smallest number of infin- ities within a primitive period-parallelogram is two, it follows that 4 oi, 2 oi' form a pair of primitive periods of C3W Art. 332. It follows at once from the formulas above that a{u + (d) _ 1 <7iM az{u -f- (J) ^/e^ _ 63 <^2W This may also be seen from the formula of Art. 326 au 1 '), and coam (Vei- ez •u,k)= am [v ei - 63 (a> + 2 goi' - w), A: J. 388 THEORY OF ELLIPTIC FUNCTIONS. We may note that n/ , , o ' \ n aij- u + 0) + 2qa)',k) cos am IVei— 63(01 + 2gaj — u), k\= — ; , n , i\ ■- ^ az{— u + CO + 2q(D ,k) _ ax{u — a) — 2qio', k) <73(m — CD — 2qa)', k) Since we have -^ L =Ve,- f2- !73(m — (H) O2U r / 7l / ahi — 2qu)',k) coscoam Vei— e~-u,K\=Vei— 62 — : r" — t-tt' ■- ^ " (72(m — 2qa) ,k) and since g(M + 2 to') _ _ (TM V" = V + v'l we substitute a),at',at"= a> + o)'; ^,f,7j"=^ + ^\ it follows at once (Arts. 276, 271) that the invariants 32, 93 and the func- tions g3M, au remain unaltered. Also owing to the equation (s?'m)2= 4[g,u - pa)][pu - g)a)"][pu - pu)'] = 4[pu - ei][pu - e2][pu~e3] the collectivity of the three quantities 61,62, €3 remains unchanged and consequently also the collectivity of the three functions (^J=»'"-fi-i (/i=l,2,3), although the indices 1, 2, 3 may be permuted. * See Schwarz, loc. cit, p. 30; or Daniels, Am. Joum. Math., Vol. VII, p. 89. THE SIGMA-FUNCTIONS. 389 We therefore have a set of more general formulas if in the preceding developments we write W, UJ", to' 1, r. r in the place of u 2ai X = cu' <^, CO , (J) n, V", r! ei, €2, 63 o\, 02, oz V = U T = - 2w where X, ft, v may take in any order the values 1, 2, 3. The corresponding changes must, of course, be made in z and h. The following table contains the values of the indices X, /i, v for each of the six different cases which may arise (see also Halphen, loc. cit., 1. 1., p. 262): Residue, mod. 2 I V 9 V' 9' k /* V 1 1 1 2 3 II 1 1 1 1 3 2 III 1 1 1 2 1 3 IV 1 1 1 2 3 1 V 1 1 1 3 1 2 VI 1 1 3 2 1 Addition-Theorems for the Sigma-Functions. Art. 335. In a similar manner as was done in the case of the theta- functions (Arts. 210) we may derive theorems for the addition of the sigma-functions. These functions like the theta-functions do not have algebraic addition-theorems. If in the identical relation (^M - ^i) (s>M2 - ^3) + (m - &U2) ipus - ipui) + {pu - gms) (pui - S>U2) = we make repeated application of the formula gw- a{u + v) aju — v) 390 THEORY OF ELLIPTIC FUNCTIONS. we have (1) a(u + Ui) a{u — Ui) a{u2+ W3) a{u2— M3) + a{u + M2) <^{U — U2) ; [G] a + u>, b + a>, c + w, d — ai, THE SIGMA-FUNCTIONS. 391 we have the following relations given by Schwarz, loc. cil., § 38: [A.] [1] aaabacad + aa'ab'ac'ad' + aa" aV ac" ad" =0, [2] OjO, a}) ac ad + afi/ai>'ac'ad' + a^a" aii" ac" ad" =0, [3] a^a aja a^ ad + afiL'afi'aji'ad' + arp."a}>"a^"ad" =0, [4] ff/i a J) aj: a^ - afi'ap' aj:! afi' + (e^-e.)<7ia"CT,6"CTc"crd" =0, [5] (e^- e„)(Tia ai> axca4 + (Cy- e;) afi'a^'afi'a^' + (ei-e„)CT^"c7^"(7„c"')^h(u)^,.v(u) ^ 1 - (ev - ex) (gy - e^f\„(M)f\„(?;) 5. Show that 1 - (e„ - e,) (e„ - ep)f^ov(w)f^oi'(i') "2 1 4?'m 2 ^M — 6; (7iU CTU dw OW 2 (pu — fi^)(j)U — e„) (7pM (7„W dw C7„M . (ex-e,){e,-e,)_^^ = A £L^ = illog a;U. pu — ex du axu dv? 7. Show that (iF{4>. k) _ F(, V) E((t>, k) k sin <^eos^ ^ a/t fc fcfc'^ &'=* AW,fc) where i''(0, fc) and E{4>, k) are Legendre's integrals of the first and second kinds; and that dE(cl>, k) _ F(, k) ^ E(4>, k) ^ dk k k Write ^ = - in these equations and note the results. CHAPTER XVIII THE THETA- AND SIGMA-FUNCTIONS WHEN SPECIAL, VALUES ABE GIVEN TO THE ARGUMENT Article 337. The theta-functions were expressed in Art. 209 through the following formulas : 7n = co m^oo '?o(m)= n (1 - g^")!! (1 - 2 52m-l cos2 7rM + g4m_2)^ 171=1 m=l m=l = 0,2,3). dz and also that . , ar Writing these values in the last equation of the preceding Article and integrating we have If both sides of this equation are expanded in powers of q, it is seen that the constant C = tz, and consequently that It is also seen from the results of the preceding Article that i}i'=2 7:qiQo^. 400 THEORY OF ELLIPTIC FUNCTIONS. Art. 341. If the formula ^3 snu = -^ '(^) ^-&, '{yk) be differentiated with regard to u, we have en udnu = !?s l(2|)_!i(ik)wjL_\ ^»(^) M?k) 1 2K If in this expression we put u = 0, it follows that 1=^3^X or l = /2_^„^3^3^. It is thus seen that From the formula if also follows that t9. ^3 = S/-- T t: ^0 ^3 «^3 ^l' = 2 X \/2 Kfcfc' Vn We note * (see also Art. 345) that QiQ2Q3= 1, ^fc =V2 = (1 -252'n-lcOs2TO +g2(2m-l))(l + 2 ^2'" - 1 cOS 2 TTW + g2(2m-l))^ we have "'"^ i _ „4m t?o(2 u, 9^) = Il7Y3t-2'^o("' 9) '^3^'^' 9)' m=i ^ y z or (1) Mu, q)Mu, q) = ^^90(2 u, q^), and similarly (2) Si{u,q)Mu,q) =^^ii2 u,q2). We also have from the product of ??o(w, o(w, 9)^i(w, g)= 2gi sinTTwH (1 - g2'")2 JJ (1-23- cos2to+ q^-^); and since 1 -2g'"cos2TO + g2m= 1 _ 2 (\/5)2"-'cos2to + (Vg)*™, it follows that ^oiu, q)^du, q) = 9*n 1 _V '^^^''' ^^' further noting that m=« m-00 m = oc II (1 - 9"')= n (1 - 9"")n (1 - 9"""'>> m=l »n=l '""I we have (3) ^oiu, q)^i(u, q) = 9*^°'='i(«' ^'^^■ * Hermite, Resolution del' equation du cirujuikme degr'e. (Euvres, t. II, p. 7; and also Sur la theorie des equations modulaires, CEuvres, t. II, p. 38; see also Webber, Elliptische Functionen, pp. 147 and 327. 402 THEORY OF ELLIPTIC FUNCTIONS. If for u we write u + i'm this equation, it becomes (see Art. 208) (4) Mu, q)Mu, q) = q^ ^^u, Vq). If for q we write ge" = - g = ge""', the quantity g* ^ becomes g*e * ^' and the equations (3) and (4) become (5) ds{u,q)Mu,q)=qie « ^^i(w, e^ Vg), (6) ,9oK3)t92(w,g)= g4e~^^?92(w,e2Vg). The six formulas above are given by Jacobi {Seconde memoire sur la rotation d'un corps. Werke, 11, p. 431). In the formula m=oo (2m+l)' 79i(w)=2 2) (- 1)"*5 * sin (2 m + 1)™, m=0 the summation is taken over positive integers including zero. If we separate the even integers and the odd integers by writing m = 2 n and m = — (2n + 1), we have n = + Qo ■&i{u)= 2gi 2) 9^"' + ^" sin (4n + l);rw, 71 = — CC and similarly n= +00 ?92(w)= 2gi 5) g*"' + ^"cos(4n + l)s-u. n= —00 Since sn 2Ku = ^ ^i^ = -L ^i(")'^3(u) ^ _1^ &i{u)&2iu) Vf^oiu) Vk ■^o{u)'&3{u) V/c «9o(w)?92(w)' cn2&. = \/^ ^^ = \/^ "^^ ^^^ = v/^ ^^^ ^2^ V k §oiu) V k do{u) ^3(u) V fc ^o(m) z?3(m)' ^o(m) ?9o(w)«?2(w) «?o(w) ^i{u) it follows from the formulas above that ,„, „„ . e~^^iu,e^Vq) _ V2gi2(-l) "g'"'^"^'^ (^^+^>" v'J snzAM— — , jp — > v2v''A;??o(2w,g2) Vfc ^(-l)«g2"' cos 4 n::u TRANSCENDENTAL CONSTANTS. 403 where the summations on the right are over all integers from n = — oo to n = + 00 . The summations are taken over the same integers in the following formulas: 1 *[VA„(ii \/7,\ _ ^/f 'X9^"'"'""cos (4n + l);rw (8) cn2Ku = -i:.V/^^^i^^i^^-2l = V2V/-5*=^ ^ —' V2^ kM2u,q^) ^k ;^(-l)"V»^cos4n;ru (9) dn2Ku=^k'e^ i>2{u, \/ q) _ _ *j ^, _^^ ^ '_ . ??2(M,e'"'2Vg) VF ^(-l)V"+"cos(4n + l);rw /- /k^A,(9i/ n2-\ /- */^ 'yg8"'+*"sin (8n + 2)n-M (11) cn2Ku = V2\/'^^^^^^^^=V2\j'^qi^^ ^ —, V /c3t9i(M,\/g) ^ k<> ^g2«'+nsin(4n + l)7rw 4, *is (v P^\/n-\ , — y(-l)V"'"'"sin(4n+l);rM §i{u, Vq) •^q2n'+n sin (4 n + 1)tiu If we put u = in (8) and (9), we have yk = V 2 q^ =- ) Jacobi (Werke II, pp. 233-235) has given several different forms for these two quotients of infinite series. If we write m = in (10) and (12) and determine the resulting indeter- minate forms, we have * Vk^ =2\/2qi —^ . X(- l)"(4n + 1)52"H» X(- l)"(4n+ Dq^'^'^'' Art. 343. By equating the expressions for the theta-funetions in the form of infinite products and in the form of infinite series we may derive interesting relations connecting the quantity q. For example, in the case of !?i(m) we have after division by qi (1) sin;rM(l-52)(l-2g2cos2;rM+24)(l-g*)(l-2g4cos2;!:u + g8)- • • = sin Ttu-q^ sin 3 nu + q^ sin 5 7tu-q^^ sin 7 Tzu + q'^'^ sin 9 TZU-- • • . * See Hennite, CEuvres, t. II, p. 275. 404 THEOKY OF ELLIPTIC FUNCTIONS. If in this equation we put u = ^ and divide by j Vs, we have * (1 - g6)(l - 5i2)(l - 5I8) ■ • . = 1 - g6_ qi2^ ^30+ g42_ or writing q^= t, it follows that (2) JJ(i -«-)= ;^ (- i)-r 3m2+m X — i j^ ^ y— xj V Upon this formula depends the trisection of the elliptic functions. If further we divide equation (1) by sin nu and then put w = 0, we have [(1 - g2)(i _ qi)(i _ ^6) . . ]3= 1 _ ^q2_^. 5g6_ Tqi2+Qq20_ .... Writing q^= t in this equation, it follows that (3) n(i -'")'= X^-l^'^^m+Dt 2 . m=l m=0 If we compare the equations (2) and (3), it is seen (cf. Jacobi, Werkej I, p. 237) that (1 - g - 52+g5+g7_ gl2+ . .)3= I _Sq + 5q3-7q^+9q^^-- ■. Further in equation (1) put \/q in the place of g. We then have (1 — g)(l — g2)(i _ ^3) _ _ sin;rM(l — 2gcos2 7rM + g^) (1 - 2 g2 cos 2 ;rM + g4) (1 - 2 g^ cos 2 ;rM + g^) . . . = sin 7TU — q sin 3 ;rM + g^ sin 5 ttu — q^ sin 7 nu + ■ • • . Write in this equation m = J and observe that QzQoQi^Q2^ = ^ ; it follows that V3 ^" = 1 + g + g2 + g6 + gio + 5I5 + . . . . V3 If we compare the two expressions for do^u), we have Qo(l - 2 g cos 2 ;rM + g2) (1 - 2 g^cos 2 ;rM + g^) . . . = 1 — 2 g cos 2 TTU + 2 g^ cos 4 ttm — 2 g^ cos &nu + • • ■ . In this equation write m = and observe that QoQ 2 = goQs . Q1Q2 It follows that Q0Q3 ^ (l-g)(l-g2)(i-g3)(i_^4^. . . =l-2g+2a*-2a«+2ci6_ . . . Q1Q2 (l+g)(l+g2)(l+g3)(i+g4). . . ^5 + ^9 2g +2g . * See Euler, Introductio in analysin infinit., ^ ^23. TRANSCENDENTAL CONSTANTS. From the formulas 2q2m-i cos2u + q ,4m-2 '-^ = 2qiJ^cosuTT l + 29^"'cos2« + '^ ■ Vfc ^j\l-252'"-icos2tt + ,4m-2 405 (in 2Ku Ku ^ ^-j^, j-p 1 + 2(72"'-i cos 2m + g*"- TT ^fij 1 - 252'»-i cos2w +5*"'- ,4m-2 It follows that (1) log sn = log ^ ,- — + 2 T, ^ cos2mM, 771=1 ^ (2) log en 2^ = log (2 gi v/f cos m) + 21" 1 2^ ■m=\ ^ - cos 2 mw. .2/i:M (3) logdn^^= log Vifc"'+ 4 y -^^ 21^^^-^ TT '^, 2m-ll- o^'^-a 7n= 1 ^ From (1) and (2) we have 2Ku\ .log2K 2_^i^2'xl^ COS (4 wi — 2)m. sn- log- sin u . 2Ku u = m= 1 + g" en ■ log cosw ^bg^^ = log 25V^+ 2 X- ! + (!)„• We also have from (1), (2) and (3) the formulas (4) —log sn du n 2Ku, 2Ku „ T, „ ,-, en an 2Ku 2K Tz T. 7Z 2Ku iii — <*j - — 4 V — ^ sin 2 mu, \ -W 1 + o'" sm w -^ 1 + 0" 771= 1 ■* 2Ku_, 2Ku . sn du 77l = X ;r sm w , ^ ■«-\ (5) log en = , ^ y, du n Ti 2 Km cosm •^ l + (— o)' en »n=i ^ + 4 y :; — f-^sin 2mu, ,„. d , , 2Ku 2K (6) -— logdn = aw 7Z 7t ,2 2 Km 2 Km K'^sn en nIm-X dn _- — L_ = 8 y 2 "' ' sm(4OT-2)M. 2 Km a i_54'"-2 406 THEORY OF ELLIPTIC FUNCTIONS. If in (4) and (5) we put u for u, we have , ,2 2Ku k ■'sn . 771=00 , ,_, 2K 7C sin M - x^ (- l)'"-io'" . „ (7) = 4 > ^ — 2— sm 2 mu, ^ ^ K 2Ku, 2Ku cosu ^, 1 + q'" en an — — m-i ^ 2Ku en m = oD ^' r. 2Ku,2Ku smu ^, l+C-a)" sn an >»= i To these we add the equations of Art. 231 (9) ^^—1— = ^ + 4 V ^^-- sin (2m - 1)m, ff 2Ku sinw .^ 1 _ o-^m-i sn '" = 1 ^ (10) ^^^^j— =^- + 4 V(-ir ? ^ cos (2m -Dm; Tz 2 Km cosw -^ 1 + g'""-^ en "1=1 and the equations of Art. 228 ,,,, 2kK 2Ku , I'v" 9""^ • .o in 11) sn = 4 gi V — a-— -— sm (2 m - 1)«, 7n=x m = l 772 = 00 (12) cn = 4:qi >, , , 27>. i cos (2 m - 1), (13) '^d,'^^=i+4XT ^ Tt Jr"', 1 ii + g 2771 COS 2 mw. In Equa. (12) write m = and in Equa. (9) put m = ^ ; it follows that m=co m=oo (14) ^32(0) = 2J^ = l+4 XT-f-^=l+4 X(-l)'"-'7^ (j2m-l Similarly writing m = -in (11) and m = in (12), we have „ , __ 7n=00 771=00 (15) ,?22(0)=^i^ = 4gi V(- 1)^-1 ? - 4gi V ;j ■^ 1 _ ^2 J7J - 1 -■ ^ r^-^ ::i,i + q 2m-l If in Equa. (13) we put m for w we have 2k' K 1 = 1+4 y (- 1)™ — 2^ cos 2 mu; '^ d;j2Ku ^1 1+5: TKANSCENDENTAL CONSTANTS. 407 and substituting m = in (10) and m = in the equation just written, it is seen that If further we differentiate (8) with regard to u and then put u =-, we have 771=00 and if Equa. (7) be differentiated with regard to u, it becomes for u = (18) I^J^\' = ^".(0) = 1 + 8 X" L^JQ!!!^. Subtracting (18) from (17) we have Jacobi (Werke, I, pp. 159, et seq.) has given forty-seven such fornaulas as those above. Art. 344. In Art. 89 mention was made of the fact that many of the properties of the 6-functions had been recognized by Poisson. For example, in the 12th volume of the Journal de I'Ecole Poly technique, p. 420 (1823), he established by means of definite integrals the formula \ X l-\ + 2 6-"^ + 2 6-^"^ + 2 6"^'^ + 2e~^^''^ + ■ + 2e-'"'^ + 2e-*'"'^ + 2 6-9"^^ + 2 6-18"/^ + K' To verify this formula by means of the elliptic functions, let ^ = ^ ■ Instead of k we take the complementary modulus k'=Vl- k^, the quan- 1 K tity X becoming- = ^. Hence if in the formula X K JlK = \ +2q + 2q'^+2q^+2q^^+ • ■ • = 1 + 2 6-"^ + 2e-*'^ + 2e-^'^ + • • • , we change fc to k', we have v/^= 1 +2e ^-|-2e ^ + and consequently the formula of Poisson.* * In this connection see a remark by Abel, CrelU, Bd. 4, p. 93. 408 THEORY OF ELLIPTIC EUNCTIONS. Art. 345. In Arts. 260 and 320 we derived the relations cTM = /Je^'-^ViC-y), [u = 2a)v] <73W = Pse^^-^'Mv), where P = ^' ^x = .^' /?2=v^. y?3- ' ^l'(O) ^' t?2(0) ^' ^3(0) "^ »?o(0) Noting that ei- 63 ej- 63 we have, if we put G = (ei- 62)2(61- 63)2(62- 63)2, T2a» 2a> ^ ^ CO B^=J^-^= = -]-■, ^67^^=l/^2giQoQi2, V 2a> -Vea- 63 «?2(0) '' 2w V 2w Vei- 63 «^3(0) » 2w S3 = v/;f Tj^^= = ^ ; ^67^^= J^ QoQs'. y 2oj Vei— 62 '^o(O) >^ 2ai If follows immediately that «i= Jf;^T[''3*(0) + ^o*(0)]= T^2Qo'[Q2«+ Q3«L 3v2w/ 12 w"' 62= if;^ y[^2*(0)-^o*(0)]= r^Qo*[16gQiS- Qa^L 3 V2 o)/ 12 ct>^ 63= - ^ f^T [^2*(0) + ^3*(0)]= - :;^Qo*[16 9Qi8 + Q2«]; /2Z _ ^2(0) ^ 2fe^ + 2fei + 2fe^ + • ■ - ^ V 7t -^62- 63 Ve2- 63 V^[^/^7=^-^^7^^]= ^3(0)- T?o(0), ^ ^ Vei— 63 — vei— 62 TRANSCENDENTAL CONSTANTS. 409 We also have /2^ _ 1 +2h + 2h*+2h^+ ■ ■ ■ ^ 2(1 + 2h*+ 2/i'6+ • ■ •) . ^__^^e,~es^d^^ ,Qj^ ^e7^^ '93(0) "^'Q2^' ^_ I A/ei-e2 _^o(0) _Q3^ ^ VeT^^ ^93(0) Q2=^' Q2«-Q3^=i6gQi8. It is further seen that (—]\-&0^m+d2^{0)+'d3^m=(e3- 62)2 + (62 -ei)2 +(61-63)2 \2ajJ = 2(61+ 62+ 63)2- 6(6162+ 6363+ 6361)= Iffa, or 92= |f;f)Vo«(0) + &2HO) + ^3«(0)], and similarly r3= A/jLyt^^^CO) + t?3*(0)]['?3*(0) + !9o4(0)][t?o*(0) - ??2*(0)]= 4 616263, 27 \2 a»/ and 4 12 /I 24 G =(63- 62)2(62- 61)2(61- 63)^= ^^i'8(0) = 9^ !L_ ^. Art. 346. The formulas of the preceding Article may be written (1) e^'"-^^i{v, t) = y — -^0(7^ = -Qo^iou, (2) e2'""'??2('y, f) = V ~ '^''62- 63(7lM = 2 QoQi2g*criM, (3) e2''^''?93('y, t) = v/ -^ "Vei - 63CT2M = QoQ2^c!2U, (4) e2"^!?o('y, ■^)= V "^ "Vei- 62CT3W = Q0Q32CT3M; or, ,5, --^•'It''-'. (6) »<«-r^«"- a- 1,2, 3; -'.-''o)- 410 THEORY OF ELLIPTIC FUNCTIONS. Noting that the coefficient of u^ in au is zero, and that the coefficient of u^ in axu is — Je^, it follows by a comparison of the coefficients on the right-hand side of equations (5) and (6) that (8) 2 7?a;^-2e,a>2-l ^"^+'(0) (>l = 1, 2, 3; ^4= »?o). 2 V 1+1(0) From (7) and (8) we have at once the relation of Art. 339, »?/"(0) _ t?o"(0) »?2"(0) I ^3"(0) 2 becomes e^ , e ^ ^"i^q*, e ^ q^. TEANSCENDENTAL CONSTANTS. 411 If for example for u in the formula for au (in Art. 291) we write u = w", we have i"""" _ 15 — — """> "=« I) £0 TT 2 gi Qo^ The formulas expressing aiu, a2U, ogu through infinite products are given in Art. 321. 1. Show that EXAMPLES t9,(0, Vg)= V2^^y/2l, ^i'(0,V9)=V2fc'Vfcv/(^ (Jacobi, Werke, II, p. 431.) ^Ky 3. Through a comparison of the coefficients in Formula (6) of Art. 346 show that '9*+i 24 ^,+ 1 2,W=/|-2eA6.^-,a. (^ = 1,2,3; r9,= t?o)- 3. Show that — h. — =1(1 -h'^+h*) (e, - 63)' 3 — L = ;± (1 + fc^) (2 - fc^) (1 - 2 fc^), (e, - 63)' 27 g ^ g2^-27g3^ _ j;.,^,, (6,-^3)° 16^-63)" 4. Verify all the formulas given in tine Table of Formulas, XLI and XLII. 5. Show that {\-k^+k*Y ^ 27 g/ ^ 1 r^/(0) + »9,^(0) + d,mf k\\ - k'r 4(g,' - 27 g,') 8 t9/(0)?93«{0)«?„«(0) CHAPTER XIX ELLIPTIC INTEGRALS OF THE THIRD KIND Article 348. In Chapter VIII we saw that the elliptic integrals of the third kind in the normal forms of Legendre and of Weierstrass were dz , r dt (22-/?2)\/(l - 22){1 - Pz^) ^"^ J (t-b)\/Afi-g2t~g3 In the neighborhood of the point z = /?, if /9 is not a root of s2 = Z (z) = (l — z2)(l — k^z^)= 0, the expansion of \/(l - z2)(l -k^z^) is by Taylor's Theorem ^ A +oo(z-/?)+ai(z-/9)2+. . . V(l - z2)(l - A:2z2) where A V(l -/?2)(1 - ;c2/?2) VZ{J3) It is evident that Legendre's normal integral becomes logarithmically infinite for z = /3 in both leaves of the Riemann surface as the two quan- tities l^A\og{z-l3) and ^ J- A log (z - ^); and for z = — /? in both leaves as -^Alog(z+/?) and ^Alog(z + ^). If /? is a root of (1 - 22) (1 - A;222) = 0, say ^ = 1, then at the point /? = 1 the integral becomes algebraically infinite of the one-half order. The integral of the third kind in Weierstrass 's normal form becomes logarithmically infinite at the point t = b in both leaves of the Riemann surface as ^ log(f-6) and ^ ^=log (t - b). Vib^-g2h-g3 V4b»- gzb - gs 412 ELLIPTIC INTEGRALS OF THE THIRD KIND. 413 Art. 349. Let us next form the simplest integral of the third kind which becomes logarithmically infinite at only two points of the Riemann surface. There must be at least two such points ai and a2, say, since the sum of the residues of the integrand must be zero. We may write the integrand in the form — ' — ■ -^"-^ = /(z, s), say. (z-ai)(z-«2)VZ(z) We shall choose the points [ai, V^Z(aO], ["2, ^Z(a2)] in the wpper leaf of the Riemann surface and we must determine the constants Aq, A\, A2 so that the integral does not becom e infinite at the two corresponding points[ai, — VZCai)], [a2, — '^Z(a2)] in the lower leaf. Accordingly we must have ,Ao+ Aiai- A2v/Z(ai)= 0, ' Ao+ Aia2- A2\/Z(a2)= 0. In the neighborhood of the point z = ai we have by Taylor's Theorem Ao + A,z+A,VZ{z) ^ A,+A,a,+A^^ VZia^^^^^^_^^^_^^^^^_^^^,^ (z- a2)VZiz) (ai -a2)V Z(ai) and consequently An + AiZ + A2VZ(z) ^ Ao + Aiai+AzVZCai) ^^^^^^^^_„^)_^ . (z-a,)(z-a2)VZ(z) (z-ai)(«i-a2) VZ(ai) It follows that Res /(z, s) = Ao + A,a,+ A2VZ(a,) ^ z=a, (ai- a2);Z(ai) 2 Ao [which owing to equations (1)] = ai— a2 In a similar manner we have Res /(z, s) = Ao+A,a2+A2V^ ^ _2M_. z-a, («2- ai) VZ(a2) "2- "1 Eliminating Ai from the equations (1), we have . A2\a2^'Z.{ai)- aiVZ(a2)\ Aq= ' '1 a2- «i and eliminating Aq from the same equations, we have ^^^ A2{VZ(a2)-v^Z(ai)} Ot2- "1 414 THEORY OF ELLIPTIC FUNCTIONS. It follows that /(z, s) ^ A2 a2V'Z(ai)-aiVZ(a2) + (\/Z(a2)-V'Z(ai))z+(a2-ai)VZ(z) a2-ai (z - ai)(2 - a2) \/Z(z) A, 02— ai VZ(ai)+VZ(z ) ^ \/Z(a2)+\/Z(z) (z-ai)VZ(z) (z-a2)\/Z(z) } When /(z, s) has this form, the integral / /(z, s)dz is of the third kind, being logarithmically infinite at the points (ai, \/Z(ai)), (a2, ^/Z(a2)). This integral may be considered the fundamental integral of the third kind and written n(z, VZlz); ai,VZ{ai); 02, V^Z(a2)) or more simply n(z; ai] 02). In a similar manner, as was proved in the case of the integrals of the second kind, we have a general integral of the third kind with the two logarithmic infinities ai, a2 if we add integrals of t he firs t kind to n(z; aj; a2). Art. 350. Take three points ai, \/Z{ai); a2, VZ(a2); as, VZiag) on the Riemann surface of Chapter VI and form the integrals n(z; fti; aa), II(z; aal as), 11(2; as] a{). Further, let A2, ^2'^' and A^(i) be the constants that correspond to A2 above. We may so choose the constants X, n, v that the expression (1) ') = T— ^f [see Art. 167], J (1 4- nsm-^0) \(p where the parameter n may be positive or negative, real or imaginary. This integral may be written It follows that Ii{n,k,u)= I " • J 1 + n sn^u z—, ^du, 1 + n sn^u where u is an elliptic integral of the first kind. Jacobi [Werke, I, p. 197] made a further change in notation by writing [see also Legendre, loc. cit., p. 70] n = — k^sn^a, where a being susceptible of both real and imaginary values, leaves n arbitrary. Multiplying the right-hand side by > the form of the eUiptic sna integral of the third kind adopted by Jacobi is n/ ; \ r^k^sna cna dna sn^u , {,u,a,k)= I — — -f- — du. Jo 1 — k''sn''a sn-'u Art. 358. In Art. 294 the following equation was derived: (y'^(M) ia-'{a) If we differentiate logarithmically with regard to a, we have 2 k^sna cna dna srflu _ @'{u — a) _ Q'{u + a) „ 0' (a) 1 — k^sn^u sn^a @{u — a) @{u + a) ©(a) from which it follows at once that n, s 1, 0(w - a) , @'{a) nfa,a)=-log ( +^?T7T- 2 0(m -I- a) 0(a) ELLIPTIC INTEGRALS OP THE THIRD KIND. 421 Interchanging u and a we further have 11 (a, u)= - log — -^^ + a -~^, from which it is seen that n(u, a)— Tl(a, u)= uEia)— aEiu). We note that this equation remains unchanged when the argument u and the parameter a are interchanged (see Legendre, loc. cit., pp. 132 et seq.). Art. 359. It is evident from the integral above through which !!(«, a) is defined, that (1) n(u, a) = — n(— M, a) and (2) n(0,a)=0. Further, since snK = 1, en K = 0, dnK = A;', we have (3) n(M, K)=0. For a = iK' we have sn a = oo = en o = dn a, so that (4) Tl(u,iK') = oo; and since sn{K±iK')=\, cn{K±iK')=T-^, dn{K±iK')=0, k K it follows that (5) U{u,K±iK')=0. From the formula expressing the interchange of argument and parameter we have (6) U{K,a)= KE{a)- aE= KZ{a) [Legendre]. These formulas follow also directly from the expression of 11 (m, o) through the theta-functions, as do also the formulas (7) mK+iK',a) = {K + iK')Z{a)+^, (8) n(2 iK', a) = 2 iK'Z{a) + |^, (9) U{u + 2K,a)=U(u,a)+2KZia), (10) U{u,a + 2K)= U{u,a) = U{u,a + 2iK'), (11) U{u + 2iK',a)= Il{u,a)+2ll{K + iK',a)- 2 n(K, a) = Uiu,a)+2iK'Z{a)+'^- From the equations (9) and (11) it is seen that the moduli of periodicity of U{u,a) are respectively 2A'Z(a) and 2iK'Z{a)+~ K 422 THEORY OF ELLIPTIC FUNCTIONS. Art. 360. From the definition of 11 (m, a) given in Art. 357 we have dll{u, a) _ k^sna cna dna sv?u du 1 — k'^sn^a srfiu = Z(a) + i Z(w - a) - i Z(m + a) [Art. 297]. We therefore have the theorem : The derivative of an elliptic integral of the third kind with regard to an elliptic integral of the first kind may be expressed through elliptic integrals of the second kind. Interchanging u and a, we also have k^snucnudnusn^a n i \ \ ii \ i rrr , \ ,„ „ 5 = Z(m)- iZ{u- a)- i Z(u + a). 1 — k'^sn^a sn^'u The addition of these two equations gives Z(m) + Z(a) — Z(m + a) = k^snu sna sn{u + a), which is the addition-theorem of the Z- function (see Art. 297). Art. 361. From the formula 0(M + K)=^e(M) Vk' we have by writing iu in the place of u e{iu + K)=^^e{iu), Vk' or, (see Arts. 204 and 220) e(0) -Vk'" '^"^"'^^0(o,ifc') k'^ 0(0, fc') If we take the logarithmic derivative of this equation, we have iZ{iu + Z) = ^^ + Z{u + K', k'). If these expressions are written in the formula n(m, ia + K)= iuZiia + K)+ ^log ®!^" " ^!" + ^1 , 2 v>{ia + %u + K) we have Ti{iu, ia + K)= uZ{a + K', k') + Uogf^-^^L±^Jfl; 2 0(a + u + K ,k) or U{iu, ia + K)=U(u,a + K', k'). ELLIPTIC INTEGEALS OF THE THIRD KIND. 423 If a is changed into ia, it follows that U{iu, a + K)=- U{u, ia + K', k'). These results may be derived directly by a consideration of the integral which defines 11 (m, o) [see Jacobi, Werke I, p. 220]. Art. 362. In Art. 227 we saw that 1 , f-,/2 Ku\ . q cos 2 m q^ cos 4 m g^ cos 6 m o^ cos 8m It follows directly from the formula XT, V Q'(a) , li„„ 9(M-a) II(m, a)= w--^ + zlog^^^^ — ; — '- 6(a) 2 fc)(u + a) that Q,(2Ka\ jl/2Ku 2Ka\ 2Ku \ n / [ n ' t: ) n Q/2ga\ , q cos 2(m + g) g^cos4(M + a) l_q2 + 2{l-q*) _ q cos2{u — a) _ q^ cos4(m — a) __ 1 - 32 2(1 - 5*) 2 Km \ tz / _ 2 r ^ sin 2 a sin 2 m , q^ sin 4 a sin 4 m ^ 0/2 ga\ L 1-9^ 2(1 -g4) q^ sin 6 a sin 6 u , "j 3(1-96) +•••]• The Omega-Function. Art. 363. Jacobi (Werke, I, p. 300) put E{u)du = logO(M). £' '0 If we integrate the formula of Art. 297 E(u + a)+ Eiu- a)=2 E{u) r^— 1 — k-'sn-'a sn^u from M = to M= M, we have at once log "(^ + "^ + log "^^ ~ ^^ = 2 log n(M) + log (1 - k^sn^a sn^u), fi(o) n(a) or n(M + g) n(M - g) ^ ^ _ ^2^^2^ ^^2^_ fi2(M)02(g) 424 THEORY OF ELLIPTIC FUNCTIONS. Further, if u and a are interchanged in the above formula, it becomes E{u + a)- E{u- a)= 2E{a) — — , 1 — k-'sn^'a sn^u which integrated from m = to w = m is log "(^ + "^ - 2 uE{a) = - 2 n(M, a) n(M — a) or TT/ \ El/ \ I 1 1 f2(w — a) ll{u,a) = uE (a) + - log -^ — ■ — ^ • 2 Q,{u + a) In Art. 251 the following formula was derived : E{iu) = i {tn(u, k') dn{u, k')+ u - E{u, k')]. We have at once log O(iM) = log cn{u, k') — — + log Q{u, k'), or -"l n{iu) = e ^ cn{u, k') n(M, k'). Art. 364. From the formula E{u + 2 mK) = E{u)+ 2 mE we have 1 n(u + 2 mK) o 771 , 1 ^/ \ log V,,o L^^ =«i • 2 wE + log n(w), 0(2 mit) n(2mK) ^ ■^" If we put M = — 2 mK in this formula, and note that n(- ii)= n(u), n(0)= i, we have n(2 mK) = e^'"'^^, and also 0(m + 2 mK) = e2'»^("+"'^'n(w), e ^^ n(M + 2mK)=e ^^Q{u). This formula shows that the function e "^^ Q,{u) remains unchanged when the argument is increased by the real period 2 K. Further, if in the formula a{iu) = e ^ cniu, k')Q{u, k'), we write m + 2 nK' in the place of u, we have _ (u+2nK') ' Cl{iu + 2niK')^{-iye ^ cn{u, k') Q{u + 2 nK' , k') , or -^(u+2nK'r (u+2nKr y?E' e 2^' n(w + 2mK') = (- l)"e ^ cn{u,k')e ^^' Q{u,k'). ELLIPTIC INTEGRALS OF THE THIRD KIND. 425 It follows that K'-E' , ,„ ^,,, u'E' „^, (m-2ng')' -TTE^, e 2^ n(iu + 2niK')={- l)"e 2*^ cn(u, k') n{u, k') (K'-E') , = (- l)"e ■''' n(w). If in this expression we put — iu for m or m for iu, we have e 2K' n(M+ 2mK') = (-l)"e 2*^' n(M), from which formula it is seen that the expression K'-E\ , e ^^' " Cl{u) remains unchanged when u is changed * into m + 4 niK'. Art. 365. We derived in Art. 263 the formula Vei— e3\<^3W / from which we have at once through logarithmic integration n(Vei — 63 • m) = ei«'"'a3M. Writing these values in the formula 2 Q(m 4- a) it is seen that rtf./ >/ ^ li „ii[Vei— eaiu — a)'\ n(Vei- es-u, Vci- 63-0;= -log 'z .-^= ^ ^ 2 Q[Vei- e3{u + a)\ + \/ei- e3wB(\/ei- 63-0) ^llog '^3(^-a) ^^g3:a. 2 (73 (w + a) (73a [See Schwarz, loc. cit., p. 52.] Art. 366. The following relations may be derived from the addition- theorems of the theta-functions given in Art. 211, formulas [C]: fc 02 (u) 02(a) k'm())Qdu+o-)Q^{u-a) ^ k^^2^cn^a + k'\ 62(1*) ©2(a) fcre2(0)H,(« + a)Hi(u-a) ^ ^^,^ ^^,^ _ ^,^^ 02 (m) 02(a) * See Jacobi, Werke, I, p. 309. 426 THEORY OF ELLIPTIC FUNCTIONS. If as in Art. 358 these expressions are differentiated logarithmically with regard to a and integrated with regard to u, the variable in the first equa- tion being less than the parameter a, we have "^snacnadna J 1, H(o — tt) , 0'(a) svP'U — srfia 2 H(a + u) fc)(a) J^^ k^snacnadna cn^u , _1, Qi(u — a) . Q'(a) A:2cn2w cn^a + k'^ " ~ 2 °^ OjCm + a) ^ e(a) ' J''" /b^sTO a cnadna dn^u , _ 1 1 „ Hi(m — a) . @'(a) drfiu dn^a - k'^ " ~ 2 °^ Hi(m + a) ^ e(a) ' These integrals * may all be expressed through the integral 11 (u, a) and an elhptic integral of the first Mnd; for example / r sn a cnadna J xr / , • i^i\ u cnadna du = 11 (w, a + iK)- sn^u — sn^a sna Addition-Theohems for the Integrals of the Third Kind. Art. 367. The addition-theorem for the elliptic integral of the third kind follows directly from the equation of Art. 358 in the form n, s , TT/ \ TT/' , \ li Q(u — a) Q(v — a)&(u+v+a) {u, a) +Il(v,a)-n{u + v,a) = - log ) ( ir.) Z i ' 2 @{u + a)@(v + a)@{u + v — a) For brevity we shall put 6(m — g) @{v — a) @(u + v + a) ^ p, •. @{u + a)@(v + a)e{u + v-a)~ ^'^''"'"■>' and we shall derive several different forms for F{u, v, a) which are due to Legendre and Jacobi.f From the formula 02(0) 0(/x + v) @{pL - v) = 02(/z) 02(v) { 1 _ k^sn^/isn^v} we have at once e^O)@{u-a)@{v-a)=e^(^y^(y^-a\ll~k^sn^^sn^(^-a\\, @HO)e{u + a)e{v + a)=@^(^y^(^ + a\\l-k^sn^y^sn^(^ + a)\, 02(O)0(a)0(M + a;-a)=02/^W/li±^-a\{l_fc2s„2li±i:,„2/?i±^_„\l 02(O)0(a)©(M+^ + a)=02(^)02/^ + a){l-fc2sn2^±^sn2(ii±^ + a) * See note by Hermite in Serret's Calcul, t. II, p. 840, t Legendre, Fonct. Ellipt., t. I, Chap. XV; Jacobi, Werke, I, pp. 207 et seq. ELLIPTIC INTEGRALS OF THE THIRD KIND. 427 and by taking the product of the first and fourth of these equations divided by that of the second and third we have 1 _ i!„2 /H^V„a /»±i: + „ V 1 _ i!„! !t±» „2/!t±i!_ a) From the formula sn(// + v) sn(j« — v) = - — ^ srfiu — sn^v 1 — k^sn^fisn^v we further have 1 — k^sn-' — ■ — sn^ \snusnv = sn'' sn-' > 2 2 J 2 2 ^ ,,, yU + V y/u + v \\ , , -, „U + V ytu + V \ 2V2/J 2 V2/ Taking the products of these two equations each multipUed by —k^ and adding a common term on either side, we have * 1 _ fc2^2 !£±lsn2 HzJi 1 - fc2sn2 ^sn2 (^ - a) multiplied by { l — k^snasnusnvsn{u+v — a) ] = jl _ fc2^2 u±v^ri^u^\ ll - k^sn^ "^sn^ (^ - a\\ ,o oU + V oU — v\\ oU + V yfu + V \ 2 2 J[ 2 \ 2 I 2 2^2/ 2 2 \ 2 I Writing —a for a in this equation, we have a second equation, which divided by the first gives 1 _ Jfc2,„2^%„2/W^ _ AU _ k^,n2U^^2hl±l + ai 1 _ k^sn^^^sn^{^L±± + a\\\\ - /fc2,„2 «^^2/?L±i! _ «)! _ 1 + k'^sn a snusnv sn(u + v -i- a) 1 — k^sn a snusnv sn(u + v — a) * See Cayley, Elliptic Functions, p. 159. 428 THEORY OF ELLIPTIC FUNCTIONS. If a is changed to —a in this expression, it is seen that „, . I — k^snasnusnv sn{u + V — a) F{u, V, a) = :; — -. ; ; — e- 1 + k^'sn asnusnv sn{u + v + a) Art. 368. It follows also from the expressions given in the preceding Article that /^9/ \r\9f \ cn2/n\ ®(w — v)&{u + V — 2a) @2^u - a)@^iv - a) = e^CO) _i-^_^__^_^^— _^, e^a)@Hu + v-a)= 02(0) i^,.^.^,^.(, + ,_,) ' @2(«)02(„ + . + a) = 8^(0) Q(^ + y(^ + ^ + 2a) 1 — k^sn^a sn^iu + v + a) From these equations we have „, , r ! 1 - fc2sn2(M + a) sri^iv + a) } { 1 - k'^sn^a sn?{ u + v - a) } 1*. F{u, v,a)= ^ ^^ '- =^ '-'-^ ^^ !r I Li 1 — k^sn^{u — a) srfi{v — a) } { 1 — k^sn^a sn^{u + v + a)\J Art. 369. Since 11 (m, a) — 11 (a, u) = uZ{a) — aZ{u), we have U{u, a) + n(w, b) - U{u, a + b) = U{a,u)+U{b,u)-U{a + b,u)+ u{Z{a)+ Z(5)- Z(a +6) | = 2 log F{a, b,u)+ u k^sn asnb sn{a + b), which is a theorem for the addition of the parameters. Art. 370. In the formula (see Table (B) of Art. 211) j?o(0)«9(?/ + z)^(x + z)d{x + y)= ^(x +y + z)^(x)d(y)^{z) + diix + y + z)^i(x)^i(7j)^i{z) write X = ?-^, y = ?-^ and a = - ?-^ and + ?^ respectively. :z K 71 7Z Divide the first result by the second and we have ^ _ H(a)H(M)H(i;)H(M + v - a) @(u - g) 9(^ -a)@{u + v + a) _ @{a) @{u) Qjv) @(u + v - a) e{u + a)&{v + a)@{u + V - a) ^ H(a)H(M)H(i;)H(u + v + a) ' @ia) e{u) @{v)e{u + V + a) or jp, s _ 1 — k^sna snu snv sn{u + v — a) 1 + k^sna snu snv sn{u + v + a) Remark. — By writing as we have done n = — fc2 sin2^, and allowing 6 to take imaginary values, the expression on the right-hand side of the addition-theorems above is always a logarithm. Legendre * * Legendre, Traite desfonctions elliptiques, t. Ill, p. 138. ELLIPTIC INTEGRALS OF THE THIRD KIND. 429 considered the following two cases, to the one or the other of which by means of real transformations the parameter n may always be reduced: (1) n = - k^sin^O, (2) n = 1 + k'^sin^d, where 6 is real in both cases. Owing to the fact that tan -'^ it = -i log , 2 ^l-t the inverse tangent appears in the second case instead of the logarithm.* Art. 371. If we put to= ^Uq, ti= pUi, <2=j?W2, <3=»>W3, -\/S(h)= P'UO, -V-S(il)= S^'Wl, -VSit2) = ^'U2, -Vj{t3)=^'U3, we have from Art. 355 n(fi; to; 00)= log ^(^°-^i) + mzuo, auiauo n(i2; to; 00) = log^(HoiU£2) ^ ^^^.^^^ aU2auo nfe; to; 00)= log '^^^^"-^3) ^ ^3^^^ augouo If U3= «i+ U2, it follows that n( — ax{u -V w) ax{v + w) auav, and the formulas given in the Table of Formulas, No. LXII, combined with the formula „.2y^ It follows that n('u [Liouville's Theorem]. (3) Representation in the form of a quotient of two products of theta- functions or sigma-f unctions. (4) Representation in the form of a sum of rational functions. (5) Representation in the form of a sum of rational functions of an expo- nential function. Art. 373. The first representation mentioned above and due to Her- niite has been made fundamental throughout this treatise; upon it, as already stated, the other representations all depend. We shall produce it again in a somewhat different form so that the dependence upon it of the other representations may be more readily seen. In Art. 87 Hermite's intermediary function of the first order was denoted by X(m) and was defined through the equation 7w=+oQ 2inmu .b X(m)= 2 Q""e~^, where (7M. We further have •«/r(u)= e'"'+''"Xi(M) ■f (m + 2w) = e«"+2-)^+''(«+2'")Xi(M + 2w). It follows that Comparing this result with (t(m + 2 oj) = - e2,(«+<.,) (^^ it is seen that we must write \Xu> = 2-r\ and 4 Aa;^ + 2 /iw = 2^w + m, where ni has been added to change the sign. We have at once X = — L and u. = — , 2w '^ 2^ and consequently also 1 2 + Jlf This function satisfies the first of the functional equations which au satisfies. We have further 2ij — u+2t] l-jTl— «— 2ffi — ■«/r(M + 2w')=-e " - "-e - '"■f(M); , , -i or, smce jjo) — tj uj = —, we have •<|r(u + 2w')= - e2''("+'"''>|r(M). It is thus proved that "(/"'(le) satisfies also the second functional equation satisfied by au. We may therefore put ylr(u)= Bau, where B is an arbitrary constant, and Zi(m)= = Cw- au DOUBLY PERIODIC FUNCTIONS OF ANY ORDER. 435 Art. 376. It is evident from above that we may write F{u) in the form * k Ik = 1, 2, k k V /k = l,2, . . . ,n; \ U = 1, 2, . . . ,4- 1; Bi<"+'> = bk,.^J' We here have F{u) expressed as a sum of terms each of which is a complete derivative. This formula is therefore especially useful in all applications of the elliptic functions that involve integration. The constant Ci may be determined if we know the value of F{u) for any value of the argument different from the quantities M4. Art. 377. We saw in Art. 299 that 2 pu - gnik where we assumed that Uk is not congruent to a period; otherwise l^Uk and gnck would be infinite. We therefore first exclude in this discussion all the quantities Uk which are congruent to periods and attach a star to the sum- mation sign to call attention to this fact. We have accordingly, if we note the formulas of the preceding Article, X*g*'»c(^- uk)=T,*^>''''z^ -^*Bk^^Kuk + \y; Bk^^^ ^'"" + ^'^^ - ■^ ^ ^ ^ 2^ ^U - pUk We note that the second summation on the right is a constant. Two cases are possible: (1) None of the quantities Uk is congruent to a period; or (2) Some of the quantities Uk are congruent to periods. In the first case we may remove the star from the summations. We then have C^X ^t^^' = ^- ^* ^^^^ follows at once that "^BhWi^iu-Uk) is rationally expressed in terms of pu and p'u. In the second case only one of the quantities Uk can be congruent to a period and therefore also to zero, since the quantities Uk form by hypothesis a complete system of incongruent infinities. This infinity may be transformed to the origin. We must consequently add Bi'^'C" to ^*Bi^^>C(^ - ■"*) that we may have y 5^(1) ^(m — Uk)- But here also it is seen that k 2* Bfc^^Cw + Bfc'^'CM = 0, since J ^'t^^' = 0- Thus without exception it is seen that ^ Bfc^^>C(w-Mfc) is rationally express- ible through g?M and p'u. * * See Kiepert, Crelle's Joum., Bd. 76, pp. 21 et seq. 436 THEORY OF ELLIPTIC FUNCTIONS. Further, since the derivatives of Z{u — Uk) are all rationally expressible through pu and p'u, it follows that F{u)= Riipu,tp'u), where R denotes a rational function of its arguments. This theorem is due to Liouville (see Art. 155). Corollary. — If a doubly periodic function has the property of being infinite only at the point m = and congruent points, then this function F{u), say, is an integral function of ^u and p'u. To prove this note that since w = is the only infinity within the first period-parallelogram we have k = 1 and Ui = 0. Further, since X Bk'-^l = 0, it follows that 5i(»=0. We thus have ^ ml dw"» jn By definition we had * d ^ du and consequently — CW = - jp"M = - 6 fp^M + ^ ^21 fp"'u = 6{pup'u + p'upu), It follows that F{u) is an integral function of p(u) and p'iu). Art. 378. Let F(u) be a doubly periodic function of the second sort so that F{u + a) = vi^(M), F{u + b)= v'F{u). The logarithmic derivative of F{u), f!g = <^(«), say, is a doubly periodic function of the first sort. The function ^(m), as seen in Art. 4, becomes infinite on the zeros and on the infinities of F{u). Let Ml", M2°, ■ • • , Um^ be the zeros of F{u); and at ufi let F{u) be zero of the * See Kiepert, Dissertation (De curvis quarum arcus, etc., Berlin, 1870). DOUBLY PERIODIC FUNCTIONS OF ANY OEDEE. 437 Xi order (t = 1, 2, . . . , m). Let m, uz, . . . , Un be the infinities of F(m); and at u, let F{u) be infinite of the fi, order (j = I, 2, . . . , n). We may therefore write F(m) = (m - u/>)^Fi{u) (i = 1, 2, . . . , m), where Fi{u) is neither zero nor infinite for u = Wi°- It follows that ^' F{u) u - ufi Fi(u) and consequently Res.^(M)= A,-; and similarly Res<^(M) = - ;.,. U=Uy It is thus seen that are all zero, and conse- quently 4>{u) = Ci + ;iC(m - Ml") + XzCiu - U2°) + • • • + XmZiu - Mm") - ^iCC'" - ■"!)- /'2C(W - M2)- • • ■ -flm(:{u - Un). Also, since it is seen that Through integration it follows that ^^(w) = gCtu+C ^(U - Ui^y^ylrju - M2°)'' • • • ■fJU - Mm°)^ ■-«') a{u — u^), or (7(M—M'))Whereu'=Mi*', M2°, . . . , Ur^;Ui,U2, . . . , Ut. Hence, since F{u + 2oJ) = F{u) [if we suppose that F{u) is a doubly periodic function of the first sort], it follows that (B) F(m)= e'^("+2'")+'^' -2,2^0 e '"^ a (u - ui°)a(u - ^^2°) ■ ■ ■ a(u - Ur°) e '=1 a{u — Ui)a{u — U2) ■ ■ ■ a(u — Mr) The two expressions (A) and (B) must be equal. We must consequently have 2co>+2ti or e ^'=1 '=1 ' = 1, and similarly e '=' <=i = L In virtue of these relations we also have (1) 2ca>-F27?(2)Mi-X"^*')=2MOT, (i=T i= r \ 1=1 i=l / where M and M' are integers (positive or negative, including zero). DOUBLY PERIODIC FUNCTIONS OF ANY ORDER. 439 From the two equations just written it follows that 2c{jjo'- cjr]')= 2m{M'^ - Mi]'). But since i^oi' — oirf = i ni, it is seen that c = 2M'i)- 2M-q'. If c is eliminated from (1) and (2), we have l~r 1= r ^Ui - ^Ui" = 2 Moj' - 2 M'a «=1 i-l For the sake of greater simplicity we may write — m' for M and m for M'. We then have c = 2 mi) + 2m'-t)' , 1= r t= r 2)''*i'' ~ 2)^» ^ ^ "'^ "*" ^ m'w', i-1 i-l where m, m' are positive or negative integers or zero. This theorem is due to Liouville.* From the latter relation it is seen that if the r infinities of a doubly periodic function of the rth order have been chosen, then only r — 1 of the zeros are arbitrary. As we saw above, we may write for a zero another zero that is con- gruent to it. We may therefore increase u^ by u^ + 2 tow + 2 rn'm' . If this is done, then for the new system of zeros and infinities we have 7?i = = m' and consequently 1= r t= r Xwi°=]XMi and c = 0. i-l >=i We then have puj\^ C "^(^ ~ "i°)<^(^ - ^2°) ^ ■ ■ aju- Ur°) a{u — ui)a{u — U2) ■ ■ • a{u — u^ It is thus seen that F{u) depends upon the quantities 2 w, 2 oi', C; ui, U2, ■ ■ • , Ut] and upon r — 1 of the quantities m'^ (we note in partic- ular that of the r quantities m," there are only r — 1 arbitrary). It follows that the function F{u) depends upon 2 r -|- 2 constants.! * Liouville (^Lectures delivered in 1847, published by Borchardt, Crelle, Bd. 88, p. 277, or Liouville, Comptes Rendus, t. 32, p. 450) proves this important theorem and also the two fundamental theorems already given, viz.: a doubly periodic function of the nth order may be expressed rationally through an elliptic function of the second order and its derivative; a doubly periodic function must become infinite at least twice within a period-parallelogram. Prof. Osgood, Lehrbuch der Funktiontheorie, p. 412, uses these three theorems as the foundation of his treatment of the doubly periodic functions. t See Schwarz, loc. cit., p. 20, or Kiepert, Crelle, Bd. 76, p. 21; or Appell et Lacour, Fonct. Ellip., p. 48. 440 THEORY OF ELLIPTIC FUNCTIONS. The expansion of the function F{u) through H(m) in the place of au may be derived in a similar manner (see Riemann-Stahl, Elliptische Func- tionen, p. 110). Corollary I. — We note that the function F{— u) is an elliptic function of the same nature as the function F{u) considered above. It is also evident that i[F{u)+ F{— u)]= 1^0 (w), say, is an even function, and that i[F{u) — F{— u)]= ■V^i(m) is an odd elliptic function. That every elliptic function my be expressed as a sum of an even and an odd elliptic function is seen from the identity F(u)= i[F{u)+F{- u)] + i[F{u)- Fi- u)], or F{u)= ■ylro{u)+ ■<}ri{u). Corollary 11. — We may next prove that every even elliptic function of order say 2 r may be rationally expressed through pu. Such a function may be represented in the form V^o(m) = a{u-Ui)a{u-U2) ■ ■ ■ a(u — Ur)a{u + Ui)a{u + U2) • ■ a{u + Ur) We may also write a{u- UiO)a{u + Mi<') a{u - Ui'^)a{u + Mi°) a^ua^up o^uP a(u — Ui)a{u + Ui) a(u — Ui)a{u + m) a^Ui a^ua^Ui gm — pui^ a^Ui° jpM - pui a'^Ui We therefore have \=T J -r H'^'wiTTcs'" - pwi") ^o(t*)= c;-^^ — ift ' J = l i-l a formula by which it is shown that V^oCm) is rationally expressed through gni. We may therefore write ■^o{u) = Roipu), where Ro denotes a rational function of its argument. Further, if ""^iCm) is an odd elliptic function, then, since p'u is also an odd elliptic function, ' ', is an even elliptic function = Ri{pu), say, so that i^ 1 (w) = p'uRiipu), where 72 1 denotes a rational function of its argument. DOUBLY PEEIODIC FUNCTIONS OF ANY ORDER. 441 Art. 381. As an interesting application of the above representation of an elliptic function we note the following: In determinantal form we write the formula ^^u)~^ = -2^2i±^>£<^J)^. 1, g)M 1, gro We may also express through sigma-quotients such expressions as 1, ^U, ff'u 1, pv, p'v = A(m), say. 1, pw, jp'w The infinities of fpu and p'u are congruent to the origin, pu being infinite of the second and p'u of the third order for m = 0. The determinant is a doubly periodic function of the third order in u with zeros Ui^= v,U2^= w and M30= -V -w. Further, mi0 + u2°+ "3°= = S (infinities), the infin- ities being the triple pole zero. It follows then that the determinant must be of the form * Q a{u + V + w)a(u — v)a(u — w)a{v — w) _ . Multiply both sides of this expression by v? and then make m = 0, and we have q C!{v + w)a{v — w) n 1' 1, pw a{u + V + w)a(u — v)a(u — w)(j{v — w) so that C — — 2. It follows that 1, pu, p'u 1, pv, p'v 1, pw, p'w Appell and Lacour {loc. ciL, p. 63, Ex. 2) give an incorrect value to the constant C. Further, since p'oj = = p'o)', if we write in the expression above v = aj and 10 = cd', it becomes ) >-, cruatoau) + w')') a^ua^{a)+w') a^ua^u) a^ua^u}' = 4 [jfm - j?(w + w')] [pw - S'w] [m - 9^'\ = 4 (j>u — e2){pu - ei){pu - eg). * See Daniels, Am. Joum. Math., Vol. VI, p. 266. 442 THEORY OF ELLIPTIC FUNCTIONS. Art. 382. The fourth method of the representation of the doubly peri- odic functions is as follows: We had in Art. 376 F{u) = Ci + ^BkW ciu - Uk) + X^**'^ Piu - Uk) ^{h- 1)! In Art. 272 we saw that J U'^(U — WWW'') and consequently B.^^^au - ..) ^ ^^ + V' S ^^^^ + ^ + ^^''\- "^^ ( • u — Uk „ ' ■" ~ "if WW W^ ) If we take the summation over this expression with regard to A; and note that the summations with regard to w and with regard to k may be inter- changed, we have tBk^^Kiu-Uk)^X^^^X'%'\-^ ^4 ^ 'J'U-Uk ^■^(U-Uk—W M>2 ) We further note that p{u-Uk)= + X ' ) / T5 5 ( ' p'(u-Uk) = -2\y- ^ -, 17 (w — Uk- wy p"iu - Mi) = 3! X -, 72' ^*C. 17 (m - W4- m;)4 It follows at once that -f^ (m - Wt)2 -f^ A ^ (it _ ^tj^_ ^„)2 „2 ^ + ■•■+22^73 Bfc(^*> r "f (" - "*- '^V" DOUBLY PERIODIC FUNCTIONS OF ANY ORDER. 443 If for brevity we put ^ (U- Uk (U - Wfc)2 (m - Uk)^ ) the above formula may be written F(u) =C, + fin) + 2;- j fin -w)- 2) ^^'''"^^ ^^'" j • Art. 383. We may next consider the fifth kind of representation of the doubly periodic function F{u). We saw in Art. 287 that ^ oj^ 2ojU-z-^ ^,l-h^"'z-2 ^1-A2'«22^' where z^= t = e" . We have at once z - 2-1 < - r If we write {u-utl— t , "' — e "" = —! where A; LJ /A;=l,2, . . . ,« \ \v= 1,2, . . . ,4- 1/ We have the following expansion (Art. 286) : «^'~a. W h^-2-')^ \ij(l-''""^-')'\it (l-/i2'"2^)2 444 THEORY OF ELLIPTIC FUNCTIONS. It is further seen that ^ ^ and ** *^ (2-3-1)2 (<-l)2 (--iX ('-**)^ Next let It is evident that /2(0) = =/2(<»). The terms in F{u) which correspond to y = 1 are The terms in F{u) which correspond to y = 2 are T 2! -[- i?'(w - Mi)]. If we differentiate the formula above for pu we have a suitable expres- sion for p'm in the form of an infinite summation, which may be written m= —00 where /sCO is a rational function in t having the property that /3(0)=0=/3M. We continue this process and finally write /(0 = /i(0 + /2(0+ • ■ +A(0, the function /(i) being a rational function in t such that /(0)=0=/(^). We therefore have F(w)= Ci- i y(BfcWwfc+ Bu(^^) + f{f) + fihH)+fihH)+ ■ ■ ■ + nh-H) + f{h-H)+ ■ • ■ . Since t has the period 2w, it is evident that F{u) has the period 2w; also noting that t becomes hH when u is increased by 2 a*', it is seen that 2 w' is also a period of F(m) provided the above series is convergent. DOUBLY PERIODIC FUNCTIONS OF ANY ORDER. 445 Art. 384. We may establish the convergence of the series in the pre- vious Article as follows: Since /(0= for f = 0, we observe that t =0 is a root of f(i) = 0, so that we may write fit) = t 0-1+ 0'2t + ■ ■ • + ap-it"-^ ■^ 1 + bit + b2t^+ ■ ■ ■ +by It is always possible to choose t so small that \bit\ +\ bifi I + • • • + I b^" I < J. It follows that the denominator in the fraction above is greater than ^, while the numerator is finite. We may therefore write m That the product on the right-hand side is absolutely convergent may be proved by writing /i«)=l+/o(0, where /o(0 = = /o(oo); it then follows by Art. 17 that the above product is absolutely convergent if m= — CO is absolutely convergent. The convergence of this series is easily estab- lished by using a geometric progression whose ratio is h^. Art. 386. We saw in Art. 377 that every one-valued doubly periodic function which has everywhere in the finite portion of the plane the char- acter of an integral or (fractional) rational function may be expressed rationally through pu and p'u, say ^(m)= i2i(gm, jp'-u), where Ri denotes a rational function of its arguments. It follows that ,,, \ dRt , , dR\ ,1 {u) = — -^ j?'M + —J- P u. OJ|)M Ofp U Writing for p"u its value jp"m = 6 p^u — \ g2, it is seen that (f>'{u) may be rationally expressed through pu, ip'u. We therefore write 4>'{u)= R2{pu,ff'u), where i?2 is a rational function of its arguments. Any rational function of pu and p'u may be written in the form Ri{pu,p'u)= ■pTT-'-^' G2Wu,pu) where G\ and G2 are integral functions. DOUBLY PERIODIC FUNCTIONS OF ANY ORDER. 447 Further, since (jp'm)2= 4 jp3y _ g^pu - gs, it is evident that we may write p / , N -S + Tp'u Riipu, p'u)= ;;7*^' W where S, T and W are integral functions of pu; or finally Ri{sm,p'u)=U+Vs>'u, where U and V are rational functions of pu. We have accordingly (1) cl>{u)=U{pu) + V{pu)p'u and similarly (2) d>'(u)=Ui{pu) + V^ipu)p'u, where Ui and V-i are rational functions of pu. We note that V and Vi cannot be simultaneously zero; for U{pu) and Ui{pu) are both even functions of u, while if <^(m) is even 4>'{u) must be odd and vice versa. From (1) and (2) it follows that (3) p'u = ^^^ and (4) p'u = ^^f^^. In general both of these equations (and always one of them) have definite forms, since V and Fi cannot both be simultaneously zero. If then the values {u) and pu are known, then p'u is uniquely determined. If in the equations (1) and (2) neither V nor Vi is zero, by eliminating p'u, we have (I) g\pu,4,{u),'{u)\^^), where g denotes an integral function of its arguments. If further we square the equation (3) and give to p'u^ its value in terms of pv., we have (II) g,{pu,4>{u)\ = Q, where ^i is an integral function. On the other hand if V, say, is zero, we have from (1) the equation (I') 9 \ pu, (f){u)}= 0, and from (4) (ir) 9i{pu,4>'iu)}=0, where g and §1 are integral functions. We thus always have two algebraic equations among the three functions pu, {u) and <^'(m) has been chosen and suppose that the equations (I) and (II) have two common roots, say gw = Si and pu = 82- Suppose that mi is the value of u which satisfies the equation Then also, since gru is an even function, the value — Mi satisfies the same equation. From the equation (3) above we have , 4>(u) — U(si) /„\ 8? Ml = ^^ '. , '^ (a) V{si) The two values that are had through the extraction of the root are +g>'M and —^'u and there is only a choice of u between +mi and — Wi. We shall suppose that +mi gives + ^^(^,) = ^(«iii^[M. (b) By a comparison of (a) and (b) it is seen that 4>{u)= {u2), and consequently corresponding to (f){u) to which a definite value was given at the outset, we have shown that {ui)=4,iu2). (i) In the same way from the value of (f)'{u) which was chosen at the outset we have '(.u)=cj>'{u{)^4>'{u2). (ii) . DOUBLY PERIODIC FUNCTIONS OF ANY ORDER. 449 In Art. 37a it was seen that if the relations (i) and (ii) are true, then Ml — W2 is a period of ^(w). It follows that, if 2'(u)], where R3 is a rational function of its arguments. In this connection note the proof due to Briot and Bouquet in Art. 156, It follows then as was shown in Art. 158 that every transcendental one- valued analytic function which has an algebraic addition-theorem is necessar- ily a simply or a doubly periodic function. Art. 387. It follows from Art. 376 that fF{u)du = Co+ C,M +2Si(i> log aiu - Uk)-^B„(^^Ciu - Uk) k- / ^^ [g)(o>(M - Uk) = p(m - Mfc) ; y = 2, 3, . . . ,4-1]. Since ]x Bt^'' = 0, we may write V Bjd) log a{u -Uk) = y\ Bk^^^ log ''^'"' ~ ""^ + Constant. ■^ 'Y auauk We also saw in Art. 299 that — V J5i;(2)|r(M _ yj.) = — (^tt V Bi(2) -f an elliptic function of u. k k It follows that fF{u)du = CiM - Ct* VBfc(2) + VB,(i)log^^^^^^^ + .^i(M), where ^i(m) is a doubly periodic function with periods 2w, 2w'. Further, since (Art. 356) , o{Uk- u) ^ 1 fp'u +fp'uk ^^ ^ ^^^^^ au aUk 2Jpu— pUk * See Baltzer, Theorie der Determinanten, p. 109. 450 THEORY OF ELLIPTIC FUNCTIONS. we have fF{u)du = wFCi + ^^Sfc^^'cJ - C^X^*^^^ ^ J 2{pu -^Uk) The moduh of periodicity of the general integral / F{u)du are therefore had at once; at the same time it is seen that this general integral may be expressed through (see also Chapter VIII): 1. An elliptic integral of the first kmd; 2. An elliptic integral of the second kind; 3. A finite number of elliptic integrals of the third kind; 4. A rational function of jfm and ip'u. EXAMPLES 1. Show that any integral function of g>u and p'u may be written in the form ^, , _ , a{u - M,") a(u -u^ . . . a(u - «„") (t(u)" where u^, u^, . . . , v^ are the zeros of the function. 2. Show that any rational function of j?m and ^'u may be written F{u) = A bo + bipu + h^u + • • • + 6»^("-'>M 3. Write A(u, Wi, Uj, . . ,M„) 1, ^, ff'u, . . , j?(" '>u 1, S>"n, JP'": g)("-l)M„ Show that . , , /^{u) has the character of an integral function at the point uq. We may therefore write (1) cj^{u)= {uo)+'^^^cl>' {no) + ^^ ~,^°^V "(^o)+ • • ■ . Next put u^uo+W, V = uo+ v', Uq being a constant. Since <^(m) has by hypothesis an algebraic addition-theorem, we have an equation of the form G{<}>{u), iu + v)}=0, where G denotes an integral function of its arguments. We therefore have GU{uq+ u'), (f>{uo+ V), <}>{2uo+ u'+ v')\ = 0. Further, if we write u = uq V = Uq + u' + v', it is seen that G{(t>{uo), iuo+ u'+ v'), ^(2wo+ u'+ v') ] = 0. * See Forsyth, Theory of Functions, Chap. XIII; or Phragmen, Acta Math., Bd. 7, p. 33; I wish to mention in particular the Berlin lectures of Prof. H.A. Schwarz, which have been used freely in the preparation of this Chapter. 451 452 THEOEY OF ELLIPTIC FUNCTIONS. If (j){2uo+ u'+ v') be eliminated from these two equations, there results an algebraic equation of the form H\cj>(uo), {uo+v'), (wo+ m')= (t>i{u'), (f>{uo+ u' + v')= 4'iiu'+ V), we see that If in equation (1) we write Uo+ u instead of u, we have ^(«o+i^')=^i(w')=<^("o) + ^^w'+^^w'^+ • ■ •, from which it follows that by a change of the origin the function {u) may be changed into the function 4>\{u') in such a way that the function 4>i{u') has the character of an integral function at the point w'= in the branch of the function under consideration. Hence without limiting the generality of the given function (f>{u), we may assume that the point m = in the branch in question of the function 4>{u) is a point at which ^(w) has the character of an integral function. Making this assumption suppose next that p is the radius of the circle of convergence of the series expressing <^(m) in the neighborhood of m = 0. If then I w I < |0, the function (f){u) has the character of an integral func- tion in the branch considered. If I w I < i |0, I D I < J jO, then is \ u + v \ < p, and we have G{(j>{u),{v),(u + v)}==Q for the region considered. If in this equation v is put =m it follows that G{4>{u),(u),4,{2u)}=0, which is an algebraic equation between 4>(u) and (^(2m) with constant coefficients. We may write this equation (2) (?i{<^(w),<^(2u)}=0. If in this equation the value of u is limited so that | m | < i p, then within this region ^(2 u) has the character of an integral function, since \2u\ < p. MANY-VALUED ELLIPTIC FUNCTIONS. 453 Suppose that for {2 u) under consideration if | w | < ^ ^. But the coefficients of this equation may be analytically continued throughout the whole region of the circle with the radius p. In this extended region with the radius p the function ^(2 u) retains the character of an algebraic function. Hence the definition of the function may be extended to a wider region than the original and indeed to a region with the radius 2 p. By writing 2 w for m in the equation (2) we have EUminate ^(2 v) from this equation and equation (2) and we have an algebraic equation of the form G2{<^(w),{u),{2-^u)\=Q, from which it is seen that the original functional element may be con- tinued over an arbitrarily large portion of the plane without the function {u) ceasing to have the character of an algebraic function. It is also easily shown that by this continuation of the function the addition-theorem is true for the extended region (see Art. 51) and that all the properties originally ascribed to the function remain true through- out the analytical continuation. Art. 389. Suppose that the equation which expresses the addition- theorem G{(u),4>{v),4,{u + v)\=0 is developed in powers of 4>{u + v). It takes, say, the form (3) 4>'^'{u + v)+Pi,i[{u),{v)}^'^^-\u + v) + Pi.Mi'>^),'i>i-i>)}^"''-'^{u + v) where the P's are rational functions of ^(m), 0W. In this equation write M -I- fci for u and V — k^ for ■;;, where fcx is a variable quantity which may be Umited to small values. 454 THEORY OF ELLIPTIC FUNCTIONS. By this substitution u + v remains unchanged, and the above equation becomes (4) (u + ki),(f>(v - ki)](l)'"^-Hu + v)+ ■ ■ • = 0. The equations (3) and (4) are algebraic and have at least one root in com- mon, viz., (^(m + f) which belongs to the branch of the function in question. Through a finite number of essentially rational operations we may by Euler's Method derive the greatest common divisor of the two equations and thus form a new algebraic equation whose degree is less than the degree of either of the original equations unless these equations have all roots in common. This we suppose is not the case. Let the form of the new equation be 4>-^^{u + v)+ P2.i[4>(u),4>iu + ki),4>{v),4>{v - ki)](j>"''-\u + v) + • • • + P2.mmu),4>{u + k^),^{v),4>{v - ki)]=Q, where 7W2 < mi. We write in the above equation u + k^ instead of u and V — k2 instead of v. That equation then becomes ^■^{u + v)+ P2a[4>(u + ^2), 4>{u + ki+ kz), iv - k2), (j}{v-ki-k2)]'"^-'^{u + v)+- ■ • = 0. It may happen that for every value k2 this equation has all its roots the same as those of the previous equation, and consequently its coefficients do not depend upon k2. If this is not the case the two equations have a common divisor, and when we derive this divisor we have a new equation of the form l{v - ki), {v ~ ki- kz) J + • • = 0, where ms < m2- This process may be continued. Each following mj is less than the pre- ceding. Finally we must either have mt = 1, or the two equations through which a further reduction is made possible have all their roots common. We thus derive an equation of the form j.,.^ , ^, T,[iu),iu + ki),cf>iu + k2), . . . ,{V - fci), {v - fcz), . . . ,{v~ kr),i){v-ki-k2), . . . ,d){u + ki+ ■ +kr),'\ ,, , ,, W'-^{u-\-v)+ ■ ■ • +P^ [same arguments] =0, . . . , («) has the character of an integral function. We shall seek to cut out of the circle two narrow strips that are perpendicular to each other and which have the property that for all points within this cross the branch of the function {u), or of the analytic continuation of the branch of ^(m) under consideration, are known. This number of branch-points is finite, since the circle is finite and the function has the character of an integral function. A straight line is drawn connecting each of these points with the origin, and at the origin a straight line is drawn perpendicular to each of these lines. We next choose a direction from the origin which coincides with none of these lines or with the perpendiculars to them. The perpendicular to this direction through the origin does not coincide with any of the straight lines or the perpendiculars to them. Fig. 76. 456 THEORY OF ELLIPTIC FUNCTIONS. We thus have two straight Unes perpendicular to each other through the origin which within the circle pass through no branch-point of the function Through all the branch-points which lie within the circle we draw parallels to the two lines, and among all these parallels we choose those which lie nearest the two lines. The two pairs of parallel lines which have thus been chosen form a cross-shaped figure within which no branch- point is situated, excepting always the origin, which in the leaf under con- sideration of the function is not a branch-point. The functions (f){u) and {u + ki), . . . ,(p{u + ki+ ■ +kr),' ,.^(0),<^(-A;i), . . . ,<^(-fci- ■ • . -fc,) . From this it maybe shown as follows that <^(w) and ^v(u) are connected by an algebraic equation: The function •«/r„(M) is expressed rationally through ^(w),^(w-|-A;i), . . . , cf>iu + ki+ k2+ ■ ■ ■ + kr). By means of the addition-theorem ^(w -I- A: i) may be expressed algebraically through (f>{u) and 4>{k{), and similarly 4>{u + k2), etc. We thus have an algebraic equation of the form (5) H[cj>i^), ^v(m)] = 0. From the four algebraic equations G[cj>(u), 4>{v), cf,{u + v)] = 0, H[cj>{u), yj^,(u)] = 0, H[(f>{u + v), ^jr^iu + v)] = 0, we may eliminate (f>(u), 4>{v), (j>{u + v) and have the algebraic equation g[f,{u), ■Vr„(v), yjr^{u + v)] = 0. MANY-VALUED ELLIPTIC FUNCTIONS. 457 Further, if we differentiate equation (5) we have an algebraic equation (6) H,[{u), t„(u), '{u), ^'(w)] = 0. We also have the eliminant equation (7) E[cj>(u),cj>'(u)] = 0. (i) If from the equations (5), (6) and (7) we eliminate <^(-m) and 4''{u) we have the eliminant equation EilyjrM, -f/Cw)] = 0. (ii) It follows then that "^viu) has an algebraic addition-theorem. Since the algebraic equation (5) exists connecting 4>(u) and '^^iu), it follows that (j){u) is an algebraic function of "^viu). We have thus solved the problem of determining the function ^{u) in its greatest generality. The function 4){u) is the root of an algebraic equation, whose coefficienls are rationally expressed through a one-valued analytic function -^^iu), which function has an algebraic addition-theorem. In the Weierstrassian theory the one-valued analytic functions that have algebraic addition-theorems, as shown in Chapter VII, are either I, rational functions of u, or II, rational functions of e " , or III, rational functions of pu and p'u. TABLE OF FOEMULAS (The formulas of Jacobi and of Weierstrass in juxtaposition) I. u = f \ ^^ = = f\ '^'^ = F(k, 4>). . p. 285. Jo \/(l - z2)(l - A;2z2) Jo Vl - A;2 sin2 z = smqi, (f) = a,mu p. 241. z = snw, Vl — z2 = cos (/> = en u, Vl — fc2z2= dnw. . p. 241. Vl - /b2sin2 9!.= A^, u = F{k,z)= Fik,(l>). . . . p. 285. am 0=0, snO = 0, cnO = 1, dnO = 1. . p. 245. ain{— u) = — am u, sn{—u) = —snu, cn{— u)= cnu, dn{—u)=dnu. sn^u + crfiu = 1, k^sv?u + dn^u =1. . . p. 247. II. d„ = |4, #=A<^ or i^^ = dnu. . . p. 243. A0 aw aw /^y= (sn'M)2= (1 - sn2M)(l - Fsn2u). . . . p. 247. sn'u = cnudnu, p. 247. cn'u = — snudnu, dn'u = — k^sn u en u. (sn'M)2= (1 - sn2M)(l - /c2sn2w), .... p. 247. (cn'M)2= (1 - cn2M)(l - A;2+ k^cn^u), (dn'M)2= (1 - dn2u)idnH - 1 + F). (See also No. LVI). 458 TABLE OE FORMULAS. 459 t/co — III. dt dt . p. 215. * = ^"' p. 298. ^ = 8?'w = - V4fi-92t-g3. ip'u)^=4:p^u-g2pu-g3 p. 325. = 4ipu - ei)(pu - e2)(pu - 63). ^t^~92t-93'=i{t-ei){t-e2){t-e3). . . pp. 191, 200. 61+62+63= 0, eiC2 + 6263 + 6361 = - J (ei2 + g^2 + gj2) = - ^ ^2, 616263= i ^3. ^ ^ ^^'~ g^'^^' = (61- 62)2 (62- 63)^ (63- 6i)2= G. . p. 408. 460 THEORY OF ELLIPTIC FUNCTIONS. IV. K= r-=^M=^= m^F(k,A . p. 212. -'o ^(1 -Z2)(l -P22) Jo ^(f> \ 2/ pi^'^)- r^= r^+ r^=^K p.285. Jo A^ Jo ^(p J» Aqi F(k, UTz + p)=2nK + F (k, p). a,mK = ~, a,m2 K = 7t = 2 &m K, a,m{p ± 2 nK) = a,m p ± nn. p. 241. V. snK = l, cnK = 0, dnK = k' p. 245. k2+k'^=l p. 213. VI. TT K'= r ^^ — = r-=M= = F(k',i). . p. 213. ^0 V^(l - z2)(l - A;'2z2) Jo \/l-fc'2sm2 V 2/ Jo VZ Jo VZ Ji \/Z Z= (1 -z2)(l - A;222). VII. sn(fi: + iK') = f , c»i(K + iK') = - ^, dn(K + iK') = 0. p. 246. A; k TABLE OF FORMULAS. 461 VIII. O) S = 4:fi~ g2t - 93. ,., r dt ^ r^'dt Bs ^ " 62 Vs J,, Vs ^he'-e ^0>^ pp. 93, 384. 0)"= CO +(o' p. 215. IX. S>co = ei, «?w"=e2, m^'= 63, . . . p. 216. «?'« = 0, ff'aj"= 0, g>V= 0. . pp. 315, 355. X. K = Vex- 63 w, iK'=Vei- ez(o'. . . p. 201 /(.2_ 62— 63 61-63 /(.'2_ 61— 62 61- 63 . . p. 201 92^ 108 (1 - -fc2+yfc4)3 . . p. 201 gs" [(2-/:2)(2A2-l)(l+F)]2 462 THEOEY OF ELLIPTIC FUNCTIONS. XI. sn(— w) = — swM, p. 245. cn(^— u)= cnu, dn(— u)= dnu. sn(u + K)= ^, p. 245. dnu ,.. , i^x k'snu u, -r xs.y — dnu dn(: u + K) = k' dnu sn{u + 2K) = — snu, cn{u + 2K) = — cnu, dn(u + 2K) = ■■ dnu. sn(u + iK')=- , p. 246. ksnu dn{u + iK') = - ksnu icnu snu sn(u + 2 iK') = snu, cn{u + 2 iK') = — cnu, dn(u + 2 iK') =— dnu. m{u + K+iK') ^'^'^ cn{u + K + iK') -- dn(u + K + iK') -- kcnu ik' k cnu ik'sn u cnu m{u + 2K + 2iK') = - snu, cn{u + 2K + 2 iK') = en u, dniu + 2K + 2iK') = - dnu. TABLE OF FORMULAS. 463 XII. p{u ± 2 oj) = gm, p. 317. KM±^)=ei+ ^^i~^''^(^i~^«l . . . pp.355, 369. g>u - ei (u ± a>')= 63+ (^3-ei)(e3-e2) . XIII. ei sn'^ (Vei— 63 • m) pu = C3+ ^ ^ / ^^ — r» . . . pp. 216, 298. sniVei - 63-u,k)= J^|l_fi , p. 305. VjpM — 63 ^(v^Tir^ . ,y, fc) = ^/^ - ^ i p.307. dn(\/iT^:ri^.w,ifc) = ^^^ p-307. Vpu — 63 (See also formulas LIV.) 464 THEORY OF ELLIPTIC FUNCTIONS. XIV. sn ,. jN isn(u,k') cn{u, k) cn(iu, k) ■- dn{iu, k) -■ 1 cniu, k') dnju, k') cn{u, k') , J ,-, 1 sn(iu, k) I cn(xu, k) cn{u, k') = cn(iu, k) cn(iu, k) sn(iu + K,k) = cn(iu + K,k) = 1 dn{u, k') _ ik'snju, k') dn{u, k') J /• , tr i\ k'cn(u, k') . dn{u, k ) sn{iu + iK', k) -- cn(iu + iK' , k)- dn{iu + iK', k) = — icnju, k') k sn{u, k') — dn(u, k') k sn{u, k') -1 sn{u, k') p. 247. p. 261. XV. p. 246. Function Periods snu cnu dnu 4 K and 2 iK' iK a.nd2K + 2iK' 2K&nd4iK' p. 245. Function Zeros Infinities snu cnu dnu 2 m K + 2 niK' (2m+ l')K + 2niK' {2m+l)K+ {2n + l)iK' 2mK+{2n + l)iK' it {m, n integers including zero.) TABLE OF FORMULAS. 466 sn cn-i dn XVI. u + (0, 1, 2, 3)K + (0, 1, 2, 3)iK' p. 245 1 dnu k cnu - 1 ksnu — dnu kcnu ksnu idnu ik' — idnu - ik' ksnu kcnu ksnu kcnu ikcnu ksnu — ikk'sn u ikcnu ksnu — ikk'sn u k en u kcnu 3iK' snu cnu dnu — snu — cnu dnu — cnu k'snu dnu cnu — k'snu dnu — dnu -k' dnu — dnu -k' dn u 2iK' 1 dnu kcnu - 1 ksnu — dnu k cnu k snu — idnu -ik' idnu ik' ksnu kcnu k snu kcnu — ikcnu ikk'sn u k cnu — ikcnu ksnu ikk'sn u kcnu ■ iK' ksnu snu cnu dnu — snu — cnu dnu cnu — k'snu dnu — cnu k'snu dnu dnu k' dnu dnu k' dn u K 2K 3K sn(u + 2mK + 2m'iK') = {- l)"'snu, cn{u + 2mK + 2m'iK')={- l)™+'"'cnw, dn(u + 2mK + 2 m'iK') = (- l)'"'dn u, {m, ml integers including zero.) 466 THEOEY OF ELLIPTIC FUNCTIONS. XVII. [See p. 368.] sn en dn - 1 - 1 1 1 -fc' 1 1 -1 2iK' Vl+k' -V¥ Vl +k' Vk' Vl +k' -Vk' Vl + k' -Vk' sn en dn — i Vk -Vl +k 1 Vk -iVl -k i Vk Vk - ik' Vk ~Vik' Vk Vk + ik' Vk V-ik' Vk %iK' Vl +A; Vk Vk Vk -Vl +k Vk'(k'+ik) -Vl -k -Vk'(k'-ik) -Vl +k sn en dn + I* -il -iki 1 1 k -ik' k 1 ■ - I + il -ikI iK' Vl -k' -iVk' Vl -k' -iVk' Vl - k' -iVk' Vl -ifc' tVF sn en dn i Vic 1 Vk -iVi-k — i Vk Vk + ik' Vk V-ik' Vk Vk - ikf Vk -Vik' Vk UK' Vl +k Vk -Vl + k Vk Vk Vl +k Vk'(k'-ik) Vl~k Vk'ik' + ik) Vl + k sn en dn 1 1 1 1 k' 1 - 1 1 Vl +k' Vk' Vl +k' -VF Vl +k' Vk' II Vl +k' Vk' u = iK K iK 2K * In the table I = lim • u^O ksnu TABLE OF FORMULAS. 467 XVIIL pf|j= ei+Vei - ezVei- 63, p. 369. ^'(2) ^ ~ ^^®^~ esWei-ez- 2(ei- eg) ^ei- 63, ^'(l "'"'^')"" ei-v/ei- 62 Vei- 63, P'U +w'j= 2(ei- e3)v^ei- 63- 2(ei- 63)^61- 63, «?( yj = 63 - Vez- 63 v'ei- 63, 8?' (yj = - 2 i(e 1 - 63) Vej - 63 - 2 1(63 - 63) Vei- 63, S?fy + wj= 63+ Ve3- 63 Vei- 63, P'(^ + w)= 2i(ei- es) Vcs^ 63 - 2t(e2- 63) Vei- 63, j?(^j= 62- i Ve2 - e3\/ei - 63, ^'(^)"'~ ^^^^ ~ ^^^ ^^3- 63 - 2i(62- 63) Vei- 63, ^V ^ ^ ) = «2+ i Ve2- 63 V61- 63, pY'^^^^') = - 2(61- ea) ^62- 63+ 2 1(62- 63) Vei- 63. (Halphen, i^onci. fiWip., Vol. I, p. 54.) 468 THEOEY OF ELLIPTIC FUNCTIONS. XIX. sin itu = -itu _ ^Il'Ul--V'" ' P- 18. m f \ ml ) ;rcotm^ = i + y'S— i— + ^j, p. 20. M ^-f (u — m m) m=+oo sin^jTW ■^ (u — m) XX. 0i(w)= X 9""e-« =l + 2gcos^+2g4cos^+2g9cos^+ • . • 77l = — C30 p. 220. gY^-^V 1 + 2 g cos 2 m + 2 g* cos 4 ii + 2 g9 cos 6 m + ■ • • , m-+a) (2m+iP (2m+l)irtu _ 2if!C Hi(m)= y 3 * e ^K =2V5cos;^ + 2V _rz^ 2 a. 4- ^„ 4,-r 3;j|^ cos- m=-« 5 ;rM + 2^/525eos|^+. . ., j^/2Ku\ = 2 \/g cos M + 2 -V^ cos 3 M 4- 2 'v'g^s cos 5 m + ■ ■ • , @{u)=@i(K -u), H(w) = Hi(X -m), . . p. 221. e /?^'\= 1 - 2 g cos 2 M + 2 g* cos 4 M - 2 g9 cos 6 w + ■ • • , jj/2_Km\ ^ 2\/5smw - 2 Vg^sinSw + 2 Vg^sinSu - • • • . H is an odd function; ©, ©i, Hi are even functions. TABLE OF FORMULAS. 469 XXI. u 1 uJ . w = 2 fioj + 2 (i'(i)' (i,li'=Q,±\,±2, ■ w f^O . . p. 319. . pp. 318, 324. du u ^~< (u — w w w^) ^ =_ ^'w = -|^log ■ • • p{- m)= «m, p<-''H~ u)= (- l)V"Hw)- XXII. W 1 sv^/ 1 1„7 V 1 . . p. 315. . . p. 323. . . p. 298. . p. 324. . p. 323. . p. 323. . p. 323. XXIII. C(u + 2w)==C" + 2ij, C(w + 2w')=Cw + 2'?'- pp. 303, 338. , = C^, T,' = C^', r^-ri + i- ■ ■ ■ P-301. 470 THEORY OF ELLIPTIC FUNCTIONS. XXIV. ei(M + K) = e(M), ei(w + iK') = ahi(u),, pp. 222, 223. B.i{u + K)=- H(m), Hi(M + iK')= ^@i(u), @{u + K)=@i{u), @{u + iK')=U'a{u), H(u + K)=Hi(m), K{u + iK')=iX@iu). X = A(m)= e 4K' 0i(m + K + iK')=iXIl(u), Siiu + 2iK')= fi@i{u), B.iiu + K +iK')=~iX@{u), Bi{u + 2iK')= fiRi(u), e{u + K + tK')==AHi(M), e(w + 2iK')=- fi@{u), n{u + K+iK')= k@i{u), B.{u + 2iK')=^-fiB.{u). 0i(m + 2mK)= 0i(m), 0i(m + 2miK')= A@i(,u), Hi(m + 2miC)= (- 1)™Hi(m), Hi(m + 2miK')= AHi(m), 0(m + 2 toK) = 0(m), ©(m + 2 miK') = (- 1)'»40(m), H(u + 2mK) = {- 1)'"H(m), H(m + 2miK')= (- l)'"^H(u). _ if' Trtiri mht -= =- 1 A = Aiu)=e ^ ^ TABLE OF FORMULAS. 471 XXV. H(2gi>)iz,"' \/;r^ a{u, oj, CO') = ^4:,^, ^ e^ " ,{u = 2ayv),. pp. 378, 304. i± (0) au) aw tii(O) ( o77. 02U = e-" "-^ 7~ = e" "-i^ ;^= a2iu,(o,a)')= "} ' e^" . aw aoj' 8 1 (0) _„,„a(aj'+M) ,,„a(w'-M) , ,v 9(2 Kt)) g::"' iT/jl /T/.l' MMll XXVI pp. 340, 380. a{u + 2aj)=- e2'("+'"> au, a{u + 2 w") = - e2''"("+"'"> au, (7i(m + 2w) = - e^'t^+^^CTiw, <7i(m + 2w")= e2 '"("+"""> (TiW, <72 (m + 2 w) = e2 '(»+<") aaw, ^2 (« + 2 w") = - e^ ■>"(«+-") (TjU, <73(m + 2 w) = e2'>("+'') azu, asiu + 2 w") = e2'"(«+<"") ajw- a(M + 2 w') = - e2''(»+-') (m, + 2 nw' a2U (2 wi + l)w + (2 n + l)w' asU 2m^ +(2n + l)w' au 2 mu) + 2 ncj' (to, n integers including zero.) \/jpM — ei=^^; VgJM — 62 = ^^ ) Vgm — 63 = (T3U au ?'m = - 2 CTlM (72M g3tt cru i7U cm 473 p. 373. p. 380. XXXII p. 384. Vei- 62= -^ = -' ^ei- €3= -^ = r ' aw acDao) au) awau) Ve2- ei= -i-— = ;;; Ve2- 63= -^-77 = ; — 77- aujau) , / ffiw e I" au) ./ — a^oj (72<^^ _ ei'^'au) a(i) au)auj \/e3-e2= — i Ve2— 63, Ves- ei= — i Vei — 63, Ve2- ei = — i v ei — €2, where r(^)>0. 474 THEORY OF ELLIPTIC FUNCTIONS. XXXm p. 230. t9o(M + i)= Mu), Mu + 1) = Oo(u), «?l(w + i)=!?2(w), ??j(M+l) = -t?i(u), Mu + i) = -^i(.u), Mu + l)=-Mu), t?o (" + l) = '=^'!*i("). ''^oCw + r) = - BMu), ^i (" + ^) = ^^«^o(w), «?i(M + t) = - B«?i(w), t?2 (m + l) = -i'^a (m) , ^2 (w + t) = St?2 (w) , «93 (" + l) = ^''^^(w), t93(M + t) = 5^3(m), do (u + ^-^) = A«?2(m), «?o(m + m + nr) = (- l)»C«?o(«), «9i (m + i-|-^) = Adsiu), t?i(M + m + nr) = (- 1)»+'"C«?i(m), t?2 (m + ^-^) = - ^^«^o(w), Mu + m + nr) = (- l)'»C«?2(w), t93 (w + ^-Y^) = iAt?i (w), «?3(m + w + nr) = CTgCw)- C = q-n'g-2nnu TABLE OF FORMULAS. 476 XXXIV. p. 386. a{u±(o) = ± e-''"ffW(TiM= ± 7— ^:^^^^===;e*'("-i'"'<7iM, Vei-e2 Vei — 63 4/ (72(u±w)= Vei-e2e*"'(TU. [Schwarz, Zoc. ci^, p. 26.] 476 THEORY OF ELLIPTIC FUNCTIONS. XXXV. . . . pp. 220, 229, 378, 397. 7n = l m=a) (2m + l)' «?i(w)= X(-l)"'2g * sin{2m+l)nu = 2<^Bm.i:u-2(^s\n^Tzu+ ■ ■ ■ , m=0 m=ot> (2m+l)' t?2(M)= ^2q ■* COS (2m+l)7rM = 2gi cos7rM + 2 5^cos3 ;rMH- m=0 «?3(w) = l+ X 25'"'cos2mmi = H-22Cos2;rM + 2o4cos47rw+ • m=l XXXVI p. 230. m=oo «9o(w) = QO U (1 - 2 ^^m-lcog 2 TTM + ?*"-2), 771 = 00 t?i(M)= 2Qo9*sJnTO U (1 - 2 52m cos 2 m^ + g^"), m = l t92(w) = 2 Qo9^ cos ;rM JJ (1 + 2 52m cqs 2 ;rM + g*"), 771 = 1 m = oo t^sCw) = Qo H (1 + 2 g'Zm-i cos 2 TTM + g^^-Z). m = l XXXVII p. 396. m = CO 771 — 00 m=-l 771 — 1 m — 00 7n=>i» "1 = 1 m = l QiQ2Q3= 1, 16gQi8=Q2»-Q3^. • . pp.396, 409. TABLE OF FORMULAS 477 XXXIX. If » = ;r^' 3 = e-"% t = — , o = e"'= h, 2(1) (o OJ CO^iiz- 3-1)2 ^^ (1 _ ;i2m2-2)2 ^ ^-^^ (1 _ ^12^22)2 p. 336. \ 2fe2mg-2 ^ ^ -; 1 -/l2m2-2 2/2 _fi2mz2\' i m=l ' p. 337 1 771=00 TT 2t A-L 1 _ (>2m ±i 1 _ g2 m = l ^ m^l i 2(il Z — Z-'- „ - 1 — /,2m--2'" = '" 1 _ „2m~2 ^_£C<;£ £_g2,»t.» JT J^ g ^ TT J: 2L_^ . p. 341. ,1 m-cB . „2m.-2 ' "=°° 1 I „2m.2 ( QAO ai(w) = ?Jl^_ e2„^' -rj 1+? / XT ^+? ^ . pp. \ ^'^^' 2 11 l+g2". 11 l+(;2m ^^^ ) -^yfl. „2,«.^ V^ 1 + 2(72 r. COS 2;r^ + (; „.=! - ■ r" m=i i + e^"" <379. m-" 1 1 n ^•>.m o — J. (.4m (1 + g2".)2 7n = l cosTTve^'""' TT -1 m=l ' I m-1 I p2ij(ify^ '"=«' 1 +^2m-lg-2 "^ 1 4-?2'"-l; «2U = e-'- JJ ^^ ., n T^ «2m-l £\ (l+g2— 1)2 •"-" 1 _ ^2m-l-,-2 "'°°° 1 _ „2m-1.2 <7qm = e2"^' TT ^!^ 2 ± — TT i 2 L O3U e 11 J _ 2m-l 11 1 _ g2m-l " 11 (1 _ o2m-l)2 478 THEOEY OF ELLIPTIC FUNCTIONS. XL pp. 397, 400. m=(» (2m+l)' «9i'(0)=27r 2) (- l)'"(2m + l)5 * = 2 ;r(gi- SgS + 5?* - • • •) = 2K / 2Kkk' _ t9o(0)=l + 2 ^ (-1)'»2'»'= 1-25 + 254- 2g9+ ' • • =SJ-^' m = l ^92(0)= 2 ;^g * =2gi+2gS+2g^+ ■ • ■ =yiii^, m=0 t?3(0)= 1 + 2 ^g'"'=l+2g + 23*+2g9+ • • • =y m = \ XLI See p. 397. t?o'(l) = = t92'(l) = i^s'd), '9i'(l) = - »9i'(0). «9o'(i) = = t9i'(i) = ^3'(i), ^2' a) = - «?i'(0). XLII See pp. 397, 411. ?92'Q= - ^>3-*t?3(0), «?2' (^)= - '^9-*'?o(0), *93'(|)= - ^•'^3-*^2(0), ,?3'(^)= ^•5-i^l'(0). ?9o'(T)=2i7rg-i?9o(0), ??o'(wi + nr) = (- l)"+i2nOT 5-»'i?o(0), ??i'(T) = -g-it9i'(0), ??i'(m + nT) = (- l)'"+«5-"^«9i'(0), ?92'(t)= - 2i;r5-iT?2(0), t92'(w + nT) = (- 1)"*+ ^2 n7!-i5-"'?92(0), «?3' (t) = - 2 inq - ' t^s (0) , ?93' (m + m:) = - 2 nmg ""'j^s (0) , t9i'(0)=2;rQo¥, '?o(0)= QoQs^, • pp. 397, 399. '?2(0) = 2 QoQi V, 'JsW = Q0Q22. TABLE OF FORMULAS. 479 XLIII. pp. 385, 410. a(o = gitlm y/ieii Vei— 63 Vei- 62 Ve2- 63 Vei— 62 «+ 1 (0) L p. 409. TABLE OF FORMULAS. 483 LI. TT Ve2-e3 Vei— 63- VeT ■62 Vei — 63 Vei— 63+ Vei — 62 :(25 + 2g9 + 2525+. . .), p. 408. (l+25* + 2gi6+. . .), o 2 1 V 4J 4/j2m 6 ^ (1 -;i2m)2^ U "A" 492-" ) 2)?W =- 261^2+ ;r2^- + 2^ (1 ^ „2m)2 ( ' "^^ 4o27n-l 2 JJW = - 2 63^2- 7:2 2^ ^J^ g2m-l)2 2lJW = - 262^2+ ;r2 2) (1 + q^rn-l)2 92 m = l ni = l p. 409. p. 336. p. 379. ,2.3, -2«i-^-i"tw=-'^^"'+ 2ri^3{u) _. v ei —e-j multiplied by a2(,u) Vei— 62 A coam(Vei —63 • u, k)^ o-iU_^- — cnjV 61-63 •u,k) sn\y 61—63 • u, k) au 02U au -\/~P — t^wCx/ei-ea • u,k) sn\y6\—63 • u, k) 1 031* . / -^— = Vei— 63 ou sn(\^ei — 63' u,k) OiU N/fii - multiplied by es a2U_ tga,m(Vei-e3-u,k), 1 (jiM sin coam (Ve 1—63 • M, A;) 03U_ a\u cosam(V'ei— 63 • M, A;) [Schwarz, loc. ciL, p. 30.] TABLE OF FORMULAS. 486 LV. Homogeneity p. 343. a{Xu, /io), ho') = Xa{u, at, co'), ^(Aw, XlU, Xco') = - (!;(m, (x), 0)'), p{}.u, ko, W) = — p{u, w, co'}, j?W(>lw, ho, W) = ^2 ^(")(M, ^, CO'), g2{ho, ho')= }.-^g2(,(o,(o'), gziXo), kco')= X-^gs{o),(o'), 486 THEORY OF ELLIPTIC FUNCTIONS. LVI p. 252. sn"u = — (1 + k'^)sn u + 2 k^srfiu, cn"u = {2A:2— i)cnu — 2k^cv?u, dn"u = (2 — k^)dn u — 2 dn^u; sn^u = (1 + 14fc2+ k'^)snu - 20F(1 + k^)snH + 24k*sn^u, cn^*^u = (1 - 16k^+ 16k*)cnu + 20k^(l - 2k^)cnH + 24 k'^cnH, dnWw = (16 - 16/c2+ k*)dnu + 20 (k^- 2)dv?u + 24 dn^u. (See also Formulas II.) LVII p. 252, et seq. SnW = M- (1 +/b2)H!+(l + 14A;2 + /c4)^_ . . . , 31 5! sn'(0)= 1, - sn"'(0)= 1 + fc^^ sn<^HO)= 1 + 14fc2+ fc*, - sn!7'(0)= 1 + 135 F+ 135/1;*+ fc^, s«(9)(0) = 1 + 1228 /c2+ 5478 A;4+ 1228 k^+ k^, cnM= 1-^1+ (1+4 fc2)|!_ . . . , cn"(0) = - l,cnW(0)= 1 + 4A;2, - cn(6)(0)= 1 + 44/b2 + 16 fc* cnW(0)= 1 + 408 /e2+ 912^*+ 64 A;^, - cn(io>(0)= 1 + 3688 A;2+ 30768 A;* + 15808 /c^^. 256^8, dnw = 1 - ^^ + F(A;2 + 4) ^ - • • • , 2! 4! dn"(0)= - A;2, dn(4)(0)= /fc2(/i;2 + 4), - dnW(0)= F(&4+ 44A;2 + 16), dn<8)(0)= P(/fc6+ 408 A;* + 912 F + 64), -dn(io>(0)= /c2(fc8+ 3688 fc«+ 30768 A;* + 15808 F + 256), [Gudermann, Crelle, Bd. XIX, p. 80.] kK 2 Ku ^ Vg siim , \/q^ sin 3 m , Vg^ sin 5 m sn + ^ 3— + ^ 5— + • • • . P- 256. 2;: ;r 1 - g 1 - g^ i _ g, A;K 2 Ku _ Vg cosu Vg^ cos 3u V'gS cos 5m — Cn-^- 1 +g + 1 ^_g3 + 1 +g5 + K 1 2 Ku ^ 1 , g cos 2m I g2 cos 4m , g^ cos 6m 2^ ^ n 4 1 + g2 1 + g* 1 + g^ "^ ' TABLE OF FOEMULAS. 487 LVIII. gj^* W (? « 22**''* 22.7^ 23.3-5'^ 24.3-5-7 [See Art. 377.] LIX. 0U = 4; + C2U2+ C3U4+ C4U6+ • • • + c„m2»-2+ ..... pp. 326-8. ^m357 2n- 1 C,= 3 f?2g3 , 22! + 23! , . . . , ^ 2* -5 -7. 11 ^ 25.3.53.13 2* -72. 13 (n — 3)(2n + l)Cn = 3[C2C„-2 + C3C„_3+ C4C„-4+ • ■ • +C„-2C2] • • •p.327. [71 > 3] LX. aai=\- \e,v? -^{& 6^2- g^)u*- ■ ■ ■ . . p. 394. 2 48 (A = 1, 2, 3.) 488 THEORY OF ELLIPTIC FUNCTIONS. LXI pp. 236, 246. (a- 1,2,3; ??4= «?o)- ^2^3^l(u + V)MU - V) = ^i{u)do{u)d2(v)l}3{v) + ^2{u)i}3(u)l}o(v)l}l(v), ^2^3Mu + V)^3{U - V)= ^2{u)dz{u)^2{v)&3{v)- ^o{u)^l{u)-9o{v)&i{v). ^2^oMu + v)§o{u - -«)= !?o(m)??2(w)!?oW«92W- !?i(m)??3(m)?9i(v)!?3(1)), ^2^0^1{U + V)^s{u ~ V)= ^i{u)d3(u)do{v)^2{v)+ ^o{u)d2(u)l}i{v)^3{v). ^3^oMu + v)do(u - V)= ■&o{u)d3{u)^o{v)^3{v)- ^i{u)^2{u)^l{v)d2{v), ^39o^i{u + v)§2{u - v)= ■»i{u)Mu)Mv)Mv) + ^oiu)^3{u)^^{v)^2{v)- !93%(2M)=t93*(M)+«?i*(M), p. 237. «93''?ot?o(2 u) = ^3Hu)^oHu) + ^2^(u)^iHu). TABLE OF EOHMULAS. 489 LXII. a{u + Ui)a(u — Ui)a{u2 + U3)a(u2 — U3), + aiu + U2)a{u — U2)(j{u3 + Ui)(r{u3 — Wi), + a{u + Us)a{u — U3)(7{ui + U2)a{ui — U2)= 0. . . p. 390. a {u + v) a (u — v) = a^u ax^v — a)^u a^v, p. 391. {ev—e^a{u + v)o{u — v)= a^ua.?v — a^^ua^v, <7(m + ■u)<7i(M — v)= a^ua^v — {ex - e^)(ei— e^a^ua^v, (e„— e^)ax{u + v)ax{u — v) = (ex— e^)a^^uaj^v ~ {ex— ej)a^^ua/v, ox{u + v) ax{u — v)= a^u a^v — {ex - e^) a^u aj^v, ai,{u + v)ai{u — v) = ai^ua/v — {ex— e^)a^^ua^v, ax{u + v)a{u — v) = axu au a^v a^v — a^'ii a^u axv av, a{u + v) ax{u — v) = axU au a^v a^v + a^u a^u ^) + 4 g)32t - q^pu - gsT p'u p'v 2{pu - gw)2 ^ _l_ (6 ffl^t; - \ q-z) {pu - pv) + 4: pH - q2m> - .93 T jp'w ffl't; ^ p{u ±v)= 2(!i?m^ - 7 ga) (^M + H - g3 T fp'up'v .... p. 353. 2(j3W - jw)2 4 L g^w - »w J 367. 1 ^ 2(ji;>Mj?i;- ig2)(^tt + ipv)- Q^ ± p'ufp'v .... p. 355. piu±v) 2{pupv + ig2)^+ 2g3is>u + pv) 492 THEORY OF ELLIPTIC FUNCTIONS. LXIII (Continued). { 1 ± sn{u + v) 1 1 ± sn(u + v) \l ± k sn{u + v) \ 1 ± ksn(u + v) j 1 ± cn{u + v) 1 1 ± cn{u + v) j 1 ± dn{u + v) { 1 ± dn{u + v) j 1 ± sn{u — v) \ 1 T sn{u — ■«) { 1 ± ksn{u — v) jl T fc sn{u — v) f 1 ± cn(M — V) \l -f cn{u — v) 1 1 ± dn{u — v) { 1 T dn(u — -y) = {cnv ± snudnv)--h D, = (cnM± snvdnu)^-^D, = (dni; ± k snucnv)^^ D, = (dnu ± k snv cnu)"^ -^ D, = (en M ± c?i ■y)^-e- D, = (snu dnv T snv dnu)^-i- D, = {dnu ± dnv)^-^ D, = k^(snucnv T snv cnu)^— D. sn{u + v) cn(u — v) = {snu cnudnv + snv cnvdnu) -h D, sn{u — v) cn(u + v) = (snu cnu dnv — snv cnv dnu) -^ D, sn{u + v)dn{u — v) = {snu dnu cnv + snv dnv cnu) -^ D, sn{u — v)dn{u + i») = {snu dnu cnv — snv dnv cnu) -e- D, cn{u + v)dn{u — v) = {cnu dnu cnv dnv — k'^ snu snv)^ D, cn{u — v)dn{u + v) = {cnjidnu cnv dnv + k'^ snu snv) ^ D. sin { am(M + v)+ am(M — v)\= 2 snu cnu dnv -^ D, sin {ani(M + v)— am(M — v) \ = 2 snv cnv dn u -i- D , cos { am{M + v) + am(M — v)\ = {cn^u — sn^u dn^v) -;- D, cos{ am(M -\- v)-a.m{u — v)\ = {cn^v - sn^v dn^u) -^ D. (Jacobi, Werke, I, pp. 83-85.) TABLE OF FORMULAS. 493 LXV. = 2 83M - r-^ log {pu - §w), g2 = 2 jfw - — log (^M - gro)- «?(w + «) - g>(w - w) = - P^^^ = _ ——log {ffU - gw), (pu — pv)^ duov P{U + VMU ~V)= (^^^^ + ig2)^+g3»U + H , \pu- pv / \p{u + V)- pv / 1, j?(M + 1;), - J?'(W + ?)) I »"*! 1, jfw, 0. . p. 354. Pi2u)= (^^^ + i^2)^+ 2g3^ ^1 |ilogg,X . . P.355. 4 j?% - gf2 gw - ff3 4 dw2 C(2w)=2CH + i^, 2 p'u a{2u) = aHi ^^ ^°^ '^ = 2 ffM((/w)3- 3 )= tt^^ - mq — + (2n + l)m. . p. 420. OUq OU Addition-theorem on p. 429. LXX. E{u)du = log a{u), p. 423. ii(iu)=e ^ cn{u, k') Cliu. k') , .... p. 424. (Vei- e3-u)= ei«-«' , . . p. 424. 2 fi(M + a) n(Vir^3 . u, ViT^^ ■ a) = 1 log '^^i^ '"l +u ^. p. 425. 2 ... p. 439. a{u — Ui)a{u — U2) ■ ■ ■ a(u — Ur) where Ui°+ M2^+ • + u/'^ ui+ U2+ ■ • ■ + Vr. .ni^cuLUNiVERSriYLIBRAH- OCT U '^ 1991