.
LIB
LECTURES
ON THE
THEORY OF ELLIPTIC FUNCTIONS
BY
HARRIS HANCOCK
PH.D. (BERLIN), DB. Sc. (PARIS), PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CINCINNATI
VOLUME I
ANALYSIS
FIRST EDITION
FIRST THOUSAND
NEW YORK
JOHN WILEY & SONS
LONDON: CHAPMAN & HALL, LIMITED
1910
-
COPYRIGHT, 1910,
BY HARRIS HANCOCK
Stanbope ipress
H. GILSON COMPANV 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. Abel in a published letter to Legendre wrote: "Si je suis assez heureux pour faire quelques decouvertes, je les attribuerai a vous plutot qu' a moi "; and Jacobi wrote as follows to the genial Legendre: "Quelle satisfaction pour moi que rhomme que j'admirais tant en
iii
781472
IV THEORY OF ELLIPTIC FUNCTIONS.
de"vorant ses ecrits a bien voulu accueillir mes travaux avec une bonte* 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 upon 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 PREFACE. V
matics. In this volume the theory becomes 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 functions; 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 into 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 developed 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 elliptic 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 whole 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 denning 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. Poincare 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 THEORY 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 functions 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
INTRODUCTION. 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 Fortneln 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-operation which has minimized my labor during the progress of printing.
HARRIS HANCOCK. 2415 AUBURN AVE., CINCINNATI, OHIO, Nov. 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 Analytical 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
xi
xii CONTENTS.
Analytic 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 such functions in general. The one-valued functions. Rational functions of the unrestricted argument u. Rational functions of the
•niu
exponential function e w .
32. Examples of functions having algebraic addition-theorems ........ 33
33. The addition-theorem stated .................... 34
34. Meray's eliminant equation ..................... 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 Meray's equation ........... 43
The Discussion Restricted to One-valued Functions
40. All functions which have the property that <j>(u + v) may be rationally ex
pressed through 4>(u), <£'(w)j <£(v), <}>'(v) are one- valued ........ 44
41-45. All rational functions of the argument u; and all rational functions of the
Urd
exponential function e w have algebraic addition-theorems and are such that
where F denotes a rational function ............... 45
46. Example showing that a function <!>(u) may be such that <j>(u + v) is ration
ally expressible through <}>(u), <j>'(u), <j>(v), <j>'(v) without having an alge braic addition-theorem ...................... 52
Continuation of the Domain in which the Analytic Function <j>(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 <j>(u) may be extended to all finite values of the argu
ment u, without the function $(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
CONTEXTS. 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 93
80. Equivalent pairs of primitive periods. Transformations of the first degree . 95
81. Preference given to certain pairs of primitive periods 96
82. Numerical values 97
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 Kh order 109
91. The zeros 110
92. Their number within a period-parallelogram Ill
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 "problem of inversion " 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 Surf ace s2 = R(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-f unctions again introduced 171
149. The sum of two integrals whose upper limits are points one over the othei on
the Riemann Surface . 172
xvi CONTENTS.
ARTICLE PAGE
150, 151. The upper limit expressed as a quotient of Theta-f unctions 172
152. Resume 173
153. Remarks of Lejeune Dirichlet 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 Normal 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 co variants 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 and K' 212
CONTEXTS. 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 JACOBI 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-Functions 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 fundamental theorem 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 a, dn u
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
V
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 not 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-f unction derived from those of the zeta-f unc
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
CONTEXTS. xix
ARTICLE PAGE
257. Derivation of the sigma-f unction from the theta-f unction 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-f unctions and Jacobi's elliptic
functions expressed through sigma-functions 304
263. Jacobi's zeta-function expressed through Weierstrass's zeta-function . . . 307 Examples , . . . 303
CHAPTER XV
THE WEEERSTRASSIAN FUNCTIONS ^n, £n, 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 i 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 pah- of primitive periods . . .* 317
The Sigma-Function
272. The expression through which the sigma-f unction 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-f unction 323
The fzz-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 u 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-f unction 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 addition-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-function 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 elliptic 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-functiona 381
CONTENTS. xxi
Differential 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 sigma-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-f unction ................ 391
Examples ............................. 394
CHAPTER XVIII
THE THETA- AND SIGMA-FUNCTIONS WHEN SPECIAL VALUES ARE 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
xxii CONTENTS.
ARTICLE PAGE
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 series 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 FINITE 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 fr.r.ction, 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 + iij 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 denned.
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
=/(a)
the exponents 1, 2, . . . , n, . . . being positive integers.
The power series on the right is denoted by P(u — a). Any such point a is called an ordinary or regular point of the function, and the function is said to behave regularly f in the neighborhood of such a point. At these points the function has the character of an integral function.
Zeros. — If the function f(u) 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) ^ 0, the point a is a simple zero, or a zero of the first order. If the derivatives f'(a), /"(a), . . . , f(n~^(a) are all zero, while /<n)(a) ^ 0, the zero u = a is of the nth order. In the latter case the function f(u) may be written
f(u) = (u-a)ng(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 monogines uniformes, Acta Math., Bd. IV, p. 3.
t " Ich sage von einer eindeutigen definirten Function einer VeranderUchen u, dass sie sich in der Ndhe eines bestimmten Werthes UD der letzteren regular rerhalte, wenn sie sich fur alle einer gewissen Umgebung der Stelle u0 angehorigen Werthe von u in der Form einer gewohnlichen Potenzreihe von u—u^ darstellen lasst." Weierstrass, Werke, 2, p. 295, 1883.
1
2 THEOEY OF ELLIPTIC FUNCTIONS.
whe/e g(u) i« 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)
ART. 2. Singular points. — If the one-valued function f(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
where n is a positive integer and where <f>(u) is a regular function at the point a and <f>(a) ^ 0. The function <j>(u) may be expanded in a con vergent power series of the form
so that/(w), when expanded in the neighborhood of u = a, is
(^ - a)n
where F(u) is a regular function in the neighborbood of u = a. The constants An, An-i, . . . , AI are determinate, .An = $(«), etc. The integer n is the order or degree of the pole.
The coefficient AI of • - is the residue relative to the pole a and
u — a
An ! An-i + . . . _j_ Al
(u — a)n (u — a)71-1 u — a
is called the principal part of the function relative to the pole u = a.
ART. 3. Essential singular points. — In the neighborhood of such a point, the function is completely indeterminate. Consider,* for example, the function
__
__
eu-a U u-a 21 (u- a)2 3! (u - a)3
in the neighborhood of the point u = a.
* Cf. Hermite, Cours redige par M. Andoyer (Quatrteme edition, 1891), p. 97.
PRELIMINARY NOTIONS. 3
If a + ip be any arbitrary point whatever, then it is always possible to give to u — a a value £ + it) as small as we wish, such that
i 6*+*' = a + 1.
For writing a + i0 = ep + l'9, the preceding equation becomes
It follows at once that
sr\
and
From this it is seen that £.and y are completely determined. On the other hand the proposed equation is satisfied if for q we write q + 2 kit, where k is an arbitrary integer, since 2 IT: is the period of the exponential function. Thus since q may be increased beyond every limit, the quan
tities £ and T? are susceptible of becoming as small as we wish.
i
The origin is an essential singularity of the function eu. A character istic distinction between the poles and the essential singularities is: If we take the inverse of the proposed function, the poles are transformed into zeros; while an essential point remains an essential point, the recip rocal of the function in the neighborhood of such a point being as the function itself completely indeterminate.*
In the present theory we have to treat such functions which have poles as the only singular points in the finite portion of the plane.
ART. 4. Remark concerning the zeros and the poles. — If the point a is a zero of order n of the function f(u), it is a simple zero with residue n
in the logarithmic derivative ^ •
/W For in the neighborhood of u = a we have
f(u) = (u - a)n g(u), where g(a) j^ 0. It follows that
f(u) u - a g(u)
** being a regular function at the point u = a. 9W Similarly it is seen that if u = a is a pole of order m of the f unction /(u),
it is a simple pole of residue — m for ^ ^ ' •
* Briot and Bouquet (Fonctions EUiptiques, p. 94) employ what seems a more appro priate name, "point d'indcter mi nation."
4 THEORY OF ELLIPTIC FUNCTIONS.
For writing
(«-•)•'
we have £$ + -!*.+%&,
f(u) u - a G(u)
C" (n^\
where • — ^-~ is a regular function at the point u = a. G(u)
ART. 5. The point at infinity. — If we write u = — , a definite point in the
v
w-plane corresponds to a definite point in the v-plane, and vice versa. The infinite point in the ^-plane corresponds to the origin in the v-plane. Hence if the function f(u) is regular at the point u = GO , the function
/(- ) must be regular at the point v = 0. It must consequently for small
W values of v in the vicinity of v = 0 take the form
= #0 + a\v + a2v2 + • • • = P(v), say,
where the a's are constants. It follows that for large values of u we must have
u u2 un~l un
If the function is regular in the neighborhood of the point oo , the infinite point is a zero of the nth order, when a0 = 0 = ai = • • • = an_1;
an 7^ 0. This function then vanishes at infinity as — (where u =00).
un
The point at infinity is a pole or an essential singularity of the function
f(u)y when v = 0 is a pole or essential singularity of /( - ). If u = oo is
\v/
a pole, we must have for small values of v
^2 + . . .+An + CQ+cv + cv2 + . V2 Vn
where the A's and c's are constants; or, for large values of u,
f(u) = Am + A2u2 + - - - + Anun + c0 + ^ + ^ + • • • .
u Ur
The part AIU + A2u2 + • • • + Anun, which becomes infinite at the pole u = oo , 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 u\(z) + U2(z)+ u%(z)+ - • - , the individual terms of which are functions of z defined for a fixed interval, converges uniformly within this interval, provided there exists an absolutely convergent series,
Ml + J/2 + - • ' ,
where the M's are quantities independent of z and are such- that within the fixed interval the following inequality is true:
j iin(z) , = Mn, where n = «.
/JL being a fixed integer. (See Osgood, Lehrbuch 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 u = 0 is developable in the series
f(u) = a0 4- a\u + a2u2 4- • • • = P(u), 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
This function being by hypothesis regular in the neighborhood of infinity, can contain no negative powers.
It follows that a i = 0 = a2 = a3 = . . . , and consequently
/(") -
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 Poly technique, l^re Partie. Analyse Algcbrique, Chapitre 9, § 2, Theoreme 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. In the latter case by "within the circle of convergence" we understand "within any interval that lies wholly within this circle."
6 THEORY OF ELLIPTIC FUNCTIONS.
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 a 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 f(u) 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 u = 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
/(«) =
(u - 1) (u - 2)
\ / \ - /
which is regular at all finite points of the plane except the points u = 1 and u = 2. These points are poles of the first order. The principal part
=
.
of f(u) relative to the pole u = I is
say,
* See Appell et Lacour, Fonctions Elliptiques, p. 7.
PRELIMINARY NOTIONS. 1
as is seen by noting that the difference
f(u) - 0i (M)
is regular at the point u = 1. The residue relative to the pole u = I is — 1. Similarly the principal part relative to the pole u = 2 is
*»(«) = —^-5,
i£ — 2 with the residue 2.
At the point u = «D the function is regular, for
(1 - v) (1 - 2 v)
is a regular function at the point v = 0.
It is further seen that v = 0 or u = oo is a simple zero. The function f(u) has then two simple poles u = 1, u = 2 and two simple zeros u = 0, u = QC . The function is said to be of order or degree 2.
It may also be observed that the equation
/(«) = c
has two roots, whatever be the constant C. Further, since the functions 0i(u) and 02 (u) are everywhere regular except at the poles u = 1, u = 2, the difference
/OO - 0i 00 - 0200
is a function that is everywhere regular. It is therefore a constant, and since /(M), 0iOO> 0200 all vanish for M = oc , this constant is zero. We therefore have
/OO = 0iOO + 0200,
a formula, which gives immediately the decomposition of the rational function f(u) into its simple fractions.
ART. 10. The general case. — A rational function
/OO '
1 + • • • + bn Q(u)
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 QOO = 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 4- 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 m zeros and m poles. We say that it is of
8 THEORY OF ELLIPTIC FUNCTIONS.
order or degree m. The equation/(w) = 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 Q\(u) = 0 and n — m zeros at infinity. The function is of order n and the equation /(ti) = C has n roots.
3°. If m = 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 f(u) = C, where C is an arbitrary constant, has a number of roots equal to the order of the func tion f(u). In particular we note that the rational functions have only polar singularities.
PRINCIPAL ANALYTICAL FORMS OF RATIONAL FUNCTIONS.
ART. 11. First form: where the poles and the corresponding principal parts are brought into evidence. Decomposition into simple fractions.
Let 01, a2, . . . , av be poles of order ni, n2, . . . , nv of the function f(u) and let the principal parts with respect to these poles be
u — av (u — av)2 (u — av}nv
Further for the most general case, suppose that the point oo 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
(f)(u) = AIQU + A2ou2 +••••+ Asous,
where s = m — n is the order of the pole.
Since each of the principal parts is everywhere regular except at the associated pole, the difference
f(u) — <£i O) — 02 (w) — • • • - 0(w)
is regular everywhere including infinity and consequently is a constant, = A, say.
PKELIMINARY NOTIONS.
It follows that
f(u) = A + Alou + A2ou2 + • • • +
ffi \« - ai (u ~ °f)2 (u ~ a>i]
where the index i refers to the indices of the poles ai, a2, . . . , av. This
formula may be written in a somewhat simpler form if we symbolize - by
u
u — a0, where a0 = », and let no = s. We then have
f(u) =A
where the summation index i refers to the indices of the poles ai, a2, . . . , a*, a0.
If we put - = v,-, we have finallv u — ai
- A
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 Qi(u) and Q(u) of the preceding article into their linear factors, so that
f(u) = C (^ ~ ci) (M - c2) . . . (M - cm) (u - 61) (M - 62) ... (M - 6,)'
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
.
f(u) u — GI u — c2
1 1
u — bi u — b2 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 f(u) has an algebraic addition-theorem; that is, if u and v are two independent variables, f(u + v) may be expressed algebraically in terms of f(u) and/(v).
* Cf. Appell et Lacour, loc. cit., 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 a0 + a\u + a2u2 + • • • + anun = 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,
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
U = 0, ± 71, ± 2 7T, ± 3 7T, • - • .
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
sin nu
7tU
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 f(u), 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 ep:u\ where P(u) is a power a
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 a\, a%, a3, . . . 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
mod di
PRELIMINARY NOTIONS. 11
formed by the inverse of the moduli of the roots, is convergent. The same will (as shown below) also be true of the series
mod (a; — u) whatever the value of u, excepting the values u = (LI, a2, . . . , which
l=X>
i
make the series infinite. It follows then that V - will represent an
fl u - at
analytic function in the whole plane.*
To prove the above statement consider the two infinite series 2un and Stfn, of which the first is convergent.
The second series will also be convergent if we have
vn < kun (k constant)
for all values of n starting with a certain limit. If we write Un =
i
, the condition just written is
mod (an — u)
mod an < j.
mod (an — u) From the inequality
mod an < mod (an — u) + mod u, we have
mod On , , mod u
mod (an — u) mod (an — u)
which demonstrates the theorem since — — decreases indefinitely
mod (an — u)
when n increases.
It is seen at once that
, , , u — an
is a regular function for all finite points of the plane. This difference we
may represent by G'(u) = — **£•
du
We thus have
Multiplying by du and integrating, taking zero as the lower limit, we have
iu, say;
* See Osgood, Lehrbuch 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
a i 0,2
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-Le frier,* establish the important results of Weier- strass | who extended to these integral transcendents the fundamental
theorem of algebra. When the series of the preceding article V - - —
, *•* mod an
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 Pu(u) - 1 + \ + • • • + u—>
an an2 an»
so that
P.(M)
,
u — an ana>(u — On)
We may next show that by a suitable choice of CD we may render the series
— + *>.(«), or
an"(u - On) convergent.
In the first place it may happen that -- being divergent, the
*~* 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. ^4 n ^ n* ,
Hence we may fix a number a> such that the series >
^
convergent.
We may then conclude from this series the convergence of
s mod
V - - - - - - , and consequently of V — — - - - •
^ mod an"(u - an) ^ an"(u - 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, Theory of Functions, p. 335.
PKELIMINARY NOTIONS. 13
For, if we put
t vn
mod anal+1 mod anw(an — u)
we have for the ratio — the same value as before, Un
vn mod an
un mod (an — u)
We must, however, always know that we are passing to a convergent series when \ve raise each term of the divergent series to a certain power.
For example,* consider the divergent series V . It is seen that
x ^ log n
the series V is also divergent, however great aj be taken.
~* (log n)
For writing
(log 2)- (log 3)- (log n)'
it is seen that
S > n ~ l - (log n)a
Note that
n - 1 n 1
(log n)w (logn)u (logn)"'
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 co a value which changes with n. With W^eierstrass write a> = n — 1. The given series may be written
This series is convergent; for writing Un = mod
f1-
\ On
it is seen that vUn tends towards zero for n = 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
/'(«)
* This example is due to Mr. Stern and cited by Hermite, loc. cit., p. 86.
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. = — ^Hl • and for brevity write du
so that
ro Q>n £ Q<n wanw \an
We have at once
which formula gives an analytic expression, in which the roots are set forth, of the integral transcendental function.
The quantities (1 -- \e w\a») are called primary functions* by \ On/
Weierstrass.
Suppose next that f(u) has equal roots, say, of the pih 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 qth
order, we have only to proceed with the quotient £^2*1, the result differing
from the preceding only by the presence of the factor UQ. (See Hermite, loc. cit.)
INFINITE PRODUCTS.
ART. 17. It may be shown that the infinite product (1 + ax) (1 + a2) . . . (1 + a») . . . has a definite value, if
represents a converge at series.!
* See Osgood, Ency. der math. Wiss., Band II2, 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.
f Cf. Mittag-Leffler, Acta Math., Vol. IV, pp. SQetseq.; Dini, Ann. di mat. (2), 2, 1870, p. 35; Harkness and Morley, Theory of Functions, p. 82; and especially Pringsheim, Ueber die Convergenz unendlicher Producte, Math. Ann., Bd. 33; Weierstrass, Werke, I, p. 173.
PRELIMINARY NOTIONS. 15
For write
Pn = (1 + ai) (1 + a2) . . . (1 + an).
Then evidently
Pn — Pn - 1 = #nPn - 1 ,
and
Pn = 1 + ai + a2Pi + a3P2 + • ' • • + cinPn-i.
Hence when n becomes indefinitely large, the series Pn will tend towards a definite limit if the series
1 +ai +a2Pl + a3P2 + • • • + anPn-1 +an + 1Pn+ • • • (1)
is convergent, the limit, if there is one, being the sum of this series.
Consider first the case where the quantities a\t a2j . . . are real and positive or zero. The quantities PI, P2, . . . are then at least equal to unity, and consequently, in order that the series (1) be convergent, it is necessary that the series
ai + a2 + a3 + • • • + an + - • • (2)
be convergent.
Further, if (2) is convergent, it may be shown as follows that (1) is convergent.
The product
Pn = (1 + ai) (1 + a2) . . . (1 + an),
when developed is
1 + 01 + a2 -h • • • + an + aia2 + • • • + a^2 • • an.
Writing
An = ai + a2 + • • • + an and
A = al + a2 -f- • • • + an + an + i + • • • , it is seen that
A A 2 An
P S 1 _l_ ^n _l_ •"•"_ J_ _1_ ^n
IT » '^T'
or Pn < eAn < eA,
which proves the proposition.
Next let the quantities «i, a2, . . . , an, . . . , previously supposed to be real and positive, take any values.
Then the series
1 + ai + a2 -f • • • + an + a,ia2 + • • - + a\a^ • • an
is evidently convergent if the corresponding series made by taking the absolute values of the a's is convergent.
n = x
Hence the condition for the convergence of the product JJ (1 + an) is
n=x n=l
that the series 'V | an \ be convergent.
16 THEORY OF ELLIPTIC FUNCTIONS.
ART. 18. If further the series
| al | + | a2 + a3 | + • • - + an | + • • • (3)
is convergent, the product
(1 + aiu) (1 + a2u) (1 + a3u) . . . (1 + anu) ... (4)
is convergent for all values of the variable u, except infinity. For if r is the modulus of u, the series
| ai r + | a2 r + • • • + | an \ r + • • •
is convergent whatever be the modulus r.
ART. 19. We shall next show that when the series (3) is convergent, the product (4) may be expressed as an integral power series P(u) which is con vergent * for all finite values of u.
Consider the product of n real factors
Pn(r) = (1 + air) (1 + a2r) . . . (1 + anr),
ai, a2, . . . being taken as real quantities and positive. Let these n factors be multiplied together.
If si(n) denotes the sum of the n quantities ab a2, . . . an; s2(n) the sum of the products of these n quantities taken two at a time; s3(n) the sum of the products taken three at a time, etc., we will have
pn(r) = 1 + Sl(«)r + s2(n)r2 + • • • + sm(n)rm + • • • + sn(n)rn.
Since any term sm(n)rm is less than Pn(r) or its limit P(r), where n = oo, it follows that sw(n)rm tends toward a definite limit smrm when n increases indefinitely; thus the sum sm(n) of the products taken m at a time of the n first terms of the convergent series
ai + a2 H- a3 + • • •
tends toward a definite limit sm when n increases indefinitely. But since
Pn(r) > 1 + Si<nV + s2(n)r2 + . . . + sm(wVm,
if leaving m fixed we let n increase indefinitely, it is seen that P(r) > 1 + sir + s2r2 + . . . + smrm.
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 Elliptiques, pp. 301 et seq.; Osgood, Lehrbuch der Funktionentheorie, pp. 460 et seq.; Tannery et Molk, Fonctions Elliptiques, t. I, pp. 28 et seq.; Picard, Traite 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 + s^nh + s2(n)r2 + • • • + sn(n)rn being less than the corresponding term of the sum
Sn(r) = 1 + sir + s2r2 + • • • + snrn,
the sum /Sn(r) is greater than Pn(r) and consequently its limit S(r) is greater than or equal to P(r).
It follows that S(r) = P(r).
Consider next the product of n imaginary factors
Pn(u) = (1 + aitt) (1 + CL2U) . . . (1 + dnu),
where | a\ | + | a2 | + | a3 | + • • • is a convergent series. It follows as above that
Pn(U) = 1 + ff^U + <72(n)W2 + • • • + <7m(n}Um + - • • + <7n(n)l*n.
Any coefficient am(n) is a sum of imaginary terms whose moduli form quantities corresponding to sm(n) above. Consequently when n increases indefinitely, since sm(n) tends towards a limit sm, the sum am^ tends towards a limit om.
The series
I + oiu + o2u2 + <73u3 +.'.-.- P(u), say, (6)
is convergent, since the moduli 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.
ART. 20. The sine-function. — As an example of Art. 16, we note that
the function f(u) = sm ~u has for its roots all the positive and negative nu
integers ± 1, ± 2, ± 3, • • • .
The series V - is here divergent, but the series V - - - — — is ** * 2
convergent.
We may consequently put a> = 1 in Weierstrass's formula. The primary factors are therefore
u
1 - -}e*.
18 THEORY OF ELLIPTIC FUNCTIONS.
Noting that f(o) = 1, and admitting* that G(u) = 0 (see Vivanti- Gutzmer, Eindeutige analytische Funktionen, p. 163), we have the formula
sin Tin nu
n = ± 1, ± 2, ± 3, - - - .
Uniting the integers that are equal and of opposite sign we have Euler's formula :
sin nu
The periodic property of the sine-function may be deduced from this definition. For write
F(u) = Au(u — 1) (u — 2) . . . (u — n) multiplied by (u + 1) (u 4- 2) . . . (u 4- n\
where A is a constant.
Changing u into u + 1, we have
F(u + 1) = A(u + 1) u (u — 1) . . . (u — n + 1) multiplied by
(u + 2) (u + 3) . . . (u + n 4- 1). It follows that
II I -V) 1
Phi 4. n — EY?/) u "*" n ~ *- •
J. \.li> \^ A J A \wy •
u — n or, when n = <*> ,
F(M +1)=- F(u).
From this we may derive at once the relation
sin (u + n} = — sin w, or sin (u + 2 TT) = sin w.
ART. 21. We may write
sin u = u JJ < ( 1 — —
m ( V ^
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 [1 — )eW3rmay be written
V mn)
* 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 eG^ = 1.
PRELIMINARY NOTIONS. 19
and passing to the product of such terms we note that the series
are absolutely convergent.
Since m takes all integral values from -oo to +00 excepting zero, we may change the sign in the above product and have
sin u =
Next changing u to — u and comparing the resulting product with the one previously derived, we see that
sin (— u) = — sinw.
The point u = GO is an essential singularity of sin u. For if we put u = - 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
integer. It follows from what we saw in Art. 3 that i> = 0orw=oois an essential singularity.
ART. 22. The function cot u. — This function may be derived from the sine-function from the formula
cot u = — log sin u. du
It follows from the above formula * for sin u that
u
1 1U/-J- J_U
.+ * */ \U + 2* 27T/
From this expression we have at once
cot ( — u) = — cot u.
We also note that the points 0, ± -, ± 2 n, ... 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 = TT, the difference
cot u --
u — ~
is a regular function in the neighborhood of u = x.
* Eisenstein (CreUe'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
where the summation extends over all integers from — oo to + oo excepting zero.
The function
sin2 u du u2 m (u — mx)2
is an even function which has as double poles the points 0, ± n, ± 2 x, - • • . The principal part relative to the pole u = mn is
(u — mil)2
From the preceding formulas the periodicity of the circular functions is easily established.
The expression of - is seen to remain unchanged when n is added sin2 u
to u.
For the cotangent consider the difference cot (u + TT) — cot u. . We find that the expression
/ i _ I\ + /! _ i \ + /_! ___ !
\U + 71 Uj \U U — 71 / \U — 7C U —
— 2 7T/
/ _ L_ i ^ + / _ i_ _1 _ \
\U + 2 7T U + 71 1 U + 37T U + 2n)
is zero.
Further, from the relation
COt (u + 7t) = 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 + TT) = log sin u + log C, or sin (u + TT) = C sin w.
In this formula put it = — - , and we have C = — 1.
2i
ART. 23. Development in series. — If we note that
u — mn mn m27i2 mV
it is seen from the expansion of the cotangent that
C0t«-! -,!-?!- ,„«?-„«£- ...
U n2 7T4 7T6
PRELIMINARY NOTIONS. 21
where «i = 2'-^, s2 = Y'l- s3 = X'^, etc. The sums V'-I, ** m2 ** Hi4 ^ w6 *~* rra
V' — , etc., are evidently zero, since the positive terms are destroyed
^4 m3
by the corresponding negative terms.
To determine the values si, s2, . . . , multiply the above formula by du and integrate.
We thus have
i • i ,1 Si U2 So W4 SQ U6
log sin u = log A + log u - -± — - -j — - -^ ^ - • • •
_ Si Jfr _ S2 "4 _ or sin z* = AM e ^ *' 4 ff4
Since sm u = 1, when w approaches zero, it follows that A = 1. Further, since
we have by equating like powers of u, after the exponential function on the right has been developed in series,
= *! ~4 2;r6
" 3 ' S2 32 - 5 ' 33 • 5 • 7 '
(see Bertrand's Calcul Differentiel, p. 421). Noting that
n=l
we have *
22 32 66 2!
24 34 90 30 4!
11 -6 1 05 -6
i 4 JL _i_ _L _(-...= — _ = _ . "• ,
26 36 945 42 6!
1 1 4-1 4.. -8 .J_.?L>rf
28 38 9450 30 8!
1 + _L _1_ 4 . 7T10 5 § 297T10
210 310 93555 66 ' 10!
The numbers-, — , — , — , — r • • • are the so-called Bernoulli num- 6 30 42 30 66
6ers (cf. Staudt, Crelle's Journ., Bd. 21, p. 372).
* See Biermann, Theorie der analytischen Functionen, p. 326; Jordan, Traitt d' Analyse, t. I, p. 360.
22 THEORY OF ELLIPTIC FUNCTIONS.
THE GENERAL TRIGONOMETRIC FUNCTIONS. ART. 24. We know that sin 2 u = 2 cot u and
cos 2 u =
1 + cot2 u cot2 u — I cot2 w + 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 we add to the argument u any positive or negative multiple of 2 n. We
say that 2 n is a primitive period of these functions. Writing cot - = r,
2i
we have here to consider any rational function of r. 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
j,,^ = F(sin u, cos u) (r(sin u, cos u)
where the numerator and denominator are integral functions of sin u and cos u.
Further, since
eiu _ e-iu eiu _j_ g-tti
sin u = - and cos u = - — - , 2i 2
it is seen that f(u\
B0(e2iu)n + Bite2''")"-1 +•••• + Bn-ie2™ + 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(eiu) + Q(eiu),
where P(eiu) is composed of integral (positive or negative) powers of eiu. But in Q(elu) the degree of the numerator is not greater than that of the denominator and this denominator does not contain eiu as a factor. Hence Q(eiu) = $(u), say, remains finite when u = <*> and also when u = — oo .
* Cf. Hermite, "Cours," loc. cit., p. 121; and also Hermite, Cours d' Analyse de I'Ecole Poly technique, p. 321.
PRELIMINARY NOTIONS.
23
We shall next study the function Consider the integral / $(u)du, where the integration is taken over the contour of the rectangle ABCD in which
OM = x0, MN = 2 TT, AN = NB = a.
Fig. 1.
If we denote by (A B) the value of this in tegral taken over the line AB, we have by Cauchy's Theorem (see Art. 96)
(AB) + (BC) + (CD) + (DA) - 2-z'S,
where 21 denotes the sum of the residues of <b(u) corresponding to the poles that are situated on the interior of this rectangle. Since .r0 is an arbitrary length, the sides of the rectangle may always be so taken that
they are free from the infinities of $(tf)-
For any point along the line DC we may write
u = XQ+ IT,
where r is a real quantity that varies from —a to +a. We may there fore write
(DC) m i f+° <*>(*<) J -a
r+a (AB) = i I &(x0
J-a
These two integrals are equal since
+ *T)<JT and similarly
) = <&(u + 2x). It follows that (AB) + (CD) = 0, and consequently (DA) + (BC) = 2?-S; or
r
Jo
(XQ— ia +r)d
C 7 - /
Jo
+ ia + r)dr = 2 iVrS.
(1)
Next let the constant a become very large and let the corresponding values of
®(XQ - ia +r) = Q[e*"(*o-^+t)] = Q[e+a+iUo+T) and $(xq + ia +r) = Q [e>(^+ta+T)] = Q[e
be respectively G and H. Formula (1) becomes then
G - H = iS or 2 = T^
an expression which gives the sum of the residues of 4>(u) 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
Note that
-•" -s • - ei
and that this quantity is equal to — i for u = <x> and to + i for u = — oo . Hence the sum of the residues of cotf ~ u\$(u} 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 3>(u) be ai of order ni,
a2 of order n2,
av of order nv.
We know that the residue with respect to a pole «i is, if we put = u — a i, the coefficient of - in the dev
it
ing increasing powers of h of the expression
h = u — a i, the coefficient of - in the development according to ascend-
it
By Taylor's Theorem
= cot _ _ cot _=
2 2 II dt 2
(m - 1)! dPi~l 2
Further, the expansion of ^>(ai + h) in the neighborhood of ai is of the form
+ positive powers of h. h hz hni
If we put A = Cn; Ax = ^, ^2=^; • • • ; ^-1= ^rf^ ^ is
seen that the coefficient of - in the above quotient is
h
n ±t — a\ „ d , t — Oi , , n dni~l . t — a\
PRELIMINARY NOTIONS. 25
The sum of the residues which correspond to the poles of <J>(u) is therefore represented by
d
i n d ^ t-di , ,r dn~ ^ t - a
C2i-coi _- , ±0*00 --
2 " dt 2
Further, with respect to the pole u = t, if we write u = t + h in the quotient
cos^-^
sn
it is seen that the coefficient of -, when h is very small, is — 2
/i
We thus have
-G-H = 1CU cot -=- - C2i
i=i z
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 61? 62, . . . , bn, it follows at once from the expression of f(u) above that
(e2iu- e2ibl) (e2iu— e2ib2) . . . (e2iu— e2l'6»)
= Ae^ut' sin (M ~ ci) sin (M ~ ^2) . • . sni (u - cw) ^ sin (u — 61) sin (u — 62) . . . sin (M — 6n)
where jj. 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-) =/(M).
26
THEORY OF ELLIPTIC FUNCTIONS.
If we change the variable by writing u = — , so that/(V) =/(—-] =
CO \ CO /
fi(u), it is seen that
fi(u + 2co) = fi (u) ,
and consequently 2 co is the period of the new function; and further all
. u
Ttl —
rational functions of eiu are now rational functions of e w.
In the next Chapter we shall show that any trigonometric function /(M) has an algebraic addition-theorem; or, in other words, f(u + v) may be expressed algebraically through f(u) and/(v).
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 f(u) has a finite num ber of singular points pi, p2, . . . , pn in the finite portion of the w-plane.f
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. { 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 (Funktionentheorie, 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.
J 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 w-plane which lies without this domain and at which the function has a definite value. The function /(w) 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 arid c are connected by any line which does not cross a canal. Take any point a^ on this line which lies within the circle of convergence about a. The value of the function f(u) at 'the point ai is therefore given by P(&i — a), and also the derivatives of f(u) at the point ai are had from the derivatives of this power series after we have written a± for u. It is thus seen that the values of f(u) and of its derivatives at a\ involve both a and a\.
Next draw the circle of convergence about ai where the arbitrary point «i 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 a! the function/(&) may be represented by a power series, say P\(u — a\).
We may show as follows that the coefficients of this power series involve both a and a\\
For the domain about a we have the series
(i) /(«) -/(a) + '^/»+ ("~a)V(«) + • • • = P(» -«);
and for points common to the domains of both a and ai we have
PI(U - a^ = P(u - a) = P(a1 - a + u - aj
- a) + P>! _ a) + • • •
In the domain about ai we have
(II)
where in this domain
/<*>(0l) = P<*>(a! - a), /£ = 1, 2 . . . ,
which quantities are known from (I).
Since the coefficients of PI(U — «i) involve both a and a\, the power series P\(u — ai) is sometimes written PI(U — ai, a).
28 THEORY OF ELLIPTIC FUNCTIONS.
At a point u situated within the domains common to both a and a\ the two series P(u — a) and P\(u — ai) give the same value for the function f(u). Hence the second series gives nothing new for such points. But for a point u situated within the domain of a± but without the domain of a, the series P\(u — a\, a) gives a value of 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 a2 situated within the domain of «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 a\, the other portion of the domain of a2 lying without that of a\. The values of f(u) and its derivatives at a2 are offered by the power series P\(u — &i, a) and its derivatives when for u we have written a2. It is seen that for all points common to the domains a i and 0,2
P2(u — o2) = PI(U — Oi)= Pi(a2 — 0,1 + u — a2)
In the domain about a2
(III) /(«) = /(a2) +1i^-«2/'(a2)+ (u~|a8)V(a)+. . .= P2(«-a2), where in this domain
which quantities are known from (II).
It is thus seen that the coefficients of the new power series P2(u — a2) which represents f(u) in the neighborhood of a2 involve the quantities a and a i, and it may consequently be written P2(u — a2) = P2(u — a2, a, d).
At those points u in the domain of a2 which do not lie within either of the two earlier circles the series P2(u — a2j a, d) gives values oif(u) which cannot be derived from either P(u — a) or P\(u — d)- Thus the new series is a continuation of the older ones.
Proceeding in this way we may reach all the points of the w-plane where the function behaves regularly. In an indefinitely small neighborhood of those points p which are essential singularities of the function f(u), the
* Weierstrass, Werke, Bd. I, p. 84, 1842, employed the word Fortsetzung; MeYay, 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 M6ray, Leqons nouvettes sur V analyse infinitesimale et ses applications gcomctriques. Paris, 1894-98).
PKELIMINARY 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, Allgenieine 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) = 2an cos bnu,
which, although always finite and continuous, never has a definite deriva tive, if b is an odd integer and
(1st) ab > 1 + | T. or (2d) ab2 > 1 + 3 -2, where in the first case ab > 1 and in the second case ab must be = 1.
ART. 29. If c is any point hi 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 abi 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 p, 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 delle 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, Memoire sur I' approximation, etc., Liouv. Journ., 1877; Osgood, Lehrbuch der Funktionentheorie, p. 89; Pringsheim, Ency. der Math. Wiss., Bd. II,1 Heft 1, pp. 36 et seq.
SO THEORY 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, f 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-Leffler), 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
- £. f JfiUt
2mJsz — 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-Leffler $ 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- valued 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.
f Sur la representation analytique, etc., Acta Math., Bd. 23, p. 45.
J Mittag-Leffler, Sur la representation, etc., Seconde note, Acta Math., Bd. 24, p. 157; Troisieme note, Acta Math., Bd. 24, p. 205; Quatrieme note, Acta Math., Bd. 26, p. 353; Cinquieme note, Acta Math., Bd. 29, p. 101.
PRELIMINARY 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 TT. 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 a> and 2 a/; for example,
f(u + 2 a>) = f(u), f(u + 2 a/) = f(u)
and f(u + 2 mw + 2 W) = 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 primitive periods infinite.
EXAMPLES 1. Prove that
where m takes all integral values, negative, zero, and positive. 2. Show that
Vr° \(i !L_y£iJ = sin x(u + a) e.
m= -a, ( \ m - a) ) sin -a
* 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
WTA
32 THEOKY OF ELLIPTIC FUNCTIONS.
3. Show that
m= +00
m= —oo
4. Show that
6. Show that [GaUSS']
1 3 COS 71X
(x + w)3 sin
(
and that
1 _ _4 / 2 1 1 \
a; + m)4 \ 3 sin2 TTZ sin4 TTO;/'
(2 0, x) = V * = ^gf at + <*2
^^Crr 4- w.'l2^ «in2 TTV oir.4 ^^
+ m)2<7 sin2 ^ sin4 ^ sin2*
3 5
(2g
sin ^ sin
where the coefficients a1; a,, . . . ; 6lf 62, . . . are connected with the Bernoulli 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, x)2 + 2(1,3) (3,3); 3(2, 0) = 7T2.
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
iciu
exponential function e w .
ARTICLE 32. The simplest case of a function which has an algebraic addition-theorem is the exponential function
<t>(u)= eu. It follows at once that
eu + v= eu -ev,
or <j>(u + v)= (j)
Such an equation offers a means of determining the value of the function for the sum of two quantities as arguments, when the values of the func tion for the two arguments taken singly are known.
It is called an addition-theorem.
In the example just cited the relation among $(M), <j>(v) and <j)(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 eu is perhaps best studied by deriving its properties from its addition-theorem.
The sine function has the algebraic addition-theorem
sin (u + v) = sin u cos v + cos u sin v
= sin u \/l — sin2 v + sin v Vl — sin2 u.
The root signs may be done away with by squaring. We also have ' tan u + tan ^ ^
1 — tan u tan v
We note in the above algebraic addition-theorems that the coefficients connecting <j>(u), <j>(v), and <j)(u + 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.
38
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
and further we shall write
We may assume either that the function (f> is defined for all real, imaginary, and complex values of the argument, or that this function is denned for a definite region, which, however, must lie in the neighbor hood of the origin. Further it is assumed that (f> has an algebraic addition- theorem. We therefore have, if G represents an integral function with constant coefficients,
G(t, 1, 0 = 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 77, we have
^£+^£ = 0, and similarly d£ du d du
dif] dv d£ dv
Write u + v = h and note that fC »££• 1 _«'££
du dh dv
We consequently have by subtraction
d dv
There are two cases possible :
dC1 dC1
First. The quantity r may appear in the coefficients — > — ; or
3£ drj
Second. The quantity £ does not appear in these coefficients. Consider the first case. We have the two equations
G(?> 1, 0 = 0,
du difj dv
ALGEBRAIC ADDITION-THEOREMS. 35
The first of these equations may be written
where the a's are integral functions of £ and y; the second equation may be written
where the A's are integral functions of £, fl, — and ^ . If ^ is elimi-
du dv
nated from these two expressions, we have
In the second case where £ does not appear in the coefficients— and
d?
- , we have at once an equation connecting ?.—-,• and ^ . °ri 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.:
// the function (f> 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 v a
j,ju
special value, and have consequently between c and — an equation of
du the form
where / denotes an integral function of its arguments.
This equation we shall call the eliminant equation* We may write it in the form
, » - 0.
We have therefore proved that if for the analytic function (j>(u) there exists an algebraic addition-theorem, we also have an algebraic equation between the function and its first derivative, the equation being an ordinary
* The equation is due to Meray, see Briot et Bouquet, Theorie des Fonctions Ellip- tiques, p. 280; Picard, Traitc d'Analyse, t. 2, p. 510; Daniels, Amer. Journ. Math., Vol. VI, pp. 254-255.
36 THEOEY OF ELLIPTIC FUNCTIONS.
differential equation of the first order. The argument u does not appear explicitly in the equation.
ART. 35. As the above theorem is made fundamental in many of the following investigations, it is of great importance to note that it is true without exception.
- In the equation H = 0 we may write any arbitrary value for v which belongs to the region considered. If after the substitution of this value
of v there remains an equation between £ and — , then our conclusions
du
above are correctly drawn; but if after the substitution of this value of v the equation were to vanish in all its coefficients, the theorem remains yet to be established. We take the following method to prove that the theorem is always true:
Develop the function H in powers of £ and — . The coefficients in this
du
development are either zero or functions of in and -2 1 including constants.
dv
It is evident that all of the coefficients are not zero, for then the function H would be identically zero.
We represent one of the coefficients which is not zero by
There must be such a coefficient which contains in and -2 ; for otherwise
dv
all the coefficients would be independent of these quantities, which there-
aXTj
fore would not enter the function H. But since — is not always zero,
677
these quantities must appear.
In this coefficient /i (77, — -* j 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 fji), ~^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 /if n, -5 ] is zero for every value of v, we have
V dv/ an equation of the form
/i0W, </>»] - 0,
where f\ is an integral function of its arguments. This equation, how ever, expresses the same thing as the equation
, </>'(")] - 0,
ALGEBRAIC 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 7) alone, /2( TJ) 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 7), and we have only to give TJ a value such that/2(^) 7^ 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 <j>(u) has the property that between the function <f>(u) and its first derivative <j>'(u) there exists an algebraic equation whose coefficients are independent of the argument u, the function has an algebraic addition-th eorem.
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 £, y, £ is one necessarily implying the existence of an algebraic addition-theorem. We must not suppose that every algebraic equation
G(^ i, 0=0
necessarily exacts the existence of an algebraic addition-theorem; neither does the relation
<t>(u + v) = F{<l>(u), 4>'(u), <f>(v), <t>'(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
/—
with respect to — , we have du
f-*®-
where ^(£) is an algebraic function of £. This equation may be written
or ..... <fe
f1
u — UQ = I
Ja
where I^Q and a denote constants.
It is thus seen that in the case of every analytic function £ = <j>(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 u0 = 0, thus having
In a similar manner
»
On the other hand we have
We thus have the equation
a formula which is of fundamental importance.
To illustrate the significance of the above formula consider the follow ing examples:
1. Let e = <f>(u) = eu; <j>f(u) . eu.
We therefore have as the eliminant equation
«-?•
rfw
and also ^(l") = <f
Since ^ = 1 when a0 = 0, we may write
'
On the other hand, £ = 0(M + v) = eu+v= eu-ev= </>(u) 6(v) = £ . » It follows that
t , t
or log £ + log ^ = log t.y.
2. Let f = <f>(u) = sin w; 0/(?/) = cos w = Vl - sin2 w - Vl - ^2. It follows that i/r(f) = x/1 - <f2, and consequently since w = 0 for £ = 0
'o VI - t2 Further, since
we have
'o v/1 - t2 Jo Vl - t2 Jo Vl - or sin-1*? + sin-1 y = sin-1 [£ Vl - -rf + v V
ALGEBRAIC ADDITION-THEOREMS. 39
3. If £ = tan u =<j>(u), we have
l±i p dt p dt p-fr dt
Jo 1 + t2 Jo 1 + t2 Jo 1 + *2'
or tan - l £ + tan ~ l y = tan ~ 1 * ^ J' •
l_l ~~ ^7/J
ART. 37. We have seen that for every function £ = <£(M) for which there exists an algebraic addition-theorem, there exists without excep tion a differential equation of the form
/[</>(«), </,»] = 0, or/(f,£)=0,
where / denotes an integral function of its arguments and where u does not appear explicitly in the equation.
If c = <f>(u) is known for a definite value of u, then from the above
si-
equation we ma}7 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 £ = <j>(u) has an algebraic addition-theorem, the values of all the higher derivatives of <f>(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
/(£, £') = 0, or, say,
where n is a positive integer and the a's are integral functions of £.
We may assume that /(£, £') is an irreducible function, that is, it cannot be resolved into two integral functions of £, £'; for if this were the case, one of the factors put equal to zero might be regarded as the integral equation connecting £ and £'.
We form the derivative ^ ^^ , which is an integral function in £, £'.
The degree of this derivative in £' is one less than the degree of /(£, £') in £'.
Further, the equation ^ ^\, = 0 is not satisfied for all pairs of values £, ^' which satisfy the equation /(£, cr) = 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 /(£, £') is irreducible. The two equations
/(£, f) = o,
are satisfied by only a finite number of pairs of common values £, if'. For their discriminant with respect to £' 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 £. 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 u = UQ of the argument, for which </>(UQ) = £o> <t>'(uo) = £o', where £o> £o' satisfy the equation /(£, £') = 0 but not the
equation = 0.
d$
By differentiation we have
We further assume that the point in question is such that the function has for it a definite derivative. We may write
d? = ?'du. It then follows that
or
u
a?'
From this it is seen that f" = 0"(u) is rationally expressed through f, f. Since the singular roots have been excluded, the denominator
In a similar manner it may be shown that £'" = <j)fff(u') may be expressed in the form of a fraction whose denominator is a power of the denominator
ALGEBRAIC ADDITION-THEOREMS. 41
which appears in the expression for £" and consequently is different from zero. The same is true for all higher derivatives.
ART. 37a. Suppose that u\ is a value of the argument u different from UQ and such that
<£ (U0) = CO = <t>(Ui),
Further let <}>(u) be an analytic function with an algebraic addition- theorem, and in the neighborhood of UQ and u\ let the function (f>(u) be regular. Finally, it is assumed that
that is, £V does not belong to the singular roots of /(c, £') = 0.
We assert that <f>(u) under these conditions is a periodic function and that UI — UQ is a period of the argument.*
Since the function <f>(u) is regular in the neighborhood of UQ, it may be developed by Taylor's Theorem in the form
<t>(uQ) + - In a similar manner we also have
By hypothesis we have
The derivative 0" (MO) may be expressed as a rational function of (/)(UQ), <J>'(UQ) with constant coefficients; (/>"(ui) has the same form in </>(MI),
It follows that
$" (UQ) = (f>"(u\), and in a similar manner
Let MQ + v be a point that lies within the region of convergence of the first of the above series and let HI + v be a point situated within the region of convergence of the second.
* Cf . Biennann, Theorie der analytischen Funktionen, p. 392.
42 THEORY OF ELLIPTIC FUNCTIONS.
Instead of u write u0 + v and u\ + v in the two series respectively. They become
Consequently, owing to the relations above,
<f>(UQ + V)=<l>(Ui + V).
Next write Ui — u0 = 2 at or u\ = u0 + 2w, and we have 0(^o + v) = (f>(u0 + v + 2a>).
The quantity v may be regarded as an arbitrary complex quantity, and must satisfy the condition that UQ 4- v belongs to the region for which </>(u) has been defined.
The quantity 2 a> is called the period of the argument of the function, less accurately the period of the function.
We may therefore conclude that a function <f>(u) is periodic, if it has an algebraic addition-theorem and if there are two points, UQ and u\, that are not the singular roots of f[(f>(u), <£'(»]= 0, for which
4>(uQ) *~ <f>(tii) and <J>'(UQ)= <t>'(u\)>
ART. 38. If we have only the one condition that <J>(UQ) = $(MI), we cannot without further data draw the same conclusions about periodicity. If the equation connecting </>(u) and <t>'(u) is of the first degree in <f>'(u), as is the case of the exponential function, then the second condition, viz., <p'(uo) = (t>'(ui) follows at once. In general this is not the case.
We may, however, effect a conclusion if the assumptions are somewhat changed: Suppose that n is the degree of the equation f[$(u), <f>'(u)] = 0 with respect to <f>'(u). To every value of <f>(u) there belong at most n values of </>'(u).
Suppose next that n + 1 points UQ,U\, . . . , un may be found, at which
and suppose also that <fr(u) is regular in the neighborhood of each of these points,, and further suppose that £0 is not a singular root of /(£, £') = 0. Write
<J>'(U0)= ttfo,
<t>'(un)= CJn.
These n + 1 values of <f>'(u) belong to one value of £0 = <t>(uo) = <i>(u\) = . . . = (j)(un). But as there can only be n values of (j>'(u) belonging to one
ALGEBRAIC ADDITION-THEOREMS. 43
value of <j>(u), it follows that two of the above values of (j>'(u) must be equal, and consequently
^(ti«) -£'(*,),
where a and ft 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 <f>(u) is periodic, u* — up being a period of <t>(u). We have then the following theorem: *
// it can be 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
the quantity u does not explicitly appear.
Suppose that ~ = <f>(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 constant.
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
I//
~ * (u
from which it is seen that the differential equation is satisfied by </>(u + v). We may therefore write
f[<t>(u + v), <j>'(u + v)] = 0.
Further, since by hypothesis <f>(u) has an algebraic addition-theorem, there exists an equation of the form
As (/>(v) is a constant, we may determine (j>(u + v) as an algebraic function of </>(u) from this equation. It is thus shown that the general integral of the differential equation
</>'(u)] = 0 * See Daniels, loc. cit., p. 256.
44 THEORY OF ELLIPTIC FUNCTIONS.
is an algebraic function of the particular solution <j>(u). 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:
7 > '
dv
, 0 = 0. (2)
The first of these equations may be written in the form
- • + Ak-i C + Ak = 0,
where the A's are integral functions of £, 77, £', T/, while the second equation has the form
+ ' ' * + Om-lC + am = 0,
the a's being integral functions of £, 77.
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 £, say
go:, f, r,, r, 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 £, r), £', tf 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
= F(? d£. V 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 £ = cf>(u + v) may be expressed rationally in terms of <j)(u), <j>'(u), <l>(v), fi(v). All these functions have the property of being
ALGEBRAIC ADDITION THEOREMS. 45
one-valued analytic functions of the independent variable. The reciprocal theorem is also true: All analytic functions for which there exists an algebraic addition-theorem and which at the same time are one-valued functions of the independent variable, have the property that <J>(u + v) may be expressed rationally through <fi(u), <j>'(u), (f>(v), (f>'(v). 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 <j>(u + v) in terms of <j)(u), </>'(u), (j>(v), (f>'(v) is not characteristic of all analytic functions with algebraic addition-theorems, but only of one-valued analytic functions. To such functions for example the remarks of Prof. Forsyth at the con clusion of Chapter XIII of his Theory of Functions must be restricted.*
ART. 41. We shall show (cf. Schwarz, loc. ciL, p. 2) that
I. All rational functions of the argument u, and
um
II. All rational functions of an exponential function e w , where a) is different from zero or infinity, have algebraic addition-theorems and have the property that </>(u + v) may be expressed rationally in terms of <£(w),
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 (f>(u) be a rational function of finite degree and let
By means of these three equations we may eliminate u and v and then have an equation of the form
(A) (?(£, i, C) = 0,
where G denotes an integral function of its arguments. Writing
(1) £ = £(«), (2) £-p(u),
du
* Cf. also Biermann, Theorie der analytischen Funktionen, p. 393, and Phragmen, Act. Math., Bd. 7, p. 33.
46 THEORY OF ELLIPTIC FUNCTIONS.
we note that both of these expressions are algebraic in u, and by the elim ination of u we have the eliminant equation
(B) 4 1) • °-
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
£=0(u) and ^-=<t>'(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 £ and — ; or it is of a higher degree in u, of the
du
form, say
a0um + aium~l + a2um~2 + • • • + am = 0,
where the a's are functions of £ and — •
du
Since this equation must represent the multiple root, it must be of the form
a0(u - uo)m = 0.
This expression developed by the Binomial Theorem becomes
It follows from the theory of indeterminate coefficients that
a\ or UQ = — — — •
Since a\ and a$ are integral functions of £ and — . it is seen that UQ
du
may be expressed rationally through these quantities. We may there fore write
where R denotes a rational function.
We thus see that for the case where the equations
$ = <j>(u) and *(%- = <t>'(u) du
ALGEBRAIC ADDITION-THEOREMS. 47
have only one common root, we have
Further, since (/> and R both denote rational functions, it is seen that
where F denotes a rational function.
ART. 43. We shall next show that the two equations
, |^- = *'(*) du
cannot have more than one common root. For assume that they have the common roots HI and u2. It follows that
(1)
(2) au
/7—
Since these two expressions exist for continuous values of c and -^
du
we may regard u\ and u2 as two variable quantities. Taking the differential of (1) it follows that
If we exclude as singular all values of u for which
0'(wi) = 0 = 0'(w2), then owing to the relation (2) we have
du\ = du2, or, ui = u2 + C,
where C is a constant.
If therefore the two equations have two common roots, these roots can differ only by a constant.
We thus have
C).
This expression is true for an arbitrarily large number of values of MI, and since the degree of (j)(u) is finite we must have the identical relation
C).
Further, for MI we may write any arbitrary value in the identity, say u\ + C, and we thus have
<£(MI + C) = ^(MI + 2 C) =s
48 THEORY OF ELLIPTIC FUNCTIONS.
Hence the roots of the identity are wx, m + C, m + 2 C, • • • . If then C ^ 0, the equation
£ = 4>(u)
has an infinite number of solutions. This, however, is not true, since the equation is of finite degree. If follows that the constant C = 0 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 <j>(u + v) may be rationally expressed through (/>(u), <j>'(u), <j>(v), <f>'(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-
tun
tial function e " .
Let /* be a real or complex quantity different from 0 and oo and write
t = e^u, and ^(0 - 0(w), (1)
where ^ denotes a rational function. Further, let
s = e^v and
1?(s) =<t>(v). (2)
It follows that
W - s) = <t>(u + v). (3)
From the three equations (1), (2) and (3) we may eliminate £ and y, and have
(A) G{<t>(u),<t>(v), $(u + v)} =0,
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 co, say. P We further have
If t is eliminated from these equations, we have the eliminant equation (B) ' /(*,£)- 0,
where / denotes an integral function.
ALGEBRAIC ADDITION-THEOREMS. 49
It follows from equations (A) and (B) that the function <j>(u) has an algebraic addition-theorem.
ART. 45. It may be shown as in the case of the rational functions that when the equations
£ = V(0 and ^- = ^(t)fit du
have one common root * in t, then we may express t in the form
duj
where R denotes a rational function. It also follows that ^(u + v) = F[</>(u), P(u), <}>(v), <t>'(v)],
where F is a rational function.
Suppose next that the two equations
have more than one common root.
Suppose that t\ and t2 are two roots that are common to both equa tions, so that
(1)
(2) du
If we consider ^i as the independent variable, then t2 is an algebraic function of t\, since
and ^ is a rational function.
From equation (1) it follows that
which divided by the expression (2) becomes
dt± = dtz ti " t2 '
or log ti = log ^2 — log C, so that
t2 = Ch.
It is thus seen that if the two given equations have two common roots, these roots can differ only by a multiplicative constant. Since
^(tz), it follows that
which is an algebraic equation of finite degree.
* By equating the discriminant to zero, we may always effect the condition that there is one common root.
50 THEORY OF ELLIPTIC FUNCTIONS.
As this equation can be satisfied by an infinite number of values of it must be an identical equation and consequently
It follows at once that
But the equation yfr(ti)=TJr (t2) being of finite degree with respect to t2 can only be satisfied by a finite number of different values of t2. It therefore follows that in the series of quantities
Ctlf C2*i, C%, . . . , CP*I, . . . , C%, . . . , (1)
some must have equal values. If the degree of the equation is n in t2 then among the first n + 1 of these quantities two must be equal, say
CP = C« (q> p). Writing q — p = ra, a positive integer, we have
Cm = 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, C2, C3, . . ., Cm~\Cm
are all different. For if
&-&(i,3£m),
then is C*'~J' = 1 (where i — j = m' <m).
This, however, contradicts the hypothesis that m is the smallest integer which satisfies the equation (2). There are consequently only m different quantities in the series (1).
We may use this fact and employ the identical equation
to show that the rational function ^r(Ji) may be transformed into another rational function ty\(tm). If then we write r = tm, we may substitute the function ^i(r) in the above investigation in the place of ty(t), where
the degree of the equation in T is — , n being the degree of the equation in t.
m
The function ^r(t) may be expressed as the quotient of two integral functions without common divisor in the form
t±» A (t-ai)(t-a2) . . . (t -a,) • (t - &!) (t - 62) . . . (t - W '
where // is an integer and where none of the a's or 6's is equal to zero.
ALGEBRAIC ADDITION-THEOREMS. 51
Further, since ^(t) = ^(Ct), we must have
\ f + u 1 ft" fll) (* - ^ • • • (* - a")
(t - 60 (* - 62) . .
* - QI) (Ct - o2) .
(C/ - &0 (C* - 62) . . . (Ct - bff)
The left-hand side of this equation is zero for t = a\; it follows that the right-hand side must also vanish for this value of t. But Cai — 0,1 ^ 0, if we assume that C 5^ 1. Hence one of the other factors must be such that Cai — a A = 0, where A is to be found among the integers 2, 3, . . . , p. As it is only a matter of notation, we may write A = 2, so that
Cai - a2 = 0, or a2 = Cax.
In a similar manner, since the left-hand member of the equation van ishes for a2, one of the factors on the right-hand side must vanish for t = a2, say Ca2 — av = 0, where v is to be found among the integers 1, 3, 4, . . . , p, say v = 3.
We thus have
Ca2 — a3 = 0, or a3 = Ca2 = C2ai. Continuing this process we derive the relations
a>i = a.i, a2 = Cai, a3 = C2ab . . . , am = Cm-ldi. Further, since C, C2, . . . , Cm~l are all different, it is seen that
01, a2, . . . , am are all different.
The quantities ab Cai, C2a1? . . . , Cm~1ai form a group of roots of the equation, and after Cote's Theorem
(t - ai) (t - Cai) (t - C2a!) . . . (t - Cm~lal) = tm - a^.
This factor tm — aim 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
so that Cft = 1. It follows at once that jj. must be a multiple of m and consequently
A (t
±— (tin
m ( ~
(tm - 6^) (tm -
52 THEORY OF ELLIPTIC FUNCTIONS.
We have thus shown that if the two equations
? = *(«), I1 = <£'(«)
du
have more than one root in common, there exists an integer m, such that fat) may be expressed as a rational function of tm. •
Writing t = e»u, it follows that tm = em(iu and
In the further discussion we may use ty\(tm) in the place of It may happen that the two equations
£ = ^i(emftu) and — = fa (em^u) m u.emtiU du
have more than one common root. By repeating the above process we may diminish the degree of ^i and replace the function ^1(em'iU) by the equivalent function fa(emm'ftu)J where m' is an integer, etc.
Since the original function -^ 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 -^, fa, fa, . . . , we must
come to a function, say £ = iK; such that £ and — have no common
du
root for the new variable that has been substituted in fa. Hence with out exception the following theorem is true:
I. All rational functions of the argument u; and u^i II. All rational functions of the exponential function e u have algebraic addition-theorems and are such that
<j>(u + v) = where F denotes a rational function.
Example. — Apply the above theory to the examples sin u, cos u, tan u. Write
piu _ p - iu i /2 _ 1
smu = - - - - = — " - -. where t = elu. 2i 2 1 t
ART. 46. It may be shown by an example that a function <j>(u) may have the property that <p(u + v) is rationally expressible through (/>(u), <f>'(u), <f>(v), <t>'(o) without having an algebraic addition-theorem.
Take the function
cj)(u) = Aeau + Bebu, (1)
where A, B, a, b are constants and a ^ b. It follows that
<t>'(u) - aAeau + bBebu. (2)
ALGEBRAIC ADDITIOX-THEOKEMS. 53
From (1) and (2) we have
b - a
b - a We further have
u + v)= Aeaueav + Bebuebv
A b — a b — a B b —a b — a
from which it is seen that (j>(u + v) may be expressed rationally in terms of 0(M), 0'(u), 0(v), 0'(t>).
We shall now show that 0(u) has no2 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 <j>(u}, we may infer that (f>(u) does not have an algebraic addition-theorem.
Suppose for the function <f>(u) there exists an equation of the form
where / denotes an integral function.
Since (j>(u) and (f>'(u) may be expressed through eau and ebu where only constant terms occur in the coefficients, we may write the above equa tion in the form
fi[eau, ebu],
where /i like /denotes an integral function of finite degree. This equation must be satisfied for all values of u for which the function (f>(u) is defined. We give to u successively the values
. 2/ri . 4:7n
U0, U0 H -- , UQ -\ -- 1 • ' • .
a a
The quantity eau has the same value, viz., eau« for all these values of u. But corresponding to one value of eaUfl, the equation above being of finite degree can furnish only a finite number of different values of ebu. On
54
THEORY OF ELLIPTIC FUNCTIONS.
the other hand there correspond to the one value eau° an infinite number of values ebu of the form
ebu°, e
bu0+-2ni
b .
which are all different, since the ratio - is not rational.
a / d£\
It follows that the eliminant equation f\£. — ) = 0 does not exist for
V duj
the given function, and consequently this function does not have an alge braic addition-theorem. We have thus proved that the existence of the relation
F denoting a rational function, does not necessarily imply the existence of an algebraic addition-theorem.
CONTINUATION OF THE DOMAIN IN WHICH THE ANALYTIC FUNCTION
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 </>(u) was denned for a certain region which contained the origin. This region we may call the initial domain of the function </>(u). We further assume that <f>(u) has an algebraic addition-theorem and is such that </>(u + v) may be rationally expressed through (f>(u), <£'(^), <£M, 0'M within this initial domain.
These properties are expressed through the two equations
(II) <j>(u + v)=F{<f>(u), <f>'(u), </>(v), P(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 <j)(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, — v in the place of v,
u in the place of u + v. We thus have
- v), $'(- v) } . (1)
Fig. 3.
<t>(u)' = F\<i>(u + v), <f>'(u + v), * 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 (f>(v) and (f)(— v) belong to the initial domain. We develop (f>(u + v) by Taylor's Theorem in the form
<j>(u + v) = <f>(v) + u0'(t>) + ~0» + • • • .'
a series which remains convergent so long as v takes such values that the points u + v and u lie within the initial domain. The same is also true of the series
4>'(u + v) = P(v) + u4>"(v) + |J0"'M + •••.•
These series may therefore be substituted in formula (1). We thus have (/)(u) expressed as a rational function of u, which as the quotient of two integral functions takes the form
QO + a iu + a2u2 + • • •
where the two series are convergent so long as | u | is less than a certain quantity, say p.
If 60 T^ 0, <!>(u) has the character of an integral function in the neigh borhood of the origin u = 0; if 60 = 0 = 61 = • • • = bk
and at the same time
a0 = 0 = ai = • • • = a*,
then (f>(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 <f>(u) becomes infinite for u = 0 but of a finite integral degree. It then has the character of a rational 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 tf>(u) may be extended to all finite values of the argument u without the func tion (j)(u) ceasing to have the character of an integral or (fractional) rational function.
Fundamental in the proof of this theorem is the expression of (f>(u) as the quotient of two power series
where the two series are convergent so long as | u | does not exceed a definite limit p.
If we draw the circle with radius p about the point u = 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 THEORY OF ELLIPTIC FUNCTIONS.
this circle the function <j>(u) is completely denned. In order to extend or continue this region, we may use the equation
<t>(u + v) =
We shall at first assume that we may write u = v without the function F taking the form 0/0. We then have * for u = v,
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 <j> 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 u in this equation, we have
<j>(u) = F {</>($ u\<t>'Qu),<t>Q *,),<!>' (I u)}.
We express <j>(u), as the quotient of two power series, = 1>0^ . Further,
™i»oOO <f>'(u) may also be expressed as the quotient of two power series. These
values substituted in F give (f>(u) defined as the quotient of two new power series, say
Since \ u has been written for u in the two new power series, they are convergent so long as | u < 2 p.
We cannot apply the above method, if for u = v the function </>(u) 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\4>(u), #(u), <t>(v), 4/(v)},
we write — - — instead of u,
1 + a
and au 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 (f)(u) as the quotient of two power series in which the numerator and denominator are
* See Daniels, Amer. Journ, Math., Vol. VI, p. 255.
ALGEBRAIC ADDITION-THEOREMS. 57
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 <j>(u) as the quotient of two power series in the form
where the series are convergent for values of u such that
| u | < (1 + a) p.
Since a = J the series is convergent if | u \ < f p.
This process, as well as the one employed in the previous article, may be repeated as often as we wish, so that we have eventually
P2,n(u)' where the power series Pi§n(u) and P2,n(u) are convergent so long as
Hence (j)(u) 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 '
cos u P2u 3 15
where Plt0(u) = u - 4- -
At the points 0, n, 2 ^, 3 it, . . . , the function tan u is zero, and is infinite at -, — , — , • • • .
For the point u = °c , 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 f1 of tan u, so that we may take p = '- .
* Weierstrass, Werke IV, p. 6, says that from the fact that <£(u) has an algebraic addition-theorem we may show that it is a uniquely denned function having the char acter of an integral or rational (fractional) function and that starting with this we may derive a comul^te theory of the elliptic functions.
58 THEORY OF ELLIPTIC FUNCTIONS.
Using the formula
tan (« + „) tanit + tanr.
1 — tan u tan v
we may extend the definition of tan u to an arbitrarily large region. For writing v = u, then is
tan2«= 2tan«
1 — tan2 u Further, if we put ^ u in the place of u, we have
1 - tan2 i u 1 - P2(£ u) P2, i (w) ' where PI, i(u) and P2, \(u) are convergent so long as
i | M | < i TT or |^|<TT.
We see that here the new circle of convergence passes through the points + TT and — TT and that the old region of convergence has been extended by a ring-formed region.
By another repetition of the same process we have
tana = ^^ _^l,2i«)
1
The radius of convergence of the two series on the right-hand side is now 2 TT, 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 (f>(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 <j>(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 <t>(u) was defined, and the property expressed through the formula
is true for this domain.
* Weierstrass (Definition der Abel'schen Functionen, Werke 4, pp. 441 et seq.) empha sizes this fact.
ALGEBRAIC ADDITION-THEOREMS. 59
On the right-hand side we again write i
- instead of u
and au instead of v,
1 + a
with the limitation that the absolute values of these quantities be less than J p.
T} / - .\
Writing first (f>(u) = 1>0^ '' and then making the formal computation
as above, we have $(u) = ltl • These two quotients are identical* within the circle with radius ^ p, so that.
or
(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 | u | < J p. Now P1>0 and PI, i are convergent by hypothe sis within the circle of radius p, while P2,o and ^2,1 are convergent within the circle of radius f 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 <f>(u) through the two quotients
Pl'°^ and Pl-1^ are the same within the interior of the circle p. The
P2,oO) P&i'OO
same process may be continued so as to extend over the whole region of
convergence.
ART. 51. We shall next prove that as the definition of the function $(u) is extended to an arbitrarily large region, the properties of the original function <£(M) that are expressed through the equations,
(I) G{<j>(u),<!>(v),<l>(u + v)} =0,
(II)
are also retained for the extended region, f First take | u | < J- p and | v | < \ p so 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 elliptic 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 = 0 write ^>M for $(u) ^LoM for ^ and PI,O(U + V)
for (f>(u + 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 \u\ < \ p and v \ < \ p. If this power series is arranged in ascending powers of u, the coefficients 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 (f>(u), viz.,
This value and the corresponding values of (£>(v), (f>(u + 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 < i 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 <f>(u).
If the point u = QO is an essential singularity, the function (j>(u) 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 QO 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 w-plane.
Suppose that m is the number of points at which <p(u) = £o, say, where £o is a definite constant, and denote these points by ai, a2, . . . , am.
Let a,, be any one of these points, and with a radius r^ draw a circle 0? about 0,^ as center. Take r^ so small that within and on the periphery of Op none of the other points a\t a2, . . . , c^-i, aft+i, . . . , an lies, and also within and on the periphery of this circle suppose that (f> (u ) is everywhere regular. Next let u take a circuit around C in the w-plane; then in the plane in which <p(u) is geometrically represented (f>(u) makes a closed curve S^ say, which does not pass through the point £o-
We may write
_- - d\<t>(u) - $Q\
and expressing <j>(u] — £o in the form
it is seen that
d<fr(u) d{rei&} e^dr rieiedO i9 "
61
62 THEORY OF ELLIPTIC FUNCTIONS.
If next we integrate around S^ in the </>(u) -plane, we have
The first integral on the right is log r, which is here zero, since the curve returns to its initial point, making the upper and lower limits identical. We thus have
r J^L.= r w.
JSptpW) — <f0 JS*
On the other hand,
r d4>(u) r </>'(u)d
J8fi </>(u)— £0 JCft (f)(u) —
where the integration of the first integral is taken with respect to the elements d$(u) and consequently over S^ in the </>(u) -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 <l>(u) when developed in powers of u — a^ is of the form
or, snce
On the right-hand side a number of the coefficients may vanish. Let AU— ' j be the first of the coefficients that is different from zero and
/C !
let k = kp, say, be the order of the zero of the function <j>(u) - £0 at the point aft.
We therefore have
and consequently
C d4>(ii) __ C A^k^u — g^)*^"1 + • • • JSf!(j)(u} — £o Jcft Afl(u — a^)kfi + • - -
all the remaining terms having vanished.
or C k.,du
Since I — ^ = 2 mk^
kfffri = I idd, or k» = I rf^.
Jsu 2 rd Jsv
it follows that
PERIODIC FUNCTIONS IN GENERAL. 63
In other words, the order of zero of the function <j>(u) — co at the point u = dp, that is, kf, is equal to the number of circuits which the curve in the (j)(u) -plane makes around co corresponding to the circle C ^ made around the point a^ by the variable u in. the w-plane. The integer k^ is at least unity.
Suppose in the place of a^ another point a0 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 01, a2, . . . , an lies, nor any of the infinities of the function. We know then that the integral
I
— co
s zero,
where the path of integration is taken over the circle We have accordingly proved * that the integral
2*»V*#(«) -to'
where $(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 <£(u) takes the value £o within S, provided each point a^ say, at which <j>(u) takes the value £o, is counted as often as the order ku of the zero of <j>(ii) — co <& the point a^.
ART. 53. Next in the place of co take another value ci, which also lies within Sw so that the corresponding value of u lies within C^ . Then the number of circuits of the curve about co is the same as the number of circuits about co, since all the circuits encircle both points.
It follows that
7 J' / \ /"*
„ <t>(u) — ci '" Jcf
<f>(u) - c
and consequently that (j>(u) takes for values of u within C \ the value £1 at least once; for if this were not the case, we know that the above integral would vanish. We have shown above that it does not vanish.
The function <j>(u) by hypothesis takes the value £o on the m different points a i, a2, . . . , aw . . . , am. Around each of these points a circle is drawn with radius sufficiently small that within the interior of the circle none of the other points of the series ab a2, . . . , am lies. Let u make a circuit about the periphery of each of these circles; then <j>(u) makes closed curves about the point £o, and none of these curves passes through co- We may therefore draw a circle about co which lies within all the other closed curves. Let £1 be a point within the interior of this last circle; then it follows from above that the function </>(it) takes the value £1 at least m times. There are consequently an infinite number
* See footnote to Art. 92.
64 THEORY OF ELLIPTIC FUNCTIONS.
of values in the neighborhood of £o which are taken by the function at least m times.
Consider next the function - --
<j)(u) - £o
It has at the point ai} a2, . . . , am the character of a (fractional) rational function, and may therefore be expanded by Laurent's Theorem in the form *
u - + an integral function in u,
-L-)
\u - amj
where G\, G2, . . . , Gm denote integral functions of finite degree of their respective arguments. It follows that
i „ / l V „ / 1 S l-Y • 1 'S p(u)
where P(u) is a power series with positive integral exponents.
The function P(u) cannot reduce to a constant, for then </>(u) would be a rational function and the point u = oo 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 <j>(u) — £o as small as we wish.
If further the point £1 is taken very near the point £Q> the value £1 is certainly taken by the function <j)(u) as u is made to increase. Hence the function </>(u) takes the value £1 at least m times in the finite portion of the plane and another time towards infinity. Since by hypothesis <j>(u) is indeterminate for u = oo, it appears that </>(u) — £o is zero for some value of u such that u <QO. Call this value am+i. By repeating the above process it may be shown that we may find such values of the function <f>(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
We have excluded as being singular all values of the function £ = <j>(u) which satisfy the equation
3/g, P) - Q
ar
* 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 theorie des fonctions, Crelle, Bd. 91, and " Cours," loc. cit., p. 98; Mittag- Leffler, Sur la representation analytique, etc., Acta Math., Bd. IV, p. 8.
PERIODIC FUNCTIONS IN GENERAL. 65
In the present discussion we shall also exclude the, roots of the equation /(£, 0) = 0. In other words, the function £ = (f>(u) is not allowed to take those values of u which make £' = <f>'(u) = 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 £o are simple roots of the equation (f>(u) — £0 = 0; for this difference can only become infinitesimally small of the first order since
<t>M = £o + ^ (u - u0) + ^ (u - u0)2 + ---- ,
and by hypothesis £0' ^ 0-
It follows that the quantities ab a2, . . . , am above are simple roots of the equation (f>(u) — £0 = 0, and consequently
-f
-». J c
-fo
if the integration is taken over a closed curve in the (f>(u) -plane that corresponds to a circle made by u about any of the points ai, a2j . . . , am.
We also saw that the above integral indicates the number of circuits made by the function </>(u) about fo m the ^(^)-plane. As this integral equals unity, we see that there is one circuit made in the positive direc tion about co corresponding to the circle made in the ?*-plane about any one of the points a. All values c i which belong to the surface included by the circuit about £o are therefore taken once by the function (j>(u) if u takes all values within the corresponding circle Cft about au. We describe about co as center a circle C with so small a radius that it lies totally within the above circuit S^ about co- We shall show that every value ci within this circle is taken once and only once by the function (j)(u) when u takes all possible "values within the circle Cu.
We saw that the integral
d6(u)
where <j>(u) is regular on and within the contour C, is equal to the number of points at which the value £ i is taken within C, provided each point is counted as often as the order of the zero of <j>(u) — £1 at this point. It follows under the given hypotheses that the above integral is always a positive integer.
If then £ is considered as a variable complex quantity on the interior of the given circle, the integral
j_ r
2*»J <
- c
is an analytic function of £. For since the denominator does not vanish for any point on the periphery of the circle, the elements of the integral vary in a continuous manner when £ varies. On the other hand, we knovr
66 THEOKY OF ELLIPTIC FUNCTIONS.
that the integral is equal to a constant. This integral considered as a function of £ must also be equal to a constant. If we let £ coincide with £0, the integral is equal to unity. It follows that every value £i which lies sufficiently near £0 is taken once and only once if u remains within the circle described about a^.
We draw circles as indicated above around all the points ai,a2, . . . , am. These points are the values of u which cause <J>(u) to be equal to £0- In the </>(u) -plane we draw the corresponding circuits around the point £o- Further we draw a circle around £0 as a center with a radius so small that it lies wholly within the circuits made about this point. Let £1 be a point within this circle. Then the value £1 is taken by <j)(u) for values of u once in each of the circles around 0,1,0,2, . . . , am respectively and con sequently at least m times.
We consider the quantity
where </>(u) takes the value £0 at the points u = a\, a2, . . . , am.
By hypothesis cf>(u) — £0 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 alt is denoted by - ^ — , it is seen that
u —a,,
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 <j>(u) would be a rational and not a transcendental function, it is seen by taking values of u sufficiently removed from the origin that <j>(u) — £o may be made arbitrarily small.
Suppose that £1 is a value 'of </>(u) which lies within the interior of the circle above. It is clear that for values of u sufficiently distant from the origin the function <j>(u) is equal to £1. We have also shown that besides this value of u the function <f>(u) takes the value ci at m other points and consequently (/>(u) takes the value £i at m + 1 points. By continuing this process it may be shown that there are an indefinite number of values which do not belong to the singular values of the function <p(u), and which may be taken by (j>(u) an arbitrarily large number of times.*
It follows from what we saw in Art. 38 that <f>(u) is a periodic function.
* In this connection see Picard, Memoire sur les f auctions enticres (Ann. EC. Norm. (2), 9, (1880), pp. 145-166), where it is shown that an integral transcendental function when put equal to any arbitrary constant has an indefinite number of roots which are isolated points on the w-plane.
PERIODIC FUNCTIONS IN GENERAL. 67
ART. 55. If the function <j>(u) has the properties expressed through the equations
/[*(*), 4^(10] - o,
(v)^(u + v}\ = 0, v) =
we have seen that the region of u may be extended by analytic continu ation to the whole plane without the function (f>(u) ceasing to have the character of an integral or (fractional) rational function for all values of the argument.
If <t>(u) has at infinity the character of an integral or (fractional) rational function, then <j>(u) is a rational function of u; but if the point at infinity is an essential singularity, then <j>(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 eu. We shall first show that eu + 27:i = eu for all values of u. Writing u = x -f iy, it is seen that
eu _ ex + iy _ g*(Cos y + i sin y) = ex cos y + iex sin y.
If now we increase u by 2 -i, then y is in creased by 2 x, and consequently MO
'277
= e* cos (y + Z ~) -h ie^ sin (y - = ex cos y + iex sin y = ex + iy = eu.
It follows that if we wish to examine the function eu, then clearly we need not study
this function in the whole ^-plane but onlv ^^kx_L__ J
o x
within a strip which lies above the X-axis Fig 4
and has the breadth 2 -. For we see at
once that to every point MO which lies without this period-strip* there corresponds a point MI within the strip and in such a way that the func tion eu has the same value at MO as at u\. For example in the figure
* Cf. Koenigsberger, Elliptische Functionen, p. 210. The lines including a period- strip need not be straight, if only the difference between corresponding points is a period.
68 THEORY OF ELLIPTIC FUNCTIONS.
Suppose that p = a + ifi is an arbitrary complex quantity, and con sider the equation
eu = p = a + ifi.
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
eU = 6X C0g y _|_ fox gjn y = p _ a _J_ ^
and consequently
ex cos y = a, ex sin y = p.
It follows that
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 a2 + /?2.
Q
To determine y, we have tan y = £ •
a
Suppose that yo is a value of y situated between 0 and n which satisfies this equation (we know that there is always one such value and indeed only one).
It follows also that
tan (y0 + 7i) = tan y0.
It appears then as if y0 + TT satisfies the conditions required of y. This, however, is not the case, since we have
cos (yo + TT) = — cos yo, sin (y0 + TI) = - sin ?/0,
and consequently the equations ex cos y = a, ex sin y = /? are not satisfied by the value y0 + x.
Hence within the period-strip the equation
eu = a + ifi is satisfied by only one value of u — x + iy, and this value of u is
u = log V a2 + /?2 +
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 ?/0 by 2 TT.
PERIODIC FUNCTIONS IX GENERAL.
ART. 57. We shall next study two other simple functions, cos u and sin u. These functions may be defined through the equations
cos u = ±(eiu + e~iu),
sinM = ^-.(eiu - e~iu}.
£ i It follows at once that
cos (u + 2 ~) = cos u, sin (u + 2 T:) = 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 x measured along the lateral axis.
It is evident that to every point MO lying without this period-strip there is a corresponding point HI within the strip at which cos u and sin u have the same values as at MO. For example in the figure
+ 6 -) = COS MI,
+ 6 -) = sin MI.
COS UQ = COS
sin M = sin
Suppose next that p is an arbitrary complex quantity, and let us see whether for the equation
cos u = p
27T
27T
U0
Fig.
there is always a solution. If there is one, there is an indefinite number. For if MI satisfies the equation, then from the above it is also satisfied by the values MI + 2 r, MI -f 4 -, • • • .
We shall show that there are always two values of u within the period- strip which satisfy the equation.
For writing
COSM = i(e*'" + e~iu} = p, we have
eiu + e-iu = 2 p.
Writing eiu = t, this equation becomes
f*-2jX + l*-0. (1)
From this it follows that
We thus have two values of t = e be MI and u2, so that therefore
t\ = cl'Ms t2 = el'"2. It follows that we have for iu\ and iu2 values of the form
Let the corresponding values of u
u2 =
where k\ and k2 are positive or negative integers.
70 THEOEY OF ELLIPTIC FUNCTIONS.
Dividing by i we have at once
ui = — iyi + ki2x, u2 = — if)2 + k2 2 TT.
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
t1 . t2 = I, oreiUleiU2 = I. We therefore have
and consequently
i(ui + u2)= 0 (mod. 2m), or
HI + u2 = 0 (mod. 2 7t).
i
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 n.
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 eu 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 /(«+ -a) -/(!»),
#u
/ and if we write u = 0, it follows
/ that
f(a) - /(O),
fl I that is, the function f(u) has
| at the origin the same value as it has at the point a in the u- "L plane; and also at the points' . . . , 3 a, 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 ^-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.
PERIODIC FUNCTIONS IN GENERAL. 71
Let u be a point in any strip. There is always a point u' 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, 2 a, . . . , 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 f(u) has the same value at both points. In the above figure, for example, u = u' + 2 a, so that/(w) = f(u' + 2 a) = /(>')• 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 eu, sin Uj cos u.
ART. 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 JTt
u
Such a function was e a . Consider next the series
fr=+x 2 in
<«-\ k v
f(~.\ _ v r, P a
J \ U ) - f f lrt & )
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 o; and, since the constants Ck 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
fc^X fc^pu
^=fe? ^T = 0(w), say,
where the dks 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 u = oo the exponential function is indeterminate (Art. 21), and for all other values of u it is seen that the function <j)(u) is one-valued.
ART. 60. Suppose that f(u) is a one-valued simply periodic function with period a = 2 to, and which has only polar singularities in the finite portion of the plane.
* Cf. Briot et Bouquet, Fonctions Elliptiques, p. 161.
71 THEORY OF ELLIPTIC FUXCTIOXS.
If w* put «,
em = * = his seen that
:r
- — •
T ~
in the Mane, when 0 varies from 0 to 2 r, the variable I describes cte about the origin with radius r, while in the M-plane the variable n the straight fine AA'. where A = - u» log r — M* log r + 2 w. Furl her, when * varies from 2arto4r7 « varies from ^' to ^ where again .I'-i" = 2«», etc.
Ifext if we gjve to i the value «*, it is seen that when I dLHUJJiB m circle about the origin in the /-plane with radius «, m deauBigii the straight line Bff. where
Hg.7. It foftows that m the t^plane the rectangle AA'BB'
uiucqponds to the ringhiduded between the two circles with radii r and * in the {-plane, and corresponding to the initial period- strip m the it-plane is the entire l-plane. Further, any period-strip is, as we may say, c^ormaUy represented on the Ijianr There being an in definite number of these strips, it is evident that to any value of / in the f^iaae UHMUMBwlii an infinite T^™J^^ of ^*lpi» ^ thr • piinr differing :;v i^:c-rril :j.il::rles : 2 _.
Suppose that the rectangle AfRI* m taken so as not to include any of the •J-e-B"aS— of /(«). Then if F(l) =/(M), it is seen that F(f) is regular aft al points at which /(«) k regular and consequently may be expanded by Lament's Theorem nt a aeries of the font
for afl values of i aitnated ••Hi'ii the ring-formed
to the rectangle AA'BB'. It also follows that for aU points vttut this rectangle, /(*) may be ntm convergent series of the form
/(•)= J ^^
T. = — X
= T^a,cosm^* + 6*sinm^«^
PERIODIC FUNCTIONS IN GENERAL. 73
Prof. Osgood, loc. cit., 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 within a period-strip take a prescribed value a finite number of times.
Suppose that <f>(u) is such a function. The function $(u) behaves within the period-strip in a similar manner as do the rational functions in the whole plane. For if w = 4>(w) is a rational function of u7 then <&(u) 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 (p(u) which we are considering must be indeterminate at infinity in the direction of the line through 0, a, 2 a, . . . (see Fig. 6). For let u0 be a point within the initial period-strip. Draw through u0 a line parallel to the line through 0, a, 2 a, - • • . On this line, starting from u0? 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 <J>(UQ). Next if we start with another point u\ and proceed to infinity in the same way as before, the function will take for the infinitely distant point the value <£(MI). Hence at infinity there appear all possible values which the function <f>(v) can take, and the function is thus said to be indeterminate at infinity (cf. Art. 3).
ART. 62. Let w = <f>(u) be a simply periodic function with the period a which satisfies the three postulates made above. Further, write
so that t and ir 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 <£(u), then w = <j>(u) has a definite value. It is thus shown that to every value of t there belongs a definite value of IT. If next we consider not only one period-strip but the whole w-plane, then there belongs to the given value of t an infinite number of values of M, namely in each period-strip one value. And if u is one of these values then all the other values have the form u 4- A-a, where k is a positive or negative integer. If we write all these values in 6(u), then IT = <£(M) takes always the same value, since o(u 4- ak)= <r>(u). Hence
* See also Henri Lebesgue, Lemons sur les scries trigonomctriques.
74 THEORY OF ELLIPTIC FUNCTIONS.
also when we consider the whole w-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 ui, u2) . . . , um, and let the corresponding values of t be
2 «• 2 « 2 jri
— u{ u2 — um
ti — e , t2= e a , . . . , tm = e a
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 Ui,u2, . . . , um by integral powers of a, we still have for t always one of the values ti, t2, . . . , tm.
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 m-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 rath degree in t, say,
F(w, t) = 0.
Solving this equation we have
w = <f (0,
•.
where ifr 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
2m u
that w is a rational function of t = e a .
We have then the important theorem:
Every simply periodic function <j>(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
— M times is a rational function of t = e a , where a is the period of <j>(u).
All such functions may therefore be written in the form
k = m 2xi
-«— v k — u
W =
fc— «
, a
fc=0
where the Ck and dk are constants.
* A treatment of simply periodic functions which have essential singularities else where than at infinity is given by Guichard, Theorie des points singuliers essentiels [These, Gauthier-Villars, Paris. 1883J.
PERIODIC FUNCTIONS IX 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 dn+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
k=0 k = 0
where ^ is a rational function of t. Hence the points t = ± oo , t = 0 are not essential singularities of y]r(t) and consequently also </)(u) has definite values for u = ± °o . In other words, the end-points of the period-strips of the function (f)(u) are not essential singularities. We may write the above equation in the form
(cm - dmw)tm + (cm_i - dm_! w)tm~l + '. • • + (c0- d0w) = 0,
2m
where m represents the number of values which t = e a can take for a given value of w, or, in other words, the number of points in each period- strip at which w = <j>(u) takes a definitely prescribed value. We call m the degree or order of the simply periodic function w = (f>(u) (cf. again Art. 10).
The functions cos u and sin u must be expressible in the above form, since
u
for them a = 2 TT, and t = e 2* = eiu. Further, these functions take a prescribed value twice within a period-strip (cf. Art. 57) and are conse quently simply periodic functions of the second degree. For them we must have m = 2, which, indeed, is seen from the relations
1 /. , l\ t2'+ Q.< + 1 cosu = }(*«• -f e-*) -5V + l)= 2(0.<* + 0 5
rinu- i *>- *-* = 1 t*
2i 0-t2 + t
Owing to the relation <j>(u) = ^(0 many of the properties of simply periodic functions may be changed into properties of rational functions; for example, the function <f)(u) has as many zeros as it has infinities in each period-strip.*
* Cf. Briot et Bouquet, Fonctions Elliptiques, p. 161; Forsyth, loc. cit., p. 215; Osgood, loc. cit., p. 409; Burkhardt, Analyt. Funktionen einer komplexen Verdnder- lichen, p. 161.
76 THEORY OF ELLIPTIC FUNCTIONS.
THE ELIMINANT EQUATION. ART. 64. In the case of the function eu it is seen that if
w = eu, then ~r - u = 0; du
and if w = cos u, then {t£f - (1 - w2) = 0,
\dul
the latter differential equation being satisfied also if w = sin u. We note that these three functions have the characteristic that each of them satisfies a differential equation in which the independent variable u does not explicitly appear.
From the previous article we saw that, if w is a simply periodic function, then
w = </>(
where ^ is a rational function.
2m,
Further, since e a = t, we have
^=r(t) ,
du . a
where ^i is also a rational function.
By eliminating t from the two expressions we have the eliminant equa tion (Art. 34)
where / denotes an integral algebraic function.
In Art. 41 we said that if there existed an eliminant equation for a one-valued function w = </>(u), then <f>(u) had an algebraic addition- theorem and belonged to one of the categories of functions I. Rational function of u, or v
2rt
- 11
II. Rational function of e a (simply periodic), or III. Doubly periodic function.
In his Cows d' Analyse a VEcole Poll/technique, in 1873, Hermite observed that if the equation
dw
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 irrationalities than the square root of a polynomial of the fourth degree, if w 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
\ du/ the category of functions defined in Art. 61?
Such a function must, as we have already seen, be expressible as a rational function of t, say "^(t), and its derivative is also a rational function ^i(t).
If we put — = v, 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 ^(0 and for v a rational function ^i(£), the equation f(w, v) = 0 must be identically satisfied for all values of t. We may therefore express w and v rationally through a parameter t in the form of the equations w = ^(0, v = tyi(t). Curves in which such a rational representation of the variable t is possible are known as unicursal.^
If then an integral of the differential equation
is to belong to the category of functions which we are studying, the equa tion
f(w, v) = 0
must represent a unicursal curve.
But this condition is not sufficient. For if f(w, v) = 0 represents a unicursal curve, there is an infinite number of ways in which w and v may be expressed rationally in terms of t. But among these ways there is one which is such that t for every prescribed pair of values of w and v takes only one definite value. Further, if w is a function of our category,
it must be a one-valued function of u, and consequently 0 *» — is also a
du
one-valued function of u.
But if w and v are given, there is (as we have just seen) only one value of t which can be associated with them. Hence if w is a function of our
* Cf. Cayley, Lond. Math. Soc., Vol. IV (1873), pp. 343-345.
t The name is due to Cayley, Comptes rendus, t. 62, who derived the funda mental properties of these curves. See also Clebsch. Ueber diejenigen ebenen Curven deren Coordinaten rationale Funktionen eines Parameters sind, Crelle, Bd. 64.
78 THEOKY OF ELLIPTIC FUNCTIONS.
category, the parameter t must be a one-valued function of u. Further, since
it follows that
*. _ ±iOO _ R(t) du *'(0 ()'
t
where R is a rational function of t.
We have consequently established the following: The integrals of the differential equation
may be functions of our category, first, if the equation f(w, v) — 0 represents a unicursal curve;* second, if w and v are such rational functions of a parameter t that to every pair of values ofw, v there belongs only one value of t',
and third, if the parameter t, as determined through the equation f Iw, — ) = 0,
V du/ is a one-valued function of u. It does not, then, necessarily follow that
these integrals are simply periodic, for they may be rational functions of u. ART. 66. The parameter t determined from the differential equation
£ = R(t) du
must be a one-valued function of u.
We are thus led to the question: What is the nature of the function R(t), that t be a one-valued function of uf
If we consider first the differential equation
where the g's are integral functions of t, the condition that an integral t of this equation be a one- valued function of u is that g0 be of the 0 degree, 0i of the 2d degree, g2 of the 4th, . . . , gm of the 2 rath degree, f
We shall derive these results for the case m = 2 in Chapter V, and from this it will be seen in the simple case before us, viz.,
du
* A simple method of representing w and v as rational functions of a parameter t, when this ca"n be done, is given by Nother, Math. Ann., Bd. Ill; see also Liiroth, Math. Ann., Bd. IV.
t Cf. Forsyth, loc. cit., p. 481, where other references are given.
PERIODIC FUNCTIONS IN GENERAL. 79
that t is a one-valued function of u, if R(t) is a rational integral function of the 2d degree in t. It then we write R(t) = a0 + a it + a2t2, it follows that
r dt
J aQ + ai$ + «2^2 where 7- is a constant.
We have four cases to consider:
(1) Suppose that a2 7^ 0 and that the roots of the equation O,Q + atf •f a2t2 = 0 are not equal.
We may then write the above integral in the form
u+r=r * ._J rp u
J a2(t - a) (t - P) a2(a - P) J [t - a. t - p J
= ^~?ilog
It follows that
a2(a -/3) t - a t - p * t-a
_ -02(a-0) (u + y)
-
and consequently t may be determined rationally in terms of an exponential function of u. Since w = ^(0, where ^ is a rational function, it is seen that in this case w is a rational function of an exponential function and therefore belongs to our category of functions.
(2) Suppose that a2 ^ 0 and that the roots of the equation a0 4- <M + a2t2 = 0 are equal.
We then have
= r
J
a2(t - a)2 a2(t - a)
It is seen that in this case / is a rational function of u, and since w is a rational function of t, w; is a rational function of u and does not belong to our category of simply-periodic functions. (3) Suppose that a2 = 0. We then have
f
J
-log (a0
It follows that a0 + <M = eai(~u+y\ so that w belongs to our category of functions.
(4) Suppose that a2 = 0 = a^ It is evident then that
Cdt t
u + T = I — = — > J «o ao
or t = a0(u + 7-).
In this case w is not a simply-periodic function.
80 THEORY OF ELLIPTIC FUNCTIONS.
EXAMPLES 1. Consider the differential equation
.. dw
or, if /. v = — ->
du
f(w,v) = w* -(w- v)3 = 0.
We must first determine whether this equation represents a unicursal curve.
If we write w — v = tw,
then is w* - w3t3 = 0,
or w = t3 - ^(0 ;
and v = w(l - t) = ?(l - 0 = <Ai(0-
It is thus seen that w and t; may be rationally expressed through t.
We must next see whether t, as thus determined, has a definite value when w and v have prescribed values.
Since w = t3, to one value of w there correspond three values of t, but only one of these can satisfy the relation v = w(l - £)> when a fixed value is given to v. Hence to every pair of values w, v there belongs a single definite value of t. We further have ji'(t) = 3 t2 and
du V(t) 3?
It follows that
This is the first case considered above where a2 = — ^, « = 0, /? = 1. Integrating we have
or
I _ e$(u+y)>
all(* W = t3 = ei(u+ )-.3 '
It is thus shown that w belongs to the category of functions considered. 2. Determine the integrals of the differential equation
'duj \du I \du
that is, of dw
f (w, v}= (v- 2)2 + (v - l}w2 = 0, where v = —
du
It follows that 9L=2W(V- 1) =0,
SI = 2(v- 2) + w2 dv
and consequently w = 0, v = 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)t2 = 0,
or v = 1 - \ = ^(0
and w = - t = i,KO-
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
B0.fa.ML.-!,
du ^'(0
and consequently we have the fourth case. It follows that t = - (u + f)
and w = \- u + f,
u+ r
and being a rational function of u does not belong to our category of functions. 3. Show that the integrals of the differential equation
J \w' T~ ~ \ du
are simply periodic functions. Note that the equation
f(w, v) = (v- 2)2 (t; + 1) = (3 w2 + 2)2
is satisfied by
v =3(1 + t2) (1 + 3t2),
w = 3(t + t3). 4. Show that the integrals of
\du
3_ /A£V
) \du)
are rational functions of u. [Briot et Bouquet.]
5. Show that the integrals of
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*u, we know that ^ = 2 w, say, is its period.
The constant /i is taken real or complex and different from zero or infinity. tm
Write t = e^ = e w , and consider the function <£(w) = ^(t), where here ty 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 \u be r and | 2 a> \ be s, so that
2 CD se10 s
= -[cos (if/ - 0) + i sin (if/ - 0)].
If R denotes the real part of the complex quantity after it, then is
--WcosOA-0)=^. 2 ajJ s s
Fig. 8.
Hence for all values u within the period-strip we have
t*>.
We assume that <f)(u) = <f>(u + 2 to) and that <f>(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 <j>(u) is a one-valued function of u, it is also a one-valued function of L Let u\ be a point within the period- strip. We therefore have in the neighborhood of ui
- G
P(u -
(A)
where G denotes an integral function of finite degree (including the Oth degree) and where P is a power series with positive integral exponents.
82
DOUBLY PERIODIC FUNCTIONS. 83
Ui>n , . iri
(u-ti,) - /
Let ti =e ", so that e "=-;
t\
f (M-tti) -
further, write — = 1 + r or e w = 1 + r.
*i It follows that
(u - ui) ^ = log (1 + r) = r - ir2 + IT* - . . • - oj 23
This series is convergent for values of r, such that
0< |T|< 1.
But we had r = - - 1 = ^-^ .
«i <i
If then \t — t\\ < | £1 |, we have the convergent series
+
3
This expression for w — ui substituted in the equation (A) shows that the function <j>(u) considered as a function of t is one- valued and has the same character for t = ti as it has for u = MI.
ART. 68. With regard to the f unction ^(M) = TJr(t) two cases may arise :
(1) the two points t = 0, t = oo may be regular points of the function. In this case ^(t) is a rational function, as there is no essential singularity.
(2) At least one of the points t = 0, t = oo may be an essential singularity. In this case we shall show that the function (f>(u) has another period 2 a>' ', say,
and we shall prove that the ratio — — is not a real quantity.
2 a)
We must show that within the period-strip there are values which may be taken an arbitrarily large number of times by <j>(u). It follows then as in Art. 38 that there exists another period 2 &/.
Let £0 be a value which </>(u) may take. This point may lie anywhere in the finite portion of the period-strip excepting the singular values of u defined in Art. 37.
Two cases are here possible: (1) The function <f>(u) = ^(t) may take the value £0 an arbitrarily large number of times. The theorem is then proved. (2) The function <£(w) may take the value ?0 a finite number of times, say m, within the period-strip. Let the corresponding values of t be ti, t2, . . . , tm.
In the neighborhood of any one of these points develop - - by
r (0 ~~ £o 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
THEOEY OF ELLIPTIC FUNCTIONS.
possibly both) of the points t = 0 or t = oo. There are then values £1, say, in the neighborhood of £0 which are taken by (f>(u) = ty(t) at least m + 1 times. By continuing this process it is shown as in Art. 38 that (/>(u) must have another period 2 aj' and consequently
<l>(u + 2 to) = (j>(u), (f)(u + 2a)')= <j)(u).
ART. 69. It follows at once from the development of <j>(u) in the neigh borhood of u\ in the form (Art. 53)
that there are no points in the immediate vicinity of u\ at which (f>(u) has the same value * (Art. 8) as it has at u\. We may therefore draw with HI as center a circle with radius p which is so small (but of finite length) that within the circle the function (f>(u) does not take the same value twice. Further, since <j>(u + 2 w) = <f>(u), it is evident that \2a)\ > p, where p is a finite quantity.
The point in the w-plane which represents 2 oj we call a period-point. Since 2 a>' is also a period-point, it is evident that
2(0'
and as above
- 2o/)=
2aj-2o)' > 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 w-plane, there are only a finite number of period-points within this surface.
If 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 (S) an infinite number of parallelo grams with finite area, which is impossible. Fig. 10.
* Cf. Burkhardt, Analyt. Funkt., p. 124; Forsyth, loc. cit., p. 59; Osgood, Lehr- buch der Funktionentheorie, p. 398.
DOUBLY PERIODIC FUNCTIONS.
85
ART. 70. We consider the following question: If 2 co = a and 2 a/ = 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 a = ie
and consequently, if b lies upon the straight line Oa, then , = « . A = 0 _i_ -
j( JT
We therefore have
u —
? = ± -' Fig- U.
6 s
that is, the ratio ^ is a real quantity. The above question may consequently 0
be expressed as follows : Can the quotient of two periods a and b be a real
quantity f
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 p 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 b 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
I 6 - ka I < I a I .
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
primitive 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 , a
write - » 22, where p2 and pi are integers that are relatively prime. a pi
It follows that
b a
— = — = a, say,
P2 Pi
and consequently 6 = p2a and a = pia. To show that a is a period we determine two integers qi, q2, such that
+ 22= I-
We know that there are an infinite number of solutions of this equation. Multiplying by a we have
Piaqi + p2aq2 = a, or qid + q2b = a.
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
— -, n + l are consecutive convergents, then
Un Un + l
L_ = ]* , where e < 1.
2
Un Un + l UnUn + l Un
Hence if we expand - in a continued fraction and if ^~ is the nth conver- gent, then is ' a °»
;~?-F5. or *J>-r+-T-
a On On On
Since dn may be made indefinitely large, it follows that
| dnb — ~rna | < p, where p is as small as we choose. Further, since dn and fn 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.; Me>ay, Ann. de I'Ecole Norm. 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 a>2, 2o>i be any two periods whose
ratio is real. The ratio — — may always be taken positive; for if it were 2 MI
negative we might substitute the period — 2 aj2 in the place of -f 2 a>2.
We lay off the periods 2a>i, 4o>i, 6<t»i, . . . ; 2 a>2, 4 a>2, 6 a>2, . . . upon the same straight line (cf. Art. 70).
It is evident that 2 «,2 = 2 «,«,, + 2 a*,
where mi is a positive or negative integer, and 2 cu3 < 2 0*1. Similarly we write 4o;2 = 2 m^i + 2 o>4,
w2 being an integer, and 2 o>4 < 2 o» j . It follows that 2a,2 _ 2 mifl>i
6 o>2 — 2
and consequently the quantities 2 o>3, 2 o>4, 2 w5, . . . are all periods.
There are two cases possible: (1) These quantities are all different; or (2) they are not all different. Suppose that 2 o>3, 2 w4, . . . are all different, and consider the n quantities 2 w3, 2 o>4, . . . , 2 a>n + <2, to which we also add 2 wi, in all n + 1 quantities.
Divide the distance between 0 and 2&>i into n equal parts; then, since each of the quantities 2 0^3, 2 w4, . . . , 2 cjn + 2 is IGSS than 2 ^i, two of these quantities must lie within one of the n equal intervals. Let these two quantities be 2 w& and 2 a>i. It is clear that 2 Wfc — 2 o>j 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
2iOJq + 2 == 2 CJp + 2)
so that 2 qa>2 — 2 mqa)i = 2 paj2 — 2 mpa)i,
mq and mp being integers; and from this it is seen that — — must be a
rational quantity.
ART. 73. We mav prove as follows that the ratio - cannot be real.
2aj
For take in the period-strip of Art. 67 two points u2 and u\ such that u? — u\ = 2 a/. In that article we saw that
and
88
THEORY OF ELLIPTIC FUNCTIONS.
It follows that
If now then is
,/«2_I V O
\ *
< i.
is a real quantity,
2w < 1, or 2 a/ < 2a>.
Fig. 13.
We thus have two periods which lie along the same straight line, of which one is less than the primitive period 2 a>, which contradicts the notion of a primitive period. Hence 2 w and 2 a>' must have different directions.* ART. 74. There exist two primitive periods through which all other periods may be expressed.
Geometrical Proof.
We shall first show that it is always possible to form a period-parallelo gram which is free from periods. Suppose that in the period-parallelo gram formed of the periods a and b 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 O/? is free from periods. Of course we have assumed that Oa is not an inte gral multiple of another period.
It is evident that 7- is a period since a + /? = f] and it is also evident that there can be no period-points within or on the boundaries of afiy.
If for example A were a period-point on the side /??-, then through A we could draw the parallel to the side 0/9 which cuts the line Oa in jj.. We would then have a period-point at /i, 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 Fig 14^
on a?-.
Suppose next that a period-point v lies within the triangle fifa (Fig. 15); then by completing the parallelogram ftvajj. it is seen that JJL is also a period- point and lies within the triangle 0/fo, which contradicts what we saw above.
* Picard, Traite d' Analyse, t. 2, p. 220, gives an interesting proof of this theorem; see also other proofs in Hermite's " Cours " (4me e"d.), p. 217, and Goursat, Cours d' Analyse, t. 2, No. 314.
DOUBLY PERIODIC FUNCTIONS. 89
We thus see that within the entire parallelogram Opra, the sides included, there are situated no period-points except at the vertices. It is also evident that if the whole u-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-
o a
Fig. 15. Fig. 16.
grams, there exists in the initial parallelogram Op fa a point uf 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 + ip,
where k and I are integers.
It follows that a one-valued analytic function cannot have three inde pendent periods a, 6, c; for, as we have just seen, these three quantities are expressible in the form
a = ka + ip,
b = k'a + I'p, c - k"a + l"p,
where the fc's and I'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 (f>(u) within the initial period-parallelogram whose sides are, say, a = 2 a), t3 = 2 a)'. In this parallelogram the function <j>(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 at. (See Fig. 17).
90 THEORY OF ELLIPTIC FUNCTIONS.
We may write
|^- = T = O + if),
2 co where by hypothesis \ p\ ^ 0, since the ratio ~- is not real. All points
2oH-2w' within the interior and on the sides of this period-parallelogram may be expressed in the form
u = 2 tco + 2 t'a)', o 2u> where 0=2=1, 0 ± Z' ^ 1.
pig 17 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 mco + 2 m'aj'
where m and m' are real quantities.
It follows that ~ *
JtL=m + m>^. 2 co (o '
and since — is a complex quantity, -^- is also complex, = of + ip' , say. CD 2 co
We thus have
m' (o + ^),
or <7r = m + m! a, pf = m' p.
It follows that
m'=£-, m = a'- &-<*. p p
Since p is different from zero, the denominator does not vanish, and consequently m and m' are determinate quantities.
It is thus seen that every complex quantity w may be uniquely written in the form
w = 2 mto + 2 m'co',
where m and m' are real quantities.
ART. 76. Two points w and w' are called congruent if
w - w' = 2 kco + 2 Ico',
where k and I are integers. The fact that w is congruent to w' may be written
w = w' (modd. 2 co, 2 to') ;
or, if no confusion can arise,
w = w'.
DOUBLY PEKIODIC FUNCTIONS. 91
It is also clear that, when w and w' are congruent, then w — w' is a period of the argument of the function. If we write
w = 2 rnu) + 2 rn'oj', w' = 2 nw +2 n V, and if w = wf (modd. 2 co, 2 a/),
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 0 . . 2 co and 0 . . 2 a>' 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 a> and 2 a/.
For let 2 aj = I.
c\ /
Then, since - — = a -f ip,
2w
it is seen that
| 2 a/ |
Further, since 2 a> + 2 a/ = 2a>(l + cr + 1,0), it follows that the length of one diagonal of the parallelogram is
2 w + 2 a/
while the length of the other diagonal is
Represent by L the longest of the four sides
| 2 aj |, | 2 w' |, | 2 oj + 2 w' , | 2 a/ - 2 w |.
Next divide the two sides 0 . . 2 a> and 0 . . 2 a/ respectively into n equal parts, so that the period-parallelogram will be divided into n2 small parallelograms. The distance between any two points situated
within one of the smaller parallelograms is not greater than — •
n
If there are periods that cannot be expressed through integral multiples of 2 a; and 2 a>' and if 2 a>i is such a period, we shall construct the con gruent point which lies within the initial period-parallelogram.
We ma write
where 0 =#1 < 1 and 0 = ,«i' < 1.
This point must fall within or on the boundaries of one of the small parallel ograms.
Admitting (Art. 69) that every period has a definite length, it may be shown as follows that r«i and («i' are rational numbers.
92 THEORY OF ELLIPTIC FUNCTIONS.
We have the congruence
2 0)i = 2 fJLiOJ + 2 [JLl'o)' ',
and in a similar manner we form
2 • 2 0)1 = 2 z2w -f 2 *2'^'
2(n2 + l)o>! =
where 0 = /^ < 1 and 0 = /*/ < 1,
(& = 1, 2, . . . , n2 + 1).
If these n2 + 1 points in the initial period-parallelogram are all different, at least two of them must fall within or on the boundaries of one of the small parallelograms, and the distance between these points is therefore
less than — . As n can be made arbitrarily large, there are then periods n
that are arbitrarily small, which is contrary to our hypothesis.
It follows that at least two of the n2 + 1 points must coincide, in which event we would have
2 pcoi = 2 /j.pO) + 2 /ip'o)',
2 qo)i = 2 JUPOJ + 2 /*P'a>', and consequently 2(p — q)co\ = 0 (modd. 2 o), 2o>'),
where p and q are both integers. We have thus shown that an integral multiple of 2 MI is congruent to the origin. Since 2 o), 2 a>' are a pair of primitive periods, it follows from the theorem of the next article that 2 a> i must be congruent to the origin.
ART. 78. Jacobi (Werke, Bd. II, pp. 27-32) proves the following theorem: // a one-valued function has three periods 0)1, 0)2, ^3, such that miaji + m2o)2 + m3a)3 = 0,
where m\, m^ niz are integers, then there exist two periods of which <DI, o>2, ^3 are integral multiple combinations.
We may assume that there is no common divisor other than unity of mi, ni2, m^. Let d be the common divisor of m,2 and m^. Of course, d = 1 when m2 and w3 are relatively prime.
Then, since — a>i = — 7^aj2 — — 6t>3 and the right-hand side is an
d d d m
integral combination of periods, it follows that — ^i is a period. Since
CL
— is a fraction in its lowest terms, when expressed as a continued frac- d
tion it may be written mi _ p = . l_
d q dq'
where ^ is the last convergent before the proper value. It follows that 3
y~-aji - pcoi = ± -wi = aj, say, d d
where oj is a period.
DOUBLY PERIODIC FUNCTIONS. 93
Let ^ = m*> ^='*3/>
d d
so that mico -I- m2'a)2 + m^'ais = 0.
Change^2- into a continued fraction, taking- to be the last convergent
m3' s
before the proper value, so that
Z^2_ _ L = ±
ra3 s sm3
Then rco2 + sa)3 being an integral combination of periods, is a period a*', say.
On the other hand,
± co-2 = a)2(sm2 — 7*7/13')
= — nutria — s(mi<jj 4- m^'aj^)
also
m2ajf',
and <^i = ^/tt».
Hence two periods &>, o>r exist of which o>i, o>2, ^3 are integral multiple combinations.*
We may conclude from the foregoing that All one-valued analytic functions are either
(1) Not periodic, or
(2) Simply periodic, or
(3) Doubly periodic.
Triply 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 la). 2 a/ be a pair of primitive periods, and suppose that
• !|*.-.« + *
2 CD
where p is positive, that is,
We wish to form another pair of primitive periods 2 io, 2 o>' such that
* 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 aj = 2 pco + 2 quj', 2at' = 2p'a) + 2q'a}',
where p, q, p', qf are integers.
Further, p and q must be relatively prime, for otherwise 2 aj would be the integral multiple of a period. The integers p' and q' must also be relatively prime. It follows that
pq' - qpf
Since 2 to and 2 to' are to be a pair of primitive periods, the period 2 must be expressible integrally through them. It follows that
and
pqf - qp' pqf - qp'
must be integers. We further have
pq' - qp'
c uentl
and f P are integers.
pcf - 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 ±1. It
would then follow, since ^- 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 = -fl and A = — 1 admissible? We required that
0 and R
We have
257 2 y/qj + 2 9 V
h
Since — = a + ip, it follows that
2 5i i[p + 9(cr + ip)] (p
and consequently
ip) ~(p' + q'o}qp + (p .r ,
%*/%) (p + qa)2 + q2p<
DOUBLY PERIODIC FUNCTIONS. 95
As p is positive by hypothesis, we must have pq' — qpf positive in order to fulfill the condition
It follows then that
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 = pq' — 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
2 5 = x + iy and 2 a/ = x' + iy*, the area of the corresponding parallelogram is
± (xi/' - ?/.r')- If further,
2 cu = c + iy and 2 u>' = £' + iif,
the area of the corresponding period-parallelogram is
It follows that, if
2 5 = 2 poj + 2 quj' and 2 at' = 2 p'aj + 2 q'a)',
, x = p; + q;'. ( x* = p'* + gT,
then I and {
y = py + gy';
and consequently
But here pq' - qp' = 1.
Hence a primitive period-parallelogram is not unique.
The linear substitution
2 Z> = 2 paj + 2 qa>',
2 5' = 2 p'a) + 2 q'a)' is denoted by
•P, >', 3'.
* Cf . Briot et Bouquet, Fonctions Elliptiques, pp. 234, 235, and pp. 268 et seq.
96
THEORY OF ELLIPTIC FUNCTIONS.
One of the substitutions which satisfies the condition
A = pcf - qp' = 1
r *iii
I- i , oj
18
In this case we have
2a) = 2cof,
2 £' = -2 co. A second substitution which satisfies the same condition is
or
2 a) = 2 co, 2 £' = 2aj
2 a/
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 Pf none can be situated on any part of the periphery of the first circle except the shaded arcs P\P% and PsP±. On these arcs there may be two periods differing by 180 degrees and possibly four periods.
In the last case the period-points
Fig. 18.
must lie at the four points of intersection of the circles, viz., Pi} P2, PS 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 PEKIODIC 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 2 a/ must lie to the left of 0 . . 2a>. We also know that | 2 a/ | > \ 2 a> \ . 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 2mw, where m 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 0 . . 2 to . . 2 ujf has only acute angles, the right angle being a limiting case.
We write r = =
2co
Fig. 19.
where
< 1.
Owing to the substitution
we may so choose 2 to' that
It follows that t3 > i V§. If further we write
h = q = eTri = it is clear that I < _ v/3 <
a fact which we shall find to be very important in the development of the
Theta-functions (Chapter X).
ART. 82. We have interpreted the equa tion A = pq' — qp' = 1 as denoting that the parallelograms formed on pairs of primitive periods have the same area. Let 2 5, 2 5' be a pair of primitive periods. The quan tities 2 to and 2 to' determine a triangle, and all such period-triangles have the same area. Let 1251=7.
2Z2T
Fig. 20.
3l2
Then if
^ = « + #, 9 (it
the area of the
co
triangle is ^- and that of the period-parallelogram is /9/2. This quan- &
tity being constant for all equivalent primitive pairs of periods, we have
const.
0-
I2
98 THEORY OF ELLIPTIC FUNCTIONS.
From this it is seen that /? is a maximum when I is a minimum. If then P is to have its greatest value, we must choose the first period 2 a> so that it has the smallest possible value.
If the ratio of the periods is a pure imaginary, then a = 0 and /? = 1 . In this case
EXAMPLE
If ton co2 and WA are periods of <j>(u) and if
29w3 = 17 0)^ + 11 <*}2,
show that
co = 5 co ! + 3 w2 — 8 w3
w' = 3 ft>! + 2 w2 - 5 w3 are a pair of primitive periods of <j>(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
k = + » . 2 ?n
V i * — u
is a simply periodic function which can be developed in positive ascending powers of u — UQ, and which is not indeterminate or infinite for any finite value of u, provided the constants At 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 <j>(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 doubly 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 + Ib,
where k and / 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 within this parallelogram, it remains in the
99
100 THEOEY OF ELLIPTIC FUNCTIONS.
whole plane less in absolute value than M . But an integral transcendental
function x 9 ,•?•?,
g(u) = a0 + a>iu + a2u2 + a3u3 + • • •
which remains finite for arbitrarily large values of u is a constant, since g(u) can remain finite only if &i = 0 = a2 = #3 = • • * • The following is a more direct proof of Liouville's Theorem.
fc=+oo 2iti
If $(u)= ^ Ake a U,
the condition that ^ + &) = ^}
/c=-oo
and consequently &— z
/fc — 6
Since - is an irrational quantity, e a ^ 1, and therefore
Aft = 0 (fc = ± 1, ± 2, . . . ). It follows that
&(U) = AQ]
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 \
where &(u) and W(u) are integral transcendental functions. We may write fc=+oo 2xi /b=+<x> k2™ u
"' ^^= Bke a U>
*<«>. has
k= -oo k=-<*>
where Ak and Bk are constants, so chosen that the two series are convergent.
Since $(w) and *F(u) both have the period a, their quotient <J>(u) has the period a. We therefore have to bring it about that the quotient ^ ' also the period 6.
We must so determine the functions &(u) and W(u) that
&(u + b)= T(u) $(M), V(u +6)= T(u)V(u),
where 77(w) is a function of u. If we succeed in this, then
or ^(w) has also the period 6.
CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 101
It will be advantageous to make our choice so that <&(u + b) has the same zero as <£(&), and consequently
does not vanish or become infinite for any finite value of u. This will be effected if we write
T(u) = eG^u\
where G(u) is an integral function in u.
We have then to seek a function 4>w and a function "^u so that
&(u + a) = <J>(iO, ¥(M + a) =
$(M + 6)= e°M3>(u), V(u + b) =
We shall next bring about a further limitation in that we determine (tt) and "^(w) so that G(u) is an integral function of the first degree in u. We will then have
4>O + a)= <fr(w), ¥(K + o)= ¥(w),
$(M + 6)
where A and /* are constants which are at our disposal. We shall see that there is an infinite number of such functions.
Hermite * called them "doubly periodic functions of the third sort (espece)."
If <£>(M + a)= v$(u) and $(u + b)= v'<E>(w), where v and i/ are con stants, one or both being different from unity, then &(u) is a doubly peri odic function of the second sort; and if v = 1 = i/ wre have the doubly periodic functions of the first sort, which are properly the doubly periodic functions.
Note that the wrord 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" (4me e"d.), pp. 227-234; Hermite, Xote sur la theorie des fonctions in Lacroix, Calcul (6me &L), 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.,Vo\. Ill (1848); Hermite, CEhvres, 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; Painleve, 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 k2ni
$(w) = 2) Ake a ",
k= -oo
it follows at once that
.6
If with Hermite we write Q = e °, it follows that
*=+» k*™u
&(u + b)= 2} AkQ2ke a . , (1)
On the other hand we had
If on the right-hand side we write for 3>(u) its value and put A = =-^ g, we have
<b(u + b)=e^i\Ake£U(k+a) (2)
k= —oo
In this formula k is an integer and we shall choose the quantities so that g is also an integer. 2™
If further we write t = e a and equate like powers of t in formulas (1) and (2), we have for the determination of the A' a the formula
A (~\2m — pf-A slmty — & -n-m-g-
If we take the logarithms of both sides of this equation, we have
/£ + log Am-g = log Am + 2 m log Q + n 2 ni, (i)
where on the right n 2 m has been added, since the logarithm is an infinitely multiple-valued function.
We shall further write 6
. o *l ~ v
H = m — v, so that e^ = e c = Q", a
or jj. = v log Q.
Since the constant /* is perfectly arbitrary, v is also arbitrary. It follows directly from (i) that
= 2 m — v +
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
Cm-g -cm = 2m-v + n , — ^ . (iii)
CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 103
To determine first a particular solution of this equation, write
ck = ak2 + pk,
where the constants a and ft are to be determined. Since cm-Q = a(m — g}2 + 3(m - g) and
cm = am2 4- fim, we have from equation (iii)
— 2 amg + ag2 — pg = 2 m — v + n — '-^—-
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
— 2 ag =2, ag'2 — t3g = — v + and consequently
We may give to the arbitrary constant v a value and we shall write v = g. It follows at once that
1 - n2-i
These values written in the formula
ck = ak2 + Pk will give the particular solution of the equation
cm-g — cm = 2 m — v + n - — '^— • (iii)
We may write the general solution in the form cm = am2 + pm + Cm,
where Cm is a function of m.
Writing for cm its value, we have
l°%Am = am2 + Pm + Cm, or
J _ pam- log Q + (0m + Cw) log Q -*1 m &
Writing for a its value from above and putting pm + Cm = Dm, we have
= Q~~° eL
104 THEORY OF ELLIPTIC FUNCTIONS.
Finally, putting Dm log Q = log Bm, we have
_ m2
Am = Q ° Bm,
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
AmQ2m = e>*Am-g, it follows that
_ m_2 _ (TO-?)'
Q2mQ °Bm = Q°Q * Bm-0, or Bm-g = Bm,
where m and g are integers.
The integer g being arbitrary we shall write g = — k, where & is a positive integer. We thus have
Bm+k = Bm.
It follows at once that
Bk = B0, Bk+i = BI, Bk+2 = B2,
B2k-i = Bk-i,
B'2k = Bjf = BQ.
We thus see that the constants BQ, BI, B2, . . . , Bk-i repeat themselves but are otherwise quite arbitrary.
It has thus been shown that the function
BmQke
satisfies the functional equations
$(u + a) =
This function <&(u) is the most general integral transcendental function which satisfies these two equations. It contains the k arbitrary constants
~D D E> O
•DO, o.it k>2, - • • , *»*-!*
ART. 86. It remains to be proved that the series through which the function 4>(w) 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
m= +00
_ ^ w2 27ri m
* Bm Q ¥ e~U .
CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 105
In this series the coefficients | #0 |, | -#1 | , • • • , | #&-i | repeat them selves. We collect all those terms which contain | B0 \ and likewise all those which contain | BI |, etc., and take | B0 \, \B'i\t . . . on the out side of the summation signs. We thus distribute the above series into
B
If
k new series of which each is multiplied by one of the quantities 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, *\/^, tends towards a dejinite value which is less than unity, with increasing values of m. For if ^/vm < p < 1, then is vm < pm < 1, and vm + 1 < pm+l < 1, etc., so that 2vm is less than a geometrical series in which p < 1. The general term in the above series is
Q and the ?ftth root of this quantity is
Q
2 «
2 in
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
Q
If we put - = a + ip (where p ^ 0, since -is not real), we have a a
-•b _ ,
KI--KUZ-XP
and
It follows that
.6
TTt — I
e a =
-«*?
becomes arbitrarily small
If p is a positive quantity, the quantity e for increasing values of m, which proves the convergence of each of the above series.
106 THEORY OF ELLIPTIC FUNCTIONS.
The condition that /? be positive need not be regarded as a limitation. For if /? is negative, we form the quotient
a + a + / a
where the coefficient of i on the right is positive. We may therefore write | = a' + ;/?', where /?' is positive. If then the coefficient of i in - is nega
tive, we interchange b and a in the whole investigation and thus form a function 4>(w) of such characteristics that
jri/fc
' a)
The function $(u) is defined by the series
where Q0 « e^6.
ART. 87. If k = 1, we have (Art. 85) $(w + a) = $(u),
&(u + b) = e~"(2 which equations are satisfied by the series
m=+oo m2 2n
where
In this case, since the B's are all equal, we may write
This is Hermite's function X(w), when we make B0 = 1. It is the simplest intermediary function and is called the Chi- function. For k = 2, we have
$(u + a) = <£(w),
(w + 6) = 6 a ( h6)
2 jri
CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 107
In this case 4>(w) contains the two arbitrary constants B0 and BI. We may therefore write
where $0(w) = 2* &** ° ' ' (w = 2 /i)
fi= -X
,= +* (2, + l).2«
$iOO = Q e° , (m = 2a
The constants B0 and BI being arbitrary, we choose B0 = 1, and BI = 0, and thus have a particular solution 4>o(w) of the functional equations; writing BQ = 0 and BI = 1 we have another particular solution <&i(u).
The functions ^oC^) and $i(t*) 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
.6
Q2 = er'l~a = q. It follows then that
^= -foe 4 ;rt
Y«2 a
u= — X
Jacobi further wrote instead of a the quantity 4 K, and instead of b the quantity 2 i'K', and consequently
,-•*.
The above functions become
«= .a «tu 2
•too- HIM* ?1 2 ^2K
* 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 panze Gebiet der elliptischen Transcendenten beherrscht, dass Jacobi aus ihren Eigenschaften wichtige Theoreme der hohreren Arithmetik abgeleitet hat, und dass sie eine wesent- liche Rolle 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 n-ichste Stelle nach den langst in die Wissenschaft aufgenommenen Elementartranscendenten einraumen miissen."
108 THEOKY OF ELLIPTIC FUNCTIONS.
ART. 88. If we put
it is seen that <j>(u + a) = ^ + a) - t
a) $
and
6) -**
It is thus shown that the function (f>(u) is a doubly periodic function hav ing the periods a and 6. This function <f>(u) cannot be a constant, for if
then 4>1(w)= C<l>o (w)> which is not true since $0(u) is developable in the
2*t
- W
even powers of e a while $i(iO is developable in the odd powers.
The functions <£(w) 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 ((Euvres, 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 Poisson and Fourier (Sur la Theorie 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.
CONSTRUCTION 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 4>(w) of Art. 85 in the form
^
- 2,
2k ' '
(J = 0, 1, . . . , k - 1).
Such functions, for reasons given in Art. 92, are said to be of the kth 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
$(u + a)= $(M),
_ !Li^"(2
3>(u + b}=e a ' These functions are therefore of the form
(a = I, 2, . . . , k + 1). We have at once, if we take p(=l) as the coefficient of ¥«(w),
0 = - pVa(u)+ B0(a}3>0(u)+ Blw3>l(u)+ • • • +B^k^(u) (a = 1, 2, . . . , k + 1).
In these & + 1 equations we may consider p, <3>07 $1, . . . , ^-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
D (1) Dc\
p (1)
• • , Bk-i
D (21
M), B0
(k+1
= 0.
If this determinant is expanded with reference to the terms of the first column, we have
- - - + Ck+lVk+i(u)= 0, where the C"s are the constant minors (sub-determinants).
We thus see that there exists a linear homogeneous equation with con stant coefficients among any k + 1 intermediary functions of the kth 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' + Xa + ffb, where X and /* are integers. We have
&(u) = $>(u' + fib + Ja) = $(u' + /£&), and further,
~~(' ^
26)= e* ^(u + 6),
- — (2u+5b)
a ,$O + 2 6),
- — [2u+(2n-l)b]
&(u + /£&) = e $(u + (« - 1) 6).
When these equations are multiplied together, we have
• • +2fi-l)]
a
or >(
It follows that
Since the exponential factor is different from zero, it follows that can only vanish when &(u') equals zero. We may therefore limit our selves to the discussion of <£>(w) 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 <&(u) 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 6 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 PERIODIC FUNCTIONS. Ill
// L — L
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
of zeros of 4>(w) within this parallelo- B>
gram, it is evident that upon the line /""
QBf there is a point D such that there y / 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 QAf 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. \Ve take P as the vertex of a new parallelogram PACB. We shall see that there are no zeros of the func tion 4>(w) 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) = <b(u), so that $(u) takes the same values upon EA as upon DP. Upon PG likewise by construction there is no zero of the function <b(u) and upon GB there is also none, since
_ Fi 21
(pu + = e u.
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 3>(u) 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
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 &(u) which is infinite for no finite value of u. The region in question is the
Fig. 22. period-parallelogram PACB. We therefore have , if we write ty(u) =
2-iN = C -fy(u)du+ f ^r(u)du + \ ty(u}du + \ ^(u)d JPA J AC JCB JBP
* Cf. Forsyth, loc. cit., p. 63; Osgood, loc. cit., 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 OF ELLIPTIC FUNCTIONS.
We may transform these integrals of the complex variable into integrals of a real variable t. 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 0 = £ = 1. It follows that
/ i!r(u)du= a I JPA Jo
+ at)dt.
Further, the variable u has at A the value p + a, and since AC = b, we have
I ^r(u}du = b I JAC Jo
a + bt)dt.
Similarly u has at B the value p + b, and therefore all values of ^ on CB have the form p + b + at, and consequently
r r° ci
I yr(u}du = a I "fy(p + b + at)dt = — a I ty(p + b + at)dt.
J AC Jl Jo
Finally we have in the same manner
r r° r1
I ifr(u)du = b I y(p + bt)dt= — I y(p H- bt)dt.
J BP **\ JQ
It is thus seen that
2 niN =
fir -| rir n
a I \'^r(p + at)—'^r(p + b+at)\dt + bl \*Y(P + a + bt) — ^(p + bt) \dt. Jo L J «^o L J
Further, since $0 + a) = $(u) arid 3>(u + b) = e a &(u), it
follows at once through logarithmic differentiation that
^r(u + .a) =^r(u) and ijr(u + b) =^(u) •
a
These values substituted in the above integrals give
F0 a
or .V = k.
We thus see* that the number of zeros of the function 4>(w) which lie within the period-parallelogram is equal to the integer k which appears in the second functional equation which <f>(w) 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 kth degree. In a corresponding manner we say of our function $(u), it is of the kth degree or order, because it vanishes at k points within the period-parallelogram.
ART. 93. For k = 1, we had in Art. 87
&(u + a) = 3>(u),
--(2u+6)
$>(u + 6) = e $>(u).
After the theorem just proved we know that there is one and only one zero of the function <b(u) which satisfies these two functional equations in the period-parallelogram. We shall seek this zero in the initial period- parallelogram. We had
m= +x 2jrt'
m= —30
Writing m = — (n -f 1) in this formula, it becomes
2 ri ot
Qn*e
n= -x
n=+x ^(2n6+6-2nu-2t*)
Tea
If we give to u the value — — in the above formula, it becomes
«[n6_(n+i)0]
Tea
n=+oo
Qw2+n(-l)n.
n= —x n= —x
If we also write a in the original expression for X(w), it becomes
Comparing the two expressions thus obtained for Xj— ~ — J, it is seen that
they differ from each other only in sign, and consequently it necessarily follows that
114 THEORY OF ELLIPTIC FUNCTIONS.
Since the zero of the intermediary function <b(u) of the first order, i.e., of X(w), is the intersection of the diagonals of the initial period-parallelogram, it follows that X(w) = 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(w) 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(w) upon its boundaries.
Suppose that the period-parallelogram which has u = p as one of its vertices is so drawn that there are no zeros upon its boundaries. There c is one zero within the period-parallelogram, since <&(u) is of the first degree. The value of u at this point may be expressed in the form
p + ha + vb,
j£ 23 where A and v are proper fractions. If now we
cover the w-plane with congruent parallelograms, there does not lie a zero of X(tt) on any of the boundaries of these paral lelograms, and within each parallelogram there is always one and only one zeroo Since all the zeros are congruent one to the other and since
from above is one of them, we must have
where g and I are integers. Every zero of X(w) 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, L
say at L, where
L = b + da,
$ being a proper fraction. We would then have
a ga + lb = b + $a, Fig. 24.
and consequently
b 1 - 2# + 2#
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 absurdum show that there does not lie a zero of X(w) upon the boundary of the initial period-parallelogram.
CONSTKUCTION OF DOUBLY PEEIODIC FUNCTIONS. 115
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 u\, u^ . . • , un] and we shall further assume that these infinities are of the first order, so that in the neighborhood of any one of them, u\ say, F(u) has the form
F(u) = — - — + c0 + CI(M - MI) + c2(u -Mi)2 + • • • ,
u — u\
where d and the c's are constants.
We shall see that every such function may be expressed through the general intermediary functions 3>(u). We shall form such a function where the integer k is taken equal to n -f 1 and which therefore satisfies the two functional equations
$(M + a) = $(M),
There being n + 1 arbitrary constants in this function, we may write it in the form
The constants BQ, B\, . . . , Bn may be so determined that the function $(u) becomes zero of the first order on the points HI, u2, . . . , un. For write
= 0,
- 0,
In these equations we may consider the B's as the unknown quantities. We have then n equations with n + 1 unknown quantities, from which we may determine the ratios of the J5's so that <f>(w) becomes zero of the first order at all the points HI, u2, . . . , un. By hypothesis F(u) be comes infinite of the first order on all these points. Form the product
/(*)-*(*)*•<*).
116 THEOKY OF ELLIPTIC FUNCTIONS.
It is seen that
<j>(u + a) = $(u + a) F(u + a) = <f>(w) F(u) =f(u); and also
f(u + b) = &(u + b)F(u + b)
= e $(tt) F(u),
-(n + l)-(2w + 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 $(u). Further, since $(u) becomes zero of the first order at the same points at which F(u) is infinite of the first order, the product f(u) = &(u) F(u) is nowhere infinite in the finite portion of the plane. A one-valued analytic function which does not have an essential singularity in the finite portion of the plane and in this portion of plane is nowhere infinite, is an integral tran scendental function; and, as there are only n + 1 such functions that are linearly independent (cf. Art. 90), it follows that
f(u) = C03>o(u) + Ci$i(w) + C2$2(U) + • • • + Cn$n(u),
where the C"s are constant.
It is also seen that ff».\
F(u) = £& . *(«)
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 (it) -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(u) is zero for the value u = ?L±J! = Cf say. Hence X(u + c) = 0 for u = ha (X = 0, 1, 2, . . . ).
If we write X(u + c)=Xi(u), it is seen that Xi(u) = 0 for u = 0. We also observe that the function Xi(u) satisfies the two functional equations
Xi(u-+a) =Xl(X>,
Xi(w + b) = e ' XI(M).
We have immediately the following relations:
7T"l
- — (2u-2u2+a + b + b)
Xi(u — u2 + 6), = e Xi(u — u2),
— Uk + 6) = e
CONSTRUCTION OF DOUI3LY PERIODIC FUNCTIONS. 117
If we put V(u) = XI(M - MI) XI(M — u2) . . . XI(M — Uk), it is seen that
V(u + a) = ¥(w),
-fc — (2u+6)
= e a
provided that
A; (a + 6) - 2(ui + u2 4- • • • + uk) = 2 ma; that is, if iii -f u2 + • • • +Uk = kc — ma, (1)
where m is any integer.
Hence if k = n + 1, the function ^f(u) becomes zero on any n arbitrary points ui, U2, . . . , un, while the other zero must satisfy equation (1). As some of the points u\, 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 ¥(M). We may therefore let ¥ (u) take the place of f(u) in the preceding article and mutatis mutandis 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 = 0, u = a, u = 2 a, • • • . Such a func tion is the Zlta-f unction (see Art. 97),
This function ZQ(U) is one-valued in the entire M-plane and has an essen tial singularity only at infinity. By means of this fundamental element Her mite* 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 /Ith order say at MI, the development in the neighborhood of this point is
F(u) - 6* + **-* + - - . + -^- + P(u - 1*0,
^ ~
the 6'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 £ and form
where now u is to play the role of a parameter, being a point within the initial parallelogram, while c is the variable. We consider in the c-plane
* Hermite, Ann. de Toulouse, t. 2 (1888), pp. 1-12, and "Cours" (4th ed.), p. 226.
118 THEORY OF ELLIPTIC FUNCTIONS.
a period-parallelogram of F(£), upon whose boundaries there is no infinity of F(£).
The function /(£) becomes infinite within this period-parallelogram on the points ui, u2, . . . , un, the points at which F(g) is by hypothesis infinite; and /(£) is infinite also at the additional point £ = u, since Zq(0)'-ao.
We form the sum of the residues of /(£) with regard to all the above infinities and have after Cauchy's Residue Theorem
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 to the left. We therefore have
a
. fp+a fp+a-
,. ,T ...twites /(?) = / m<K+ I /(«
// // *s v *Jp+a
/
Fig. 25. or, as in Art. 92,
£ 5) Res/(£) = a \ f(p + at)dt + b \
- a \ f(p + 6 + at)dt - b \ f (p + bt)dt. Jo Jo
Further, since ZQ(V + a) = Zo(v)| Z0(V + 6)
it follows that
/(£ + a) = F(^ + a) Z0(^ - <f - a) =
and /(£ + 6) = F(^) < Z0(^ - 0 + -^ | =/(c)
We therefore have
o
and consequently *
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.
COXSTKUCTION OF DOUBLY PEKIODIC 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 /(£).
We had -,, ,
X(l7 + C)
The function X(y + 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(v + c) = riv + /-2v2 + • • • ,
where the fs are constant and f\ ^ 0. Through differentiation it is seen that
X'O + c) = fi + 2 ?2v + • • • , and consequently
We note that the residue of Z(v) with respect to v = 0 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 u — c, we have
c — u
which is the development of Z0(u — c) in the neighborhood of £ = u.
We next form the corresponding development of F(£). In the interior of the period-parallelogram the function F(£) becomes infinite at the points HI, u2, . . . , UH but not at u. Hence we may develop F(£) by Talor's Theorem in the form
Further, since /(£)= ^(f) Z0(u - c), it follows that
and consequently Res /(f) = _ F(u).
^ = U
We saw above that 2 Res/($) is independent of u, but as shown here, the single residues are dependent upon this quantity.
120 THEORY OF ELLIPTIC FUNCTIONS.
ART. 98. We shall next calculate the residues of /(£) with respect to the other infinities u\, u^, . . . , un. Suppose that the function F(£) becomes infinite of the ^th order on the point u\, so that F(£) when ex panded in the neighborhood of this point is of the form
where the 6's and c's are constants.
For the value £ = u\ the function Z0(u — £) is not infinite and may be developed by Taylor's Theorem in the form
It follows that the coefficient of - - in the product F(£)Zo(w — £) is
,
which is the residue of /(£) with respect to the infinity £ = u\. The resi dues with respect to the other infinities u2, u3, . . . , un are found in the same manner. The 6's and X, of course, have different values for each of these points.
Let the orders of infinity at Uk be fa (k = 1, 2, . . . , ri) and in the neighborhood of the infinity Uk let the principal part of the function
(U — Uk)^k (U — Uk)Kk~l (U — UkYk~2 U — Uk
It follows at once that
k = n p .
V Res m - 5) 1 4k, iZo(« - uk) -^ l^i, fcTi L
We also saw that Res /(c) = — F(w), which must be added to the sum just written.
On the other hand we had
Res/(£) = - / F(p + at)dt = C, say,
«/o where C is a constant.
Equating these two expressions for the sum of the residues, we have F(u)=C+ j$\bktlZ0(u-uk)- ^fz0'(u-Uk) + b-Mz0"(u-Uk)-> - -
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 Z0 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.
ART. 99. There is a restriction with respect to the constants that appear in the above development.
We saw that
Z0(v + a) = Z0(r) and Z0(r + b) = Z0(r ) -
a
It follows that Z0(v) is not a doubly periodic function; but all its derivatives are doubly periodic, since we have
Z0'(v + a) = Z0'(t>), Z0'(v + b) = Z0'(i>), etc.
Hence under the summation sign of the preceding article all terms except the first are doubly periodic. Further, since F(u + 6) = F(u), it also follows that
Since Z0(u - uk + 6) = Z0(u - uk) - — >
a
it is evident from the equality of the two summations just written that
k=n
i-0, or 6,,! = 0.
k=l k=l
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
2 ResF(u) = 0.
Take any period-parallelogram, upon the sides of which there are no infinities of F(u).
Fig. 26.
/f Then by Cauchy's Residue Theorem 2 m V Res F(u) = ( F(u)du.
^ JvACB
But from Art. 92 we have
I F(u)du = a I F(p + at)dt + b \F(p + a + bt)dt
JpACB i/O t/0
b + at)dt - b
Further, since it follows that
F(u + a) = F(u) = F(u + b), 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,it &*,2, • • • , bk, 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 u\ were such a point, then is
F(u)=
in the neighborhood of this point, and consequently
SResF(^) = 6ltl.
But as the sum of the residues is equal to zero, it also follows that 61,1 = 0 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 doubly periodic function must have at least two infinities of the first order within 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 = $>u. 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 z = <#u\ 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. W^e 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 ^th order on any point, let this point be counted as ^ 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
= 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),
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 u\ be an infinity of G(u) of the Ath order, so that
G(u) = (u - UI)~*GI(U), where GI(UI) ^ 0. We then have (Art. 4) in the neighborhood of HI,
H(u) = -- - — + P(u - m),
u — u\
so that
ResH(u) =- I,
u=u\
that is, the residue of H(u) with respect to u\ is the order of the infinity of G(u) at the point u\ with the negative sign.
Suppose next that w\ is a zero of G(u) of the jj. th order, so that
G(u) = (u — wi)nG2(u), where G2(wi) ^ 0. We then have in the neighborhood of w\
H(u) = — — + P(u - wi), or u — wi
Res#(V) = /£,
U = Wi
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
- Z/t + S/* = 0,
where SA denotes the sum of the infinities of the function G(u) in a period- parallelogram, an infinity of the Ath order counting as A simple infinities, and where H/z denotes the number of zeros of the first order of G(u), a
CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 125
zero of the /*th order counting as /* zeros of the first order. Since G(u) = F(u) — v, it follows that the number of roots of the equation
F(u) -v = 0
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- parallelogram 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 fcth order with the same periods. Neither of these functions is supposed to have an essential singularity in the finite portion of the u-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, . . . , un, for which F(u) = z. If we write these values of u in w — G(u), we have n values of G(u), say w\ = G(UI), w2 = G(u2), . . . ,wn = 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 kih degree in z.
We may next suppose that z = </)(u) is a doubly periodic function
with the periods a and b, then w = -^- = <t>'(u) is a doubly periodic function
du
having the same periods. Hence from the theorem above there is an
algebraic equation connecting z and — , say
du
4!)-°-
It is easy to determine the degree of /in z and ; for if <f)(u) = z is of 7 du
the nth degree then — occurs to the nth degree in the above equation. du
If MI is an infinity of the ^th order of (j>(u), then MI is an infinity of the / -f 1 order of <j)'(u), so that <j>'(u) becomes infinite on the same points as <f>(u), the order of infinity of (/>'(u) being one greater on each of these points than is the order of (j>(u) on the same point.
If all the infinities of </>(u) are of the first order and if n is the order of 4>(u), it follows that (/>'(u) is of the 2 nth order and consequently the
* Cf. Neumann, Abel'schen Integrate, p. 107.
126 THEORY OF ELLIPTIC FUNCTIONS.
degree of /( z, -r\is at most 2 n in z. This equation f(z, — )we have \ du/ \ du)
called the eliminant equation.
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 or z = <@u (Art. 102).
Let z be a doubly periodic function of the second order (n = 2), so that the eliminant equation is / dz\
•T Au) - °'
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) 9o(z) + gi(z) - + g2(z) = 0,
\duj du
where the g's are integral functions of at most the fourth degree.
We saw above that z and — are infinite at the same points within the
du
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
du g0(z)
and becomes infinite for those values of z which make g$(z) = 0. It follows that QQ (z) must be a constant and consequently the equation (I) becomes /, N2 j-
(io Or) + 0i<0£f,*(«)-Qp
\duj du
where the constant has been absorbed in the two functions g\(z) and g%(z).
ART. 107. If z 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 v2 du
Making these substitutions in the above differential equation we have fdv\2 1 fl\dv 1 /IN n
fc;^-^vW^P+%ra
Since 9ri(-jandg2 (-)are at most of the fourth degree in -, it follows \vj W v
CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS. 127
that v4gi(-} and v4g2l- ) are integral functions of at most the fourth
W W
degree in v, which we denote respectively by 9i(v) and 92(0).
The above differential equation is then
v2 du
We saw above that -^ is finite for finite values of z; the same must also
fiv ® ^
be true of —and v.
But in the differential equation just written -- becomes infinite for
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 (F) that
= - i 9 1 (z) ± i \/gi(z)2-4g2(z) ; du
or, if we write 4 R(z) = g\(z)2 — 4
du
where R(z) is an integral function of at most the fourth degree. It follows that f*z dz
f*z dz
* - *9i(z) ± ^R&
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.
/** rl?
If we write u = I ,
Jo V 1 - z2
we have u = sin-1;?, 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 u is a one-valued simply periodic function of u. In the more general case above we wish to consider z as a function of u. This is the so-called " problem 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 /(z) 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
t(z) = Cf(z) + Clf where C and C\ are constants.
CHAPTER VI THE RIEMANN SURFACE
ARTICLE 108. At the close of the preceding Chapter we were left with 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 = ± v
where z 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 z = oo is also a branch-point of this function;
for - = - = 0 for z = oo . Consequently — and likewise s has
s ± V z— a s
only one value for z = oo .
There are other reasons why z = a and z = oo are called branch points. Corresponding to the value z = z0, let s = s6 be a definite value of s. Along the curve (1) from z0 to zi consider the values of s at all the points of the curve which differ from one another by infinitesimally small quantities, and similarly consider the values of s along the curve (2) until we again come to z^ The value of s at this point will be the same whether we have gone over the first or second .curve, provided the branch-point a is not situated between the two curves.
This may be shown geometrically as follows:
Let OM = z, Oa = a,
aM = z — a, and | z — a \ = r. We therefore have
z — a =
Fig 27 where </> is the angle that aM makes with
the real axis. It follows that i$ i$0
s = r^e 2 and SQ
If aM turns about a, and M starting from z0 after making a circuit returns again to ZQ, then if this circuit does not include a, the values of <f>0 and SQ
128
THE RIEMANH SURFACE.
129
are the same as before the circuit and consequently s0 has its initial value. But if the circuit includes a, the quantity r0 is the same after the circuit, but <po has become <£0 + 2 -. The corresponding value of so is
We thus see that s0 has taken the opposite sign after the circuit.
ART. 109. Consider next the expression s2 - R(z),
where R(z) is an integral function of the fourth degree in z. We may write
Fig. 28.
R(z) = A(z- Ol) (z - a2) (z - a3) (z - a4),
A being a constant. We then have
s = ± VR(z) = ± A* (z - d)* (z - a2)4 (z - a3)* (z - o4)*-
The function s has two values with opposite signs for any value of z except °i> a2, ^3, «4- When z is equal to any of these values, s has the one value zero. The points a i, a2, «s and a4 are branch-points. The value a± — ZQ is the radius of the circle about ZQ wrhich goes through a\. Suppose that z is any point situated within this circle so that
\Z - ZQ\ < \di - ZQ\.
Then, since z — ai = z — ZQ — (ax — z0), we have
(z -
= - (a,
- z 1 - ^^^ *.
Since
< 1, the right-hand side may be expanded by the Binomial
Theorem in the form
- ZQ
di - ZQ
This 'series is uniformly convergent for all values of z within* the circle. In the same wTay we may develop (z — a2)*, (z — ^3)*, (z — a4)* in posi tive integral powers of z — 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. tit., p. 77 and p. 285.
130
THEORY OF ELLIPTIC FUNCTIONS.
four series together, the multiplication being possible, since the series of the moduli of the terms that constitute the four series are convergent. We thus derive the result: We may develop s = \/R(z) in positive integral powers of z — ZQ, if z0 is different from the four branch-points ai} a2, a3, a4. The series is uniformly convergent within the circle about ZQ as center, which passes through the nearest of the points «1? a2, a3, a4.
ART. 110. We may effect within this circle the same development by Taylor's Theorem in the form
2 S0
We must decide upon a definite sign of s0 = VR(z^ and use this sign throughout the development. If at the beginning we decide upon the other sign, then in the series we must write — s0 instead of s0; that is, all the coefficients are given the opposite sign.
If the sign of s0 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 z 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 ZQ cut this path at £. Then at all points of the portion of path 20£ the corresponding values of the function are known through the series. Let z\ 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 zif that is, si = V R(ZI) through the series
si = '
|2=2l
i 29.
Thus si is uniquely determined, if the sign of s0 has been previously chosen. We next take z\ as the center of another circle Ci, which also must not contain a branch-point. Then precisely as we expanded 5 in powers of z — ZQ in the circle Co about ZQ we may now expand s within C\ in powers of z — z\ about z\. This circle C\ 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 z\ is taken sufficiently near £ that the circle about z\ partly overlaps the
THE BIEMANN SURFACE. 131
circle about ZQ. That this may be the case z\ must lie so close to £ that the distance between the points is less than the radius of the circle C\, 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 PQ(Z — ZQ) while the one in C\ may be represented by PI (z — z\). 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 P0. But the development PI holds good for the entire circle Ci. We thus go in a continuous . manner to values of the function which lie without the circle Co- The series PI 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 zm of the path as center within which the point z' lies. We may develop the function within Cm in positive integral powers of z — zm and may then compute sf = \/R(z*) from this development. This process is called the " Continuation of the function along a prescribed path from ZQ to z'." 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 BI 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 zf which is had through the continuation of the function along the path B\ is the same as the one which is had through the continuation from ZQ to / alon^ B. It is clear that if the two values of \/R(zf) 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 zf 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 BI 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 T- on
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 BI at «&' where BI is taken very near to B. as shown in Fig. 31, and call ak, ak a pair of neighboring points.
We suppose that the curves B and B i 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 FUNCTIONS.
Fig. 31.
pressed through Pk(z — Zk)'} and at the same time we assume that one of the circles used in the continuation of the function along the path BI includes also the same points ctk, <x.il and that all points within this circle are had through the series Pv(z — zv). Hence we must have the same
value of s at the point z = ak from either of the power series P^ or Pkf, provided this is true of every pair of neighboring points that preceded this pair. The same is also true of the point z = a^. 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 BI. Heffter [Theorie der Linear en 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 B\ with a finite number of curves drawn from ZQ to zf which lie at a finite distance from one another 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 zf.
ART. 112. Let (1) and (2) be two paths between ZQ and z' which do not include a branch-point. If we go along (2) from 00 to z' and then back again along (1) from z' to ZQ, we come to the same initial value of the function, From this it follows: // the ^ function s = \/R (z) is continued from the point z = ZQ along a I 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 ZQ.
Fig. 32.
Fig. 33.
THE RIEMAXX SURFACE. 133
ART. 113. Suppose next that the closed curve includes a branch-point, for example ai. We again fix the sign of s0 for z = zQj and write
s = Vz - alVRl(z), where
A(z - a2) (z - a3) (z - a4).
We may allow VRi(z) to have an arbitrary sign, and so choose the sign of \/z — ai that s = s0 will have the same sign for z = z0 as has been pre viously assigned to it. _
If we make a circuit about a i, it is seen that \/Ri(z) is not affected by it, since ai is not a branch-point of \/R\(z). Hence upon making a circuit about 01 we need consider only the first factor (z — ai)*. We may make this circuit along a circle of radius r with a i as center. For the points of the periphery, it is clear that
| z - a1 | = r,
so that
z — a i = re^.
It follows that ^
(Z - Ol)i = r*e*.
Let the value of <j> corresponding to z = z0 be <£ = (f>Q, so that t£ Fig. 34.
(z0 - ai)* = r*e2,
where the point ZQ of course lies upon the periphery of the circle. When a complete circuit is made about 01, starting from z0, it is seen that <£o is increased by 2 -, and consequently after this circuit the above expres sion becomes
rie 2 _ r*e 2 ci* = - r*e 2 . It follows that after a circuit* about QI has been made, the quantity
(z — ai)* and consequently also s = VR(z) changes its sign.
Further, if we make a circuit about a! along any arbitrary curve B which does not include any other branch-point except «i, then s changes sign with this circuit; for this is the case when a circuit has been made about the circle around 01, and as there is no branch-point between the circle and the path B, it follows that starting from z0 ^"e 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 suppose that the circuit is made along the path z0a3fZQ. Let os- be a closed curve about ai and yOn a closed curve about a2. It follows immediately from the above considerations that the
*-Cf. Bobek, 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 a\ and a2 along the path o we may just as well make the circuit along the path z0d 'srzoyd KZQ, there being no branch-point between this curve and the curve z0a^z0. After the circuit z0d£TZo the function s changes sign as it again does after the circuit ZQ^KZQ, so that after the two circuits around the points a{ and a2 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 z0 with the other value of s. ART. 115. We may next see how the function
s = A* \/{z — a\)(z — o2) . . . (z — an )
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 01, 02, . . . , 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 z = — , so that when z = GO , we have t = 0. In the z-plane the point t
at infinity corresponds to the origin in the £-plane. We then have
s = t 2 A K/(l - aiQ (1 - a2t) ... (1 - ant).
Now take a circuit about a circle with the origin as center and which does not contain one of the branch-points 01, 02, - • • , on. We must therefore write
* ==
and it is seen that the function s changes sign when n is an odd integer. In this case the origin in the Z-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 a\ with a2 and a3 with 04. The paths along which the function s is continued must never cross these lines a\ a2 and 03 04. They may be called " canals." The z-plane which contains these two canals may be denoted by the z-plane, a dash being put over z (see Fig. 36).
THE KIEMAXX SURFACE.
135
If once the initial value s0 of the function s = \/R(z) is fixed for the point ZQ, then s is completely one-valued in the z-plane; for in whatever manner the continuation from z0 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-cut*.
If further the sign has been ascribed to the initial value s0 of the function s, then we may ascribe to s its proper value for every value in the 2-plane. These values we suppose have been written down on a leaf, which represents the z-plane. Again starting with - s0 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 e&2 and a3 writh a4 are also supposed to have been drawn, so that s is also one-valued on it.
We note that corresponding to the same value of z, the values of s = ±vR(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
Fig. 37.
around a\, say, and return to the point «2 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 a^ a2 and a3 a4.
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 Integrate, p. 81.
136
THEOKY 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 =
(cf. Grundlagen fur eine allgemeine Theorie der Funktionen einer kom- plexen verdnderlichen Grosse. Inauguraldissertation von B. Riemann. Crelle, 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 s is to be found, we must also indicate whether the value of z is taken in the upper or in the lower leaf.
7 a2
Fig. 38.
In the figures a path that is taken in 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; Durege, Elemente der Theorie der Funktionen. For other references see Wirtinger, Ency. der math. Wiss., Bd. II3, Heft 1,
THE KIEMANX 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 no;te 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 denned 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 z 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 w2j for the same value of z. In the special case above where w = s = ± vR(e)t we have w\ = — w2. In general, however, this is not the case. But if we consider the sum wi + w%, this sum is a one-valued function of 2, for if z is given, w\ + 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
w2 - <}>(z)w + ^0) = 0, where (j>(z) and ty(z) are one-valued functions of z, such that
wi + w2 = <j)(z) and wi • w2 =
Hence every one-valued function of position on the Riemann surface s = vR(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 <j)(z) = w\ + w2 will have a definite value for every value of z, as will also ^r(z) = w\ • 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 <p(z) and ^r(z) must be rational functions of z.
ART. 118. When we solve the above quadratic equation, we have
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 -^(2) + c/)2(z)= S(z), say, becomes zero or infinite of the (2 n + l)st order for z = b, where n is an integer.
We note that the point 6 cannot be a branch-point on the Riemann surface, for ai, a2, a^, #4 are the only branch-points on this surface.
We may write S(g) = (z _ b)2 m+ig^
where Si(z) is a rational function of z.
About b as a center describe a circle which does not inclose any other zero or infinity of S(z).
We then have 2n+i
and if z makes a circuit about the circle, the function V$ ±(z) retains its
2n + l
sign, while (z — b) 2 changes sign. Consequently the function
changes its sign with this circuit, so that w = 2SS -| does not
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 S(z) must have the form
S(z) = Si(z)2 { (z - ai) (z - a2) (z - a3) (z - o4) }- We may therefore write
w
= $ <f>(z) + % Si(z) V(z - ai) (z - a2) (z - a3) (z - o4) = p(z) + q(z) VR(z) = p + q • s,
where p = p(z) = , q = q(z)= k l^ are rational functions of z.
£ 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 sufficient condition that it? be a one-valued function of position on the Riemann surface.
THE ZEROS OF THE ONE-VALUED FUNCTIONS OF POSITION.
ART. 119. Let z = a be a position on the Riemann surface, which is different from the branch-points «i, a2, a3, a4. We may then draw a circle around a which lies entire!}7 in one leaf of the Riemann surface.
It may happen that w = 0 for z = a, while at the same time p and q are infinite for z = a. For suppose that
(z — a)*4"1 z — a
We may also develop s for points within the circle in the form s = h0 + hi(z - a) + h2(z - a)2 + • •
It is evident that s is not infinite for z = a, and it is also clear that if ^ T^ fi, then w becomes infinite for z = a; but if ^ = ,«, then we may so choose the coefficients in the development of p and q that w = 0 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^ + h0fu, or, since X = (u, we must have
e, + h0f, = 0. Further, it is necessary that the coefficient of (z — a)-^"1) be zero, that is,
ex-i +f*-ih0 +fihl =0, etc. These conditions may all be satisfied; and consequently
w = kr(z - a)r + kr+i(z - a)r+1 + • • • ,
where the k's are constant and where r is a positive integer greater than 0. Finally we may write
w = (z - a)r[kr + kr+1(z -«) + •••].
We see that w becomes zero of the rth order for z = 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 for 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, a\ say. If z makes a circuit about a\, 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 ai, 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& = (z -
Since a\ is not a branch-point of y/Ri(z)t we may expand this function in positive integral powers of z — a\ and have
= (z — ai)*[&o + bi(z — ai) + b2(z — ai)2 +•••]. We put
(z — ai)*= t or t2 = z — ai.
Let a circuit be made about a\ along a circle with radius r, so that
z - al = t2 = re**, or i(j>
t = Vre~*.
If then z describes a circle with radius r around ai in the 2-plane, then t describes a circle with radius Vr around the origin in the 2-plane. If the circuit of z begins with the initial value 0 = 0, then the circuit of t begins with the value 0 = 0, and when 0 increases by 2 n we have 0/2 increased by TT. Hence to the whole circle in the 2-plane there corresponds the half-circle in the 2-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 2-plane.
Suppose that w vanishes at a branch-point, a\ say. Further suppose by the substitution z — a\ = t2thatp(z) becomes p(t) and q(z) becomes q(t).
In the neighborhood of the point t = 0, let
p(t) = antm - and q(t) = f3ntn -f
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 z — a\ = t2, it follows that z — a2 = ai — a2 + t2, z — a3 = ai — «3 + t2, z — 0,4 = «i — 04 + t2, and consequently R\(z) becomes V(t), where V(t) = fa - a2 + t2) fa - a3 + t2) (ai - a± + t2).
We note that this function does not vanish for t = 0, so that there is no branch-point of this function within the circle t = 0, if this circle is taken
THE RIEMAXN SURFACE. 141
sufficiently small. We may consequently expand VV(t) within this circle in positive integral powers of t2 and have
= t[b0 + M2 + M4 +•••]•
It is further seen that if w becomes zero at the point z — 0,1 = ft, it may be developed in the neighborhood of t = 0 in positive integral powers of t in the form
w = Co ci
A A+l A+2
= cQ(z - di)2+ ci(z - aO 2 + c2(z-al) 2 +....'
It follows also that the function w becomes zero of the ;Uh order at the branch-point z = 01. In other words, if w becomes zero at a branch-point z = ak, 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
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- a)*' [c0' + ci'(2 - a) + c2'(z - a)2 +••-], and consequently
log w = x'log(* - a) + log [c0' + Ci'(z - a) + c2'(z - a)2 +•••].
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 = /' log (z — a) + e\ + e2(z — a) -f • • • .
If z makes a complete circuit about a, the power series e\ + e2' (z — a) + . . . does not change sign; log (z — a) is, however, increased by 2 id and con sequently /' log (z — a) is increased by 2 idX. It follows that -.
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
THEOKY OF ELLIPTIC FUNCTIONS.
number due to the change in - r log w when z makes an entire circuit
Jj 711
about a.
This same analytic property must be retained if a is also a branch point, say a i.
From the development above
w = (z — ai)2[c0 + ci(z — ai)* + c2(z — it follows that
log
log [c0
or
/I
log w = -log (z — 2
+ eo + e\ (z —
experienced in - . log w is X since log (z — 2 TCI
Now to make a complete circuit around &i we must make a double circle. By this circuit (z — 01)* does not change sign. It follows that the change
changes by 2 • 2 ni. But
here X is twice the exponent of the lowest power of z — a\ 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.
If w = f(z) = p + qs 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 taken from the point ZQ, where \/R(z) takes the value s0, to the point 2', where
Fig. 40.
takes the value s', then the inte gral lf(z)dz taken over this path
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.
THE RIEMANN 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.: If a function f(z) within a portion of surf ace tha* is completely bounded, the boundaries included, is everywhere one-valued, finite and continuous, then the integral taken over the boundaries of the surface in such a way that it has the bounded surface always 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 outer 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 6 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 aft? in the Riemann surface. We may go from a point a on the outer edge to a point b on the inner edge without
crossing the curve
Fig. 42.
which lies wholly in the up per leaf. Consequently the curve apf- must not be re garded as the c mplete boundary of a portion of surface. But if we also
.p. 43 draw a congruent curve
otftf, 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. tit., p. 155.
144 THEORY OF ELLIPTIC FUNCTIONS.
crossing one or the other of the two curves apj- or a'fi f. Hence a/??- and a'Pr' together form the complete boundary of this portion of sur face of the Riemann surface. By Cauchy's Theorem the integral taken over /(z), where the path of integration extends over both afir and ci'fi'f'i 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$? and a'jf-f as the paths of integration we may take paths which lie indefinitely near the branch- cut 0,10,2, 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 = I 2 qsdz —I 2 qsdz = 0, *J *J a-2. <J 0,2
the elements of integration taken in the opposite directions over / pdz canceling one another.
ART. 123. If a one- valued analytic function be developed in the 2-plane in the form
(z - a)A (z - a)*"1 z - a
where P(z — a) denotes a power series in positive integral powers of z — a, then we know that the residue of f(z) with respect to z = a is
61 = Res f(z),
z = a
the quantity bi being the coefficient of- - •
The same definition is given for the residue of a function of position on the Riemann surface, provided the point a does not coincide with a branch-point.
If, however, this point is a branch-point, a\ say, and if the function becomes infinite at this point, then it follows from above that the development of w = f(z) in the neighborhood of this point is
f(z) = ( 6xW2 + 7 \l n/2+ • ' •+r-^LT|+, bl u + P{(s-oi)*}-
(z-ai)A/2 (z — aiY^12 (*— 0l)* (-2 - fli)*
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 RIEMA^N SUKFACE. 145
about the infinity a of the function f(z) and if f(z) does not become infinite on any other point within or on the circumference of this circle, then is
2~rf J/(*)<fe = Res/(*)>
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 &i.
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
(z - arfdz,
where the integration is over the double circle.
Suppose that r is the radius of the double circle, so that
z — a i = re^, and consequently also
^d'ble-circle ^0 Jo
This integral is always zero, except when 1 + - = 0. In this latter case
'd'ble-circle
It follows that
I (z - a{)'2dz = i I d(f>
^d'ble-circle ^0
Res/(z) - JL Cf(z)dz = -LT
z = di * '*1 ^d'ble-Hrrle ^ '*l
d'ble-circle
where 62 is the coefficient of (z — cti)~*, since k = — 2. We thus have finall>' Res/(z) =2b2;
z=ai
or, the residue with respect to a branch-point is equal to DOUBLE the coefficient of (z — ai)"1 in the development of the function in powers of (z — &i)*.
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, simple 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 THEORY OF ELLIPTIC FUNCTIONS.
within this surface, boundaries included. The integration is to be so taken that the interior of the portion 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 id. We have therefore the theorem : // a function within a completely bounded portion 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 *
w = p + qs,
where p and q are rational functions of z and where s = v72(z). It follows that
w dz p + qs p -f qs
If the numerator and the denominator of the right-hand side of this expression are multiplied by p — qs, we have
w
where P and Q are rational functions of z.
It is thus seen that the logarithmic derivative of w = p + qs is a rational function of z 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 BIEMAXX SURFACE. 147
If /j. is the order of the zero of the function w at the point a, then in the neighborhood of a
d-^^ = -*— + P(z - a) [see Art. 4]; dz z — a
and if ^ is the order of infinity of the function w at the point /?, then in the neighborhood of p
dz z -
It follows that
and
z=3 dz
The above discussion is true when a and /9 are not branch-points.
If a is a branch-point, say a1? and if w becomes zero at this point, then in the neighborhood of this point we have
p w =(z - ai)2[#0 + 0i 0 - «i)* + 92(2 - ai)§+ • - • 1
and consequently
log w = | log (z - ai)+ log[£o + 0i (2 - «i)* + •••]•
It follows that
& ^ - -^— + flog [0o + ffl(z - fll)* + - . . 1
C?2 2 — Ox C?2
Since the logarithmic expression does not become infinite for z = a\, it maj' be developed in the form
log [0o + g\(*- ai)* + ' • '1 -*•>*!(« -«i)* + " ' '»
and consequently
ff
dz z — ai 2
We therefore have (cf. Art. 120)
z = ai dz 2
If on the other hand a! is an infinity of the Ath order of w, then is
-P d log w 3
Kes — * — = — X.
148 THEORY OF ELLIPTIC FUNCTIONS.
ART. 126. We shall now apply Cauchy's Theorem to the function
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 = P, (t) + Q0(t) Y/I - «,) (I - a2] (I - a,} (\ - a4]
' \t / \t / \t / \t I
- Pi(t)+Qi(t) (l - ait)(l - a2t)(l - a3t)(l - a40, where
The functions Pi(f) 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
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 + Qs. * Cf. Neumann, loc. cit., p. 111.
THE RIEAIANX SURFACE.
149
These residues fall into two groups: those of the one group have refer
ence to the infinities of the function
, which exist through the van-
ishing of w, while those of the other group refer to the infinities of - , which are also the infinities of w.
If by H// we denote the sum of the orders of the zeros and by 2x 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/£, while — Sx" is the sum of the residues of the second group.
It follows at once that
2 Res (P + Qs} =
- SJ = 0,
or
It has thus been shown that the sum of the orders of the zeros of w is equal 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 ath order is counted u-ply and an infinity of the Ath order is counted A-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. infinite as often as p + qs is infinite, and since the relation
It becomes
is true also here, it becomes zero as often as p + qs becomes zero. We thus have the following theorem: The equation
p + qs = k
has in the Riemann surface as many solutions as p -f qs has infinities. Hence also the function p + qs takes every value in the Riemann surface an equal 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 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 corn-
Fig. 4;
150
THEOKY OF ELLIPTIC FUNCTIONS.
Fig. 48.
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 b. We note that one-
half of the canal b 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 a^2 and a3a4. The Riemann surface con
taining the two canals a and
b we denote by T'. The sur
face which does not have
these canals is denoted by
T. The surface Tf is said
to be of order f 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 b.
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 — 0,1) (z — a2). The associated Riemann surface con sists of two leaves connected along the canal ai«2- The integrals ~ .
P = r^,
J Vr(z)
* Cf. Neumann, loc. cit., p. 146.
f In general, if TV 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) TV = 2 p + 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, t/2= R(x) is a curve of deficiency unity, if s2= R(z) is a Riemann surface of order unity.
49
THE RIEMANN SURFACE.
151
where the paths of integration are taken over the two curves (1) and (2), are equal since the function , — - is one-valued finite and continuous
for all points of the surface between these two curves.
If we let the path of integration (1) approach indefinitely near the_canal a,ia2, then, since the values of Vr(j) on the right and left banks of this canal have contrary signs, we have
dz
where in the last integral the integration is taken along the upper leaf and the left bank.
It follows that P is different 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
u = I — =
J Vr(z}
Vr(z)
will be a one- valued function of position in the surface where the cut has been made. This integral will then be independent of the path of integration, which we have just shown by going around the canal a,ia2 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 ft immediately oppo- /B site on the other bank without crossing
Fig 51 either the canal AB or the canal a,id2, but
it is impossible to make a circuit around
the canal a\a2 or around either of the branch-points a! or a2 without crossing one of these canals. It follows that the above integral in T' is one-valued.
152
THEORY OF ELLIPTIC FUNCTIONS.
ART. 130. Next let
and let
u(z, s)=
u(z,s)=
dz
Vr(z)' dz
where the path of integration is in Tf;
where the path is in T.
»,sGVr(z)
The integration in both cases is always counted from a fixed point 20, s0, 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%2j $2 j^2t $2 §2\f ^1 j^2r $2
dz
Fig. 52. It follows that in T'
o, S2 (*Z2,S2
= I
zi, Si *Jz0, S0
(1) (2)
where the integrand
Vr(z) stood with every integral.
= t*(«2, S2) -
- is to be under
Next take the integral from z0, s0 to 2, s in T where there being no canal AB we go by the way of the two points A and p. We have
(*(> r* f*z, s u(z, s) = / + / + / ,
i/JBO, S0 *J p *J *
where the distance between ^ and p being indefinitely small the middle integral on the right may be neg- lected. But from above
and
I
= u(z, s) - u(X),
1 ig. 53.
where both of these integrals are in the Tf surface. From this it is seen that u(Zj s) = u(z, s) + u(p) - u(X),
where u(p) and 7i(X) are the integrals from z0, s0 to p and from ZQ, s0 to X in the T' surface, the path of integration being taken in any manner so long as neither of the canals a^2 and'AZ? is crossed.
THE RIE3IAXX SURFACE.
153
On the other hand,
Vr(z) or, from the figure,
and since
we have
where the integration of the last integral is taken in the upper leaf and the lower bank of the canal aja2 . We have finally
u(z, s)=u(z, s)+P,
r~
Fig. 54.
where P is a quantity which does not depend upon the path ZQ, s0 to zit 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 T' only by 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. Ellintische Functionen (2d ed.), p. 370]. \
EXAMPLE
/j — \T\ -
VI -z2
REALMS OF RATIONALITY.
ART. 131. Let z be a complex variable which may take all real or complex, finite and infinite values. Consider the collectivity of all rational functions of z writh 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 (z). Consider next the one- valued functions on the fixed Riemann surface. If we denote any such function by wi = pi + qis and any other such function by w2 = P2 + ?2«, then the sum, difference, product and quotient of the two functions w\ 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 (z, 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 (z, 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 explicitly appear. This equation may be written
du '
where p(z) is an integral function of at most the second degree and R(z) is an integral function of the fourth degree. We saw in the preceding Chapter that p(z) + \/R(z) is a one-valued function of position on the fixed Riemann surface. WTe are thus led to the study of the integral
dz
p(z) + VR(z)
As the lower limit of this integral we take any point z0 of the Riemann surface, at which s has the value s0= +Vfi(«o). 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
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 uf
It is possible that the point z, s lies in the neighborhood of a branch point a i, say.
We then have the following development:
and consequently
We thus have a series which proceeds in ascending powers of (z
155
156 THEORY OF ELLIPTIC FUNCTIONS.
ART. 133. Suppose that p(ai) does not vanish.
i
We may then develop • in integral powers of (z — c^)* in
the form P.W + v /t(2)
'- ci(z- 01)* + c2(z-
p(z) + VR(z) If we put p^ ^
Jz0,Sop(z) + VR(z) it is seen that
= a,
/*«,« ^p- /»0i /7? /'Z.a
u = I az I az + /
Jz», s0 p (z) + VR (z) Jz0, So p (Z) + VR (z) t/o, p (z)
dz
= a +
r*>
*J a\
p(z) + VR(z)
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 swso It follows that
u — a =
z's By hypothesis the point z, s lies in the neighbor hood of a\j that is, on the inside of a circle within which the series developed above is convergent. We
ttl F. 55 2 may therefore integrate this series and have
u — a = I ] h ci(z — ai)^+ c2(z — ^j)l -f • • • > d(z —
Jai lp(&i) }
= z- al +2c(z
p(a1) 3°] If we put z — a i = t2, we have
It follows that
or (u — a)* =
where of course the quantities ci, c2, . . . , /2, etc., are constants. By the reversion of this series we have
But since t2 = z — a\, it is seen that z is two-valued and not one-valued in the neighborhood of u = a.
THE PROBLEM OF INVERSION.'
157
ART. 134. If ;Xai) = 0> the above development becomes 1
p(z) + VR(z) We then have
u — a = I [e-i(z — = 2e-i(z - a
e-i(z -
e0
e0(z - 01)* +
}d(z-al)
From this we conclude that t is developable in positive integral powers of u — a and consequently is one-valued in the neighborhood of u = a. It follows also that z is one-valued in the neighborhood of this point.
Hence in order that z 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) = 0 =
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
dz
N*P(Z)
were infinite. We may prove, howrever, as follows that this integral is never infinite.
We saw above that
1
p(z)
-== is developable in a power series
which is convergent within a certain circle. Let this circle cut the path of integration at the point zf, s'. We then have
dz C2''*' dz
«/*
,* p(z) dz
Uo
Fig. 56.
',*'p(z) + VR(z)
The first integral on the right is finite, since it does not become infinite for any value between z0, «o and zf, s'; while the second integral, as shown above, may be expressed through the series [_2 6-1(2 — ai)* + CQ(Z —
158
THEORY OF ELLIPTIC FUNCTIONS.
This series is finite for the values z', s' and a\. It follows therefore that
dz
/* =
*J Z
»«p(z) + VR(z)
has a finite value even when p(ai) = 0, and at the same time it has been shown that the integral
dz
is finite when the upper limit is a branch-point.
ART. 136. We may now confine ourselves to the consideration of the
integral /^s d
u = I •
Jzo,soVR(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 includes 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
VlR(zj ~ <
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.
VRCz)
which series is convergent for all values of z without the circle.
Let z', sf be the point where the path of integration starting from the point z0, s0 and leading to infinity, cuts the circle.
We have
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*-_r &»+$+... 1*
J*.*vB(d Js.s'Lz2 «3 J
T >.
2*2
an expression which is finite for both the upper and the lower limit. We have thus shown that the integral
:-s dz
f
Jz«,
VR(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 b 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 ai, 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
C 4* + C 4* = o, where s = Jc s JK 8
c
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
that in the integral / — the elements of integration cancel in pairs, so JK s n. jz
that this integral is zero. We have thus shown that the integral / -
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, s0 and 2, s, without crossing either of the canals a or b, the two curves will form a closed curve, and from what we have just seen (1)
dz r^'^dz =
8 1/2, S 8
(1) (2)
C*>sdz = Czs dz
J ZQ, SQ S J ZQ, So S
or
a>" (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
where the dash over u signifies that the integration is to be taken in the Riemann surface T' ', in which the canals a and b are not to be crossed, it follows from above that u (z, s) is entirely 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, 8) =
where the path of integration is taken in the Riemann surface T, which does not contain the canals a and b. We shall show that here the integral
u(z, s) is not a one-valued function of the upper limit z, s, but -depends upon the path of integration.
In the T-surface the inte gral corresponding to u(z, s)
s
u(z, s) =
dz
dzm
s
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 (A) u(z, s)
Fig. 59.
We have seen that in the Riemann surface Tf every integral is indepen dent of the path of integration. We note that (see Art. 130)
_f =u (z2, s2) — u(zi, si).
Returning to the equation (A), it is seen that neither of the canals a or is crossed between zQ, SQ and p, so that
f" 4* =
JZQ,SQ $
n
THE PROBLEM OF INVERSION. 161
Further-, there is no canal between A and z, s. It follows from what we have just shown that
'°dz = u(g s) - u(X) in T',
s
where we go in T' from ZQ, s0 to z, s by crossing the canals a3 a4 and a i a2 as shown in the figure. We have to make the same crossings to go from zQ) SQ to L We therefore have from the equation (A)
u(z, s) = u(z, s) + u(p) — u(X).
If the canal a had been crossed at any other point pi, AI instead of at p,A, we would have had
u(z, s) = u(z, s) + u(pi) —
Consider the difference
The points p and pi are both on the same side of the canal a, while the point X and Xi are both on the opposite bank. It is seen that
u(p) -u(Pl) - f JP1
s
and u(X) - u(^) = — in
s
where the path of integration in T may be quite arbitrary, provided only it does not cross the canals a and &. We may therefore take the path of integration from p to pi indefinitely near the right bank of the canal, while the path from X to ^ 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
{u(p) - u(^\-\u(io1) - M(^i)} = 0,
and consequently u(p) — u(X) has the same value at whatever point the crossing has taken place.
ART. 139. If we cross the canal a from z0, s0 to z, s in the opposite direction from that gone over in the previous case, we have
Cz>sriz r^ d? rz,*fj?
I *= I ™. + I «£ (i
*SZQ,SQ S JZO,SQ S *) p S
- u(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 z0, s0 to the canal joining ax and a2 and after crossing this canal into the lower leaf come out again
into the upper leaf by crossing the canal a3a4 and then pro ceed to z, 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(X) from u(z, s).
If the canal a is crossed // times in the first direction and v times in the second direction,
Fig. 60.
we will have u(z, s) = u(z, s)
We have precisely the same result if we cross the canal b. 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 : u(X) — u(p) = A, for the canal b : 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 b cannot be crossed. The quantities A and B are called the Moduli of Periodicity.
ART. 140. We have seen that if a and 6 are two quantities whose quo
tient is not real and if the coefficient of i in the complex quantity - is
positive, we may determine a function $(u) which satisfies the two func tional equations
$(u + a) = <b(u),
This function is (cf. Art. 86)
6) = e
m= +00
where Q = e
THE PROBLEM OF INVERSION. 163
If the two moduli of periodicity A and B have the property that the
•efficient of i in ~— is positii B
form a function 4>(w) so that
coefficient of i in — is positive, then we may write a = B and 6 = A and B
*(«)=
m = + oo m2- 2 m
— — fr- mu
. m= —oo
where Qo = e" Band Bm+k= Bm. We then have
B} =
xik
~~(* A
$>(u + A) Instead of the variable u we may introduce any variable quantity, say
u(z, s)= I -•
*SZQ,SO S
We then have
$00 -^«(*, «)]-¥(*,*), 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) = u(z, s) + tnA 4- nB.
Hence the complex of values "*&(z, s) which belong to one position z, s is expressed through
\p(2> s) = &[ti(z, s) + mA + nB],
where m and n are integers.
Since 4>(w) has the period B, the above complex of values reduces to
<b[u(z, s) + mA].
We saw in Art. 91 that the following relation existed for the general ^-function: _ E**(2mM+W26)
Consequently the complex of values above becomes
-^[2mu(z,s)+m*A]
<&[u(z, s) + mA] = e $["& «)]•
It is evident that &[u(z, s)] = W(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.
164 THEORY OF ELLIPTIC FUNCTIONS.
ART. 141. We saw in Art. 94 that
Let the corresponding "^-functions be denoted by
We then have, for example,
=
_xik
It follows that
- ~[2 mu(z, s) +m?A} -
B
and since ^1(2,5), ^^(z, s) are both one-valued functions of position on the Riemann surface, it is also seen that — 1 ' is a one- valued function
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
is infinite for no point of the Riemann surface, including the point at infinity. It follows that ^1(2, s), ^(z, s) are everywhere convergent and
\T/ f^ s\
consequently the quotient — 1V ' ' has definite values everywhere on the
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
) _ R( ,
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 b : u(p) — u(X) = B.
It made no difference where the point A, p was situated on the canal. We may therefore take the point a, a' where the canal b cuts the canal a and have accordingly
u(ar) — u(a) = A, or
A = P — in T', (cf. Neumann, loc. cit., p. 248],
J a S
THE PROBLEM OF INVERSION;
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
p.
A- f-i
Jb 8
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
the integration being in the negative direction. We have thus shown
' that B is the closed in-
p S" 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, a i, a2, a3, say. But here the point at infinity is also a branch-point (Art. 115). We may therefore connect a! and a2 by a canal and a3 with the point at infinity. The Riemann surface may then be represented as in the former case (see figure).
A
ART. 144. In the derivation of the function 4>(w) the ratio — cannot
n
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 u(z, 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
H(z, s) = p + iq.
* Riemann, Theorie derAbeVschenFunctionen, Crelle, Bd. 54, p. 145; see also Koenigs- berger, Elliptische Functionen, pp. 368, 369; Fuchs, Crelle, Bd. 83, pp. 13 et seq.
Fig. 62.
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 da(z>s) - du(z> s) ** = du(z> s) . 1 = ** = 1 . dx dz Ox dz dz \/R(z)
it is seen that — is infinite for z = 01, a2, a3, or a4. On the other hand,
du dp , . 00
- = _£ _|_ l—Lj
Ox Ox Ox and consequently either -2 or -^ or both of these derivatives are infinite
for z = ai, a2, a3, or a 4. 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 &. It is seen that we may go over both the banks X and p of a and b with a single trace.
The integral / pdq taken over this trace may be divided into several integrals as follows:
//*&>) ru) /*u) /*v»
pdg = I pdg + / pd# + / pdq + / pdq, t/ar£on+a •J09Jfon-» t/^/a'on-a t/a'^aon + ft v
where1 (^o) as an upper index means that we are on the right bank, means the portion of curve gone over, and + a means on the canal a in the positive direction.
ART. 145. We saw above that
du = —^= = dp + idq, VR(z)
dz^ = dx^idy = d .d VR(z) VR(z)
If we write t - = ^(x, y) + ^(x, y),
VR(z) then is
(j)(x, y)dx — ^r(x, y)dy + i{ ^(x, y)dx + <f>(x, y)dy} = dp 4- idq. It follows that j/x
W,
THE PROBLEM OF INVERSION. 167
The function ^(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 a and b are indefinitely narrow. The same is true of the function </>(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 (p) (A) r (P) (A)
Pdg = J+m*P ~ P^dq + J+>{P ~ P^dq- In Art. 139 we put A = u(X) — u(p) on the canal a;
(A) (A) (p) (p)
or A = p + iq — { p + iq }
(A) (P) (A) (P)
= P - p + t\<i - q}-
If further we write A = a + ifi, then is
(A) (P)
a = p — p on the canal a. We also had
(p) (p) (A) (A)
B = u(p) — Ti(X) = p + iq — {p + iq}
(p) (A) (p) (A)
= p- P + i{q-q\,
and writing B = r + id.
(p) (A)
it follows that 7- = p — p on the canal b. It is seen at once that the above integral may be written
/ pdq = - a I dq + r I dq.
J J+a J+b
+a «/+*
» it is clear thai
VR(z)
Since du = , Z > it is clear that
-J$ -
= I dp + i I dq.
Jb Jb
Further, since A = a + if}, we have
/? = I dq; and similarly
*) b
8 = fdq. <Ja
The integral above is finally
/ pdq = rfi - ad.
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
+ 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
.
taken over the same region*
Consider the curvilinear integral
where as above the integration is to be taken over both banks of the two canals a and b in the Riemann surface Tr. We have seen that p is one- valued, finite and continuous within this surface, since it is the real
part of v(z, s). But (see Art. 144) -2- and -2- become infinite at the points «i, Oi2, a3 and a4.
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 y
c ao , i • ap do d^a
for — the expression -£- -* + p — *-
ax ax ay axay
T <« dP ,1 • dv da d2q
and for — the expression -*- -^- + p — *-:
ay ay ax axay
and consequently
//^ Crdp do dp do~\~ paq = • \ -A- — * — — *- axoy. J J |_ax ay ay axj
* Forsyth, p. 23; see also Casorati, Teorica delle funzioni di varidbili complesse, pp. 64-69; Neumann, AbeVsche 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
But since **- = & and & = - &
Ox dy ay dx
(being the conditions that u(z, s)= p + iq have a definite derivative), and since I pdg = (5/- — ad, it follows that
As the elements under the sign of integration are essentially positive, it is seen that fa — ad is a positive quantity. But we have
B = r + i^ = (r + id ) (a - J3) = ar + pd , . ad - fa A a+ip a2 + ^ ~~ a2 + p2 ~ ? a2 + p2 '
D
Since ad — fa 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. A B
We may therefore (see Art. 86) form functions $(u) such that *(« + B) -$(*),
•01
" a A
ART. 147. In the expression
since w(2, 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
¥(*,*) =$[u(z,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 ¥(z, s) — 0. We shall now see how often the function W(z, s) becomes zero on the Riemann surface Tr.
In Art. 92 we saw that if a f unction f(z, s) is discontinuous at isolated positions within a portion of surface, but otherwise is one-valued and finite, then
log/fcs) . 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 Functionen, etc., p. 102; Falk, Ada Math., Bd. 70; Pringsheim, Math. Ann., Bd. 27.
170 THEORY OF ELLIPTIC FUNCTIONS.
diminished by the sum of the orders of its infinities within the portion of surface in question; i.e.,
\mj '
2m J dz
As the portion of surface we shall take the surface T' which is bounded by the canals a and fe, and for/(z, s) we have here ^(2, s). There being no infinities, HA = 0, and consequently
j__ rd
2 Tti J
, s) d
dz
where the integration taken over both banks of the canals a and b is equal to the sum of the orders of the zeros in T' ' .
Now on the canal a we have u(X)— u(p)= A,
or u(X)= u(p) + A.
It follows that
= <S>[u(p) + A] = and consequently that
On the canal b we have u(p)— u(X}= B, or u(X) = u(p) — B.
It follows that
or
From the figure in Art. 142 it is seen that
r d\oKv(z,8) dz = w d log v(Z, s) dz + p» d\
Jr dz Jarfon+a dz Jpdpon-6
+ /•(« d log V(z, s) dz + rv d log V(z, s)
Jy/a'on-a ^Z Ja'
I -/ aT^
J+adz[ ^(^
'd'aon+b
which owing to (M) and (N)
dz
THE PROBLEM OF INVERSION. 171
But from Art. 139 it is seen that
We therefore have finally
i pnoE
! -i J
dz
It is thus seen that the intermediary function W(z, s) has k zeros on the surface T'; and since W(z, s) vanishes on the same points as ^(z, s), it follows that W(z, s) has k zeros on the Riemann surface T.
ART. 148. We saw (Art. 87) that when k = 2
Further, write Q = q*, and it follows that
If in ®i(u) we write — « in the place of //, the summation is not thereby changed, and we have
/(=-»
From this it is seen that ®i(u) = ®i(—u), or ®i(w) is an even function. Similarly writing — « — 1 for ,« in the formula for HI(M) we have
or HI(M) = HI( — u), so that this function is also even.
ART. 149. If in ©I(M) we write u(z, s) instead of u, then ®I(M) becomes
¥<,(*, s) = %(z, s) . e B (cf. Art. 140).
Suppose that, starting from a point z0, 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 b as often as we choose. The point z, s may lie in either the upper or the lower leaf. Next starting from the point z0, — s0, which lies immediately under the point z0, SQ, let us construct a second path, which is everywhere congruent to the first path, that is, which lies in the under
172 THEOEY 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(zy 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
where (I) and (II) are used to denote the paths of integration. Suppose that 20, SQ coincides with one of the branch-points, for example with 01, then ZQ, SQ and ZQ, — s0 coincide, and we have
/ (*z>s dz rz'~s rlz
I S£+ I * = o,
«/ai S t/ai S
(I) (II)
or
u(z, s) + gA + hB + u(z, - s) + g' A + h'B = 0,
where g, g', h, hr denote integers. It follows that
u(z, s) + u(z, - s) = TA + dB,
where f and d are integers.
// then we take a branch-point as the initial point of the path of integration, the function u(z, s) has at two points situated the one over the other in the Riemann surface Tf, values whose sum is equal to integral multiples of A and B.
ART. 150. If we write u(z, s) for u in ®i(u), we have the function W0(z, s); similarly let M*i(z, s) denote the result of substituting u(z^s) for u in Hi(V). Then noting the relations existing between M*0, "^o and between M^ and "^i, it is seen (cf. Art. 141) that
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 ^0(z, - s) = &i[u(z, - s)]
'S) *!&.-») Ktu^-s)] 6i[- tZ(g,g) + rA + dB] _ 0i[- u(z,s)]
HI[- u(z, s) + rA
as is seen from the functional equations which ®i and HI satisfy. Since ®i and HI are even functions, it follows that
R(z, - ») - - *(*' 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;
or R(z, s) is a rational function of z alone.
ART. 151. Since A: = 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
(M)
i A3z + A4
where the A's are constants. This function has the two zeros of the first order
and the two infinities
Remark. — If the zero z = - ^ is a branch-point, say ai, then (see
1 A
Art. 120) twice the exponent of the lowest power of 2 — ai = z + — in
A i the development in ascending powers of z — a-i is the order of the
zero. But as the development of the numerator of the above expression
is simply AI \z + ^ I it is seen that 2 is the order of the zero for
4o L ^
z = - ^ • Such a zero is therefore to be counted as two zeros of the first
1 4.
order. The case where — ^p is a branch-point may be treated in an
analogous manner.
ART. 152. It follows directly from equation (M) above that
from which it is seen that z is a one-valued doubly periodic function of u with periods A and B. We call z the inverse of the elliptic integral u, where
dz
- r
J
- a<2}(z - a3}(z — a4)
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 ^
s = — — du
is a one-valued function of u; for the derivative of a one-valued doubly peri odic function is one-valued and doubly periodic.
ART. 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 merit iibersteigt. 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 fur 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 study the one-valued functions of position on the Riemann surface s = \/R(z)t we might have shown immediately that //7,\2
©-««•
For in the differential equation (cf. Art. 106)
' -^
when a definite value is given to z, say z0, then the sum of the two roots
of the equation is ^ v /^x ^
I -j- 1 \ =
\dz/2 A0(z0)°
On the other hand, corresponding to the value z0 there are within the initial period-parallelogram two values of u say ui and u2. Also, since u\ + u2 = Constant, it follows that
!*!•.- A (ii)
But the left-hand side of (i) is the same as the left-hand side of (ii), and consequently* A 1(2) = 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 6; also let 2 = /(«) 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(w, z) = 0,
which is of the second degree in w and of the fcth degree in z. This equation may be written
Lw2 + 2Pw + 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
L L~
where a = ±VP2 - LQ.
We therefore have 7- , p
O — Ljii ~r -t j
so that o is a one- valued function of w.
We saw above that ^
— = ±\/R(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
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 b. It follows also (see Art. 104) that an algebraic equation exists between a -r-(dz/du) and z; and consequently a -±-(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 *
- P + T(z) —
— ^ = p + qs,
p and q being rational functions of z.
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 &th 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 — •
du ,
There exists (Art. 104) between w and w' = — an integral algebraic
du
equation
(I) G(w, w') = 0,
which is of the Mh 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 /bth order, there correspond k values of u to WQ in the period-parallelogram, say
HI, u2, ..;,***. (2)
We also know that between the functions / and w there is an algebraic equation
(II) (?! (w,0=0
of the fcth degree in tt so that corresponding to the value w0 there are k values of t, say ^ t2, . . . , 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'2, . . . , tw'k~l
enjoy the same property. It follows that the sums
/i + t2 +t3 + • • - + tk = PQ,
+ t2w2' + t3w3' + • • • + tkwk' = PI,
+ t2W2'2 + t3W3'2 + • • • + tkWk'2 - P2,
fiwi'*-1 + t2w2'k~l + faM'a'*-1 + • • - + to'*-1 = P*-i
are one- valued functions of w, and have definite values for all values of w on the Riemann surface. They are therefore rational functions of w. ART. 157. If we multiply the above equations respectively by
Ak-i, Ak-2, • • • ^i,l, add the results and equate to zero the coefficients of
t-2, £3, • • • > tkj we will have the system of k equations:
~* + •-.- + Ak-2W2' + Ak-i = 0, ~* + • • • + Ak-2wB' + Ak-i = 0,
(4) and the additional equation
= PA-I + A'iPjt-2 + - • • + --U-2Pi + Ak-iPo. (5)
The equations (4) show that the quantities ^.i, A2, . . . , Ak-i are the coefficients of an algebraic equation of the ft— 1st degree whose roots are w2', w3', . . . , wkf.
We obtain this equation by dividing (I) arranged in decreasing powers of wr by w' — -M'I'. The coefficients of the quotient, which are integral functions of w0 and wi', will give the quantities .-ii, A2, . . . , Ak-i-
From equation (5) we have t expressed as a rational function of w and w'.
This theorem is a generalization of Liouville's Theorem above. In Chapter XX we shall again prove indirectly both theorems.
178 THEORY OF ELLIPTIC FUNCTIONS.
ART. 158. We shall prove in Chapter XVI that the doubly periodic function of the second order z = (j>(u) is such that <j>(u + v) may be expressed rationally in terms of </>(u), (/>'(u), <f>(v)t (£>'(v), say
<l>(u + v)= Ri[<j>(u), <l>'(u), </> (v), P(v)], (1)
where R with a suffix denotes a rational function, and consequently also
f(u + v) = Rd&(u), <j>'(u), <f>(v), $ (v)]. (2)
For the present admit the above statements.
By Liouville's Theorem it follows that w = F(u) is a rational function of 4,(u) and p(u), or p(u} _ BJ[^(M)) ^(M)]. We consequently have
F(u + v) = R3[<f}(u + v), <j>'(u + v)]
= R^(u),<t>'(u},<j>(v),<i>'(v)}. (3)
Also from Briot and Bouquet's Theorem
and <j>'(u)=R6[F(u),F'(u)].
Hence from (3) we see that
F(u + v)= R7[F(u), F'(u), F(v), F'(v)].
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 <j)(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), (£>'(u), <t>(v), (/>'(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, niu
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
s2 - R(z) = 0,
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 (z, 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, 2), 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, 2), where
s = V'A (z - ai) (z - a2) (z - a3) (z - a4) = ~,
du
degenerates into the simply periodic realm when any pair of branch points are equal and into the realm of rational functions (z) when two pairs of branch-points are equal (including of course the case where all the branch-points are equal).
Thus the elliptic realm (z, 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
R (z s) = A° + AlS + A*s2 + ' ' ' + Aksk B0 + BlS + B2s2 + • • - + Bis1
where the A's and B's are integral functions of z. Owing to the relation s2 = A(z — a\) (z — a2) (z — a3) (z — a4), 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
_C±Ds ~~E~ where C, D, and E are integral functions of z as are A0', AI, BQ' and
Writing |= p(z) and 5= q(g)j
EJ E/
it is seen that D / \ / \ / \ / \
Ri(z, s) = p(z) + q(z)*s = p(z)
s
where q(z)'S2 = Q(z) and where p(z), q(z), and Q(z) are rational functions of z. (See also Arts. 125 et seq.) Consider next the integral
C
, s)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
JSM dz which may be written / -^ dz. « J VZ
f(z) denoting a rational function of z, and s =\/~R(z).
* Legendre, Memoire sur les transcendantes elliptiques, 1794. See also Legendre, Fonctions Elliptiques, t. I, Chap. I.
180
ELLIPTIC INTEGRALS IN GENERAL. 181
ART. 161. Suppose* in general that
R(z) = C0zn + C1z"-1 + . • • -f Cn, where the C's are constants. When n is greater than 4, the integral
dz
is no longer an elliptic but a hyper elliptic 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
f(z) = ^ = G(z) + ^I&, g(z) g(z)
the g'a and G's denoting integral functions, and say
g(z) = B(z - b^ (z - b*)**(z - b^ . . • . Hence when resolved into partial fractions
/(z) = G(z) +2) — ^—>> (Ak constants),
i (z - bi)** and also
dz
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
ink •'
\/R(z}
We thus have two general types of integrals to consider,
and Hk= C
J (z-
VR(z) dz
(z-b)><VR(z) ART. 162. Form the expression
= kzk~l\/R(z) - .=zk = _[<2kR(z) + zR'(z)]
%VR(z} 2VR(z)
- 2VR(z)
iZ + 2kCn].
* Briot et Bouquet, Fonctwns EUiptiques, p. 436; see also Koenigsberger, EUip- tische Functionen, p. 260; Appell et Lacour, Fonctions EUiptiques, p. 235.
182 THEORY OF ELLIPTIC FUNCTIONS.
It follows through integration that
2zkVR(z} = (2k + n)C0Ik + n-i + (2k + n- l)Cl!k+n-2
+ (2k
If in this expression we put A; = 0, it is seen that In-i may be expressed through In-2, In-s, • • - , I Q, 1 - 1 and through the function VR(z); when k is put = 1, we mayjexpress In through In- 1, In-2, • • • , /o and through the function z\/R(z). If further we write for In-\ its value, we may express In through In-2, In-s, .. . • , /Q, J-i and an algebraic function. This algebraic function is an integral function of the first degree in z multiplied by \/R(z).
Continuing in the same manner, we may express In + X through In-2, In-3, - • • , IQ, I - 1 and an algebraic function 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
^ [" VR(z) 1 ^ k v^Tz) + - R'^
dzL(z-b)k\ (z-b)k+1 2 (z-b)kVR^zj
-R'(z)(z-b)}.
/ - 2VR(z)(z-
If we write - 2kR(z) + R'(z) (z-b)=
then is
z) + R(v + V(z)(z - 6)
or <l>W(z) = (v - 2k)RW(z) + (z -
It follows, since
that 21
=- 2kR(b)
(n — 1)1
(z -
n ~ l ~^ k (n — 1)!
ELLIPTIC INTEGRALS IN GENERAL. 183
Integrating it is seen that
(n-V (6) Hk-n+2
(n - 1)! If we put k = 1, we see that H2 ma\r be expressed through
This is correct only if R(b)j£ 0; i.e., if 6 is not a root of the equation R(z)= 0. This case is for the moment excluded. We note that
rr C dz 1 . „ f(z - 6) dz r 7 r .
HO = I — ===== = 70, H-i = I - - • • •
J VRz J \
R(z) „ C
n. -(n-2)= I
J
VR(z)
From this it is seen that the integrals HQj H-i, H -2, . . . , #_(n_2) may be expressed through integrals of the type /A-. Hence the integral HI alone offers something new.
We note that H2 may be expressed through H\, IQ, I\, . . . , 7n-2 and through an algebraic function of z. If we put k = 2, we may express H3
«
through H2, HI, . . . , #_(n_3) and through - -J-^ ; or, if for H2 we write
(z-by
its value just found. H3 may be expressed through //1? 70, /i, . . . , In-2 and an algebraic function of z. In general, we may express Hm through HI, I0, 1 1, . . . , 7,j_2 and an algebraic function of z. We thus have to consider only the integrals 70? I\, . . . , In-2 and HI = /_1? since I-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 77jt+1 drops out. Since R(z) cannot have a double root, as otherwise it could be taken from under the root sign in V 7? (2), we may in this case express
HI through the integrals 770, 77_!, . . . , 7/_(n_2^, - ~; and conse-
2 — 6
quently through integrals of the type 7*.- alone.
ART. 164. We have therefore to consider the integrals
r z*dz
I , -
J V.
T Ik
where k = 0, 1 ..... n — 2, where n is the degree of the integral func tion R(z), and in addition the integral
184 THEORY 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 6 is called the parameter (Legendre, Functions Elliptiques, t. I, p. 18) of the integral HI.
ART. 165. For the elliptic integrals, if n = 4, we have the integrals /o, /i, /2, HI', if n = 3, there are the integrals 70, /i, HI. In the first of these cases we shall see that /i reduces to elementary integrals; and with Legendre we call
70 = / — — — - an elliptic integral of the first kind, J VR(z)
/O 7 z Z an elliptic integral of the second kind, \/R(z)
H\= / — an elliptic integral of the third kind.
J (z-b) VR(Z)
LEGENDRE'S NORMAL FORMS. ART. 166. In the expression
dz dz
VR(z) VA(z - en) (z - a2) (z - a3) (z - a4)
let us make the homographic transformation
... _ at + b
ct + d ' It follows that
z — ak = —
ct + d and ad -
(ct + d)2 We then have
dz ^ (ad - bc}dt
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
[got2 + git + g2][h0t2 + hit + h2], * Legendre, loc. cit., Chap. II.
ELLIPTIC INTEGRALS IN GENERAL. 185
where go = (a — cai) (a — ca2),
g\ = (d — cai) (b — da2) -f- (a — ca2) (b — ddi), , g2 = (b — dd\) (b — dd2)j
and where h0, hi, h2 are had when we interchange a! with a3 and a2 with a4 in the expression for the g's.
That the coefficients of t3 and t disappear, we must have
hog i + go^i = 0, gih-2 + hig2 = 0. These two equations are satisfied if we put
0i = 0 and hi = 0. From the expression gi = 0 it follows that
2 ab — (dd + be) (di + d2) + 2cddid2 = 0; and from hi = 0 we have
2 db - (dd + be) (a3 + d4) + 2 cda3a4 = 0. These two equations may be written
„ \ i o d c n
'i T &2) ~r £ - - did2 = u. 6 a
i3 + a4) + 2 - — a3 &4 = 0. 6 a
From them we may determine - + - and - . - considered as unknown quantities.
If d3 + a4 = a i + d2 and a3 • a4 = ax • a2, the two equations reduce
to one and then we need only determine the quantities - + - and - • -
b a b d
so that they satisfy the one equation. When these two quantities have
been determined, the quantities - and - may be found from a quadratic
o a equation.
When these conditions have all been satisfied, then in the expression
the coefficients of t in both factors drop out. We have finally
dz (ad - bc)dt
g2) (h0t2 + h2)
Legendre further wrote ^ = - p2, = - q2
g-2 h2
80 that dz (ad - bc)dt
VR(z) VAg2h2(l - p2t2) (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)dz dz p ___
If we put £=k2 and C = ad
the above expression is
- z2) (1 -
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
CQ(z)dz u
• -v\ / becomes
J VR(z) J V(l - z2) (1 - k2z2)
where /(z) denotes a rational function of z. We may write this function in the form
,, ,_ 0(Z2) +Z(/>T(Z2)
where <£, <j>i, ^, ^i denote integral functions. If we multiply the numer ator and the denominator of this last expression by ^(z2) — z^1(z2)f it is seen that /(z) = /0(z2)+ z/^z2), where /0 and fi are rational functions of z.
The above integral correspondingly becomes
r _ f(z)dz = r Mz^dz _ + r zf^z^dz
J V(l- z2) (1 - /b2z2) J V(l - z2) (1 - A;2z2) J V(l - z2) (1 - Fz2)
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
V(l-z2)(l-A;2z2) and
d ia-z^g-Fz2)"! = q0 + ai(z2 - 6) + a2(z2 - b}2 + a3(z2 (z2 - b)k J (Z2 _ 6)fc+i \/(l-z2)(l-/c2z2)
ELLIPTIC INTEGRALS IN GENERAL. 187
it may be shown that the integral
is dependent upon the evaluation of the integrals
r dz t r z2dz
J Vl-z2l-£2z2' J V(l-z2)(l-A;
These integrals are known as Legendre's normal integrals of the first, second, and third kinds respectively.
ART. 168. The name " elliptic integral " is clue to the fact that such an integral appears in the rectification of an ellipse. Writing the equation
cy fy
of the ellipse: ^ -f ^~ = I, the length of arc is determined through
rx* /-, - AM2j r\ a4 -
s = I V l i~ j } dx = I V ;
Jo ' \<H7 J0 v a'
If the numerical eccentricity is introduced:
-
/« /a2 _ ^2 ^x g2 _ g2j.2
= I V/~; r-«— I /, , 9, , 2 2 ^dx.
Jo * a2 — ^2 Jo v(a2 - x2) (a2 — e2z2)
If further we put x = a sin 0, it is seen that
s= I Vl - e2 sin2 (j> d<f).
i/O
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
f(z)dz
where f(z) again denotes a rational function of z, we may by writing
z = mt -f n
make a23 + 3bz2 + 3cz + d = ±t*-g2t- g3,
where g^ and g% are constants.
This is effected bv writing n = , am3 = 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
F(t)dt
r F(I
J \/4 P -
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
r dt r tdt r
J V4<3 - t - 3 J V4P - g2t - 93 ' J
- g2t - 93 J V4P - g2t - 93 (t - b) V4 13 - g2t -
which correspond to the normal forms employed by Weierstrass. ART. 170. In the expression
(1) R(z) = A(z - ai) (z - 02) (z - a3) (z - a4),
make the homographic transformation
•-££•'./•. ."/:
and so determine the coefficients * that to
z = 0,1, z = a2, z = as, z = a4 correspond
i--!, « — 1, i-4-l, z = + |.
It follows immediately from (2) that
(5) .-^-, (0)
where p, 9, r, s are constants which may be determined as follows: In (4) write z = a3, z = 1, and in (5) put z - a2, z = -- 1. We thus have
2 2r
- a2 = — — >
1 - fJL 1 +
Equations (4) and (5) thereby become
z
a3 1 + /£ 1 - z
. i — — * — • •
0.3-0,2 2 1 - //z a2 - a3 2 1 - ,«z
In the same manner we derive from equations (3) and (6) the following:
l--fl l+£
#4 — a\ 2 1 — «z ai — a4 2 1 — /tz
* Koenigsberger, Elliptische Functionen, p. 271.
ELLIPTIC INTEGRALS IN GENERAL. 189
Equations (7) and (8) become through division
Z — Q2 = fJL — 1 . 1 + Z 2 — a3 ,« + 1 1 — Z
Writing in this equation z = a4, z = - , we have
q—
- + 1 A; — 1
and similarly for the values z = ai, z = — -, the same equation gives
K Ol - 02 = « ~ 1 ^ fc ~ 1
ai - a3 « + 1 ' k 4- T
The quantities A: and ,« may be determined from the last two expressions in the form
(11)
(12) 1 + f* = «i ~ «3 . 1 - k
1 — a a i — 0,2 1 4- k
From the equations (9) and (7) we have
(/z 2(1 - («z)2 rfz 2(1 - «z)2
and consequently
O / 1 -«\ '>
2(1— /iz)a
Through the multiplication of (7), (8), (9), (10) and (13), it follows at once that
dz
r^^ = JL
J x/7?^ M
-Z2)(1-/C2Z2)
where M = - (
and where 2 and z as determined from (7) and (8) are connected by the relation
_ a3 + #2 _^ <*3 ~ a2 z — /£ 2 2 * 1 - az'
the quantities u. and A: being determined from equations (12) and (13).
190 THEORY OF ELLIPTIC FUNCTIONS.
ART. 171. If in the equation
we put the right-hand side = r, then the six different anharmonic ratios which may be had by the interchange of the a's are denoted by
1. , 1 . T T-l.
rV ' 1-r' T-l' r
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
±k,
i-k/ Vl-v \i-ik
(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
A(z - ai)(z - a2)(z - a3) into Legendre's normal form.
Noting that A(z — a\)(z — a2)(z — a3)
-Limit -—(z - ai)(z - a2)(z - a3)(z - a4) 1,
a4 = oo L a4 J
we have to write in the formulas above — -in the place of A, and let
a4
a4 become infinite. We then have
r dz J_ f _=
J VA(z — a\)(z - a2)(z — a3) M J \/ (\
dz
- k2z2} where M = - y 7 >
_ a3 + a2 , a3 — a2 z — k
Z — " ' J
and
- 03
ELLIPTIC INTEGRALS IN GENERAL. 191
ART. 172. In the expression
VR(z) = VA(z - ai) (z - a2) (z - a3) (z - a4)
write
If we put V(ai - o2) (a i - a3) .4=2 Af, we have
Choose e so that 3 e - a'2 ~ °4 - °3 ~ a4 - 0.
ai — a2 ai — a2
Let ei be the value of e that satisfies this equation, and write
e We finally have
&2 — &4 i &<* —
e2 = e\ -- • - *• and 63 = e\ -- -
0,1—0,2 a i —
dz
2 J/ ^(t-ei)(t-e2)(t-e3) . dt
where eie2 + 62e3 +
It also follows that
rP(z)dz _ r J \S «/
where P and p denote rational functions.
The quantities g2 and g3 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 = C C^—>
J VR(z)
where the function R(z) may be written
R (z) = a0z* -f 4aiz3 + 6a222 + 4a32 + a4.
192 THEORY OF ELLIPTIC FUNCTIONS.
Write 2 = £!,
x2
dz - x*dxi ~ ^1^2,
X22
where the variables xi, x2 individually are not determined, but only their quotient.
We then have
-f
It is seen that /(a; i; x2) is a binary form* of the fourth degree. We have at once
If next we write
xi = <*>yi + by2}
X2 = cyi +
it is seen that/(xi, x2) becomes another binary form </)(ylt y2) of the fourth degree.
ART. 174. In general make the above substitutions in the binary form of the nth degree
, x2) = a0^in + niaixin-*x2 + n2a2x1n-2x22 + • - • + anx2n, where HI, n2, . . . are the binomial coefficients.
We thus derive another binary form of the nth degree <j>(ylt y2). It is seen at once that
— /Oi, x2) = a0zn + mai^-1 + n2a2zn~2 + ...+«„
= a0(z - ai) (z - a2) ... (z - an), say. It follows that
. . (xi — anx2), and correspondingly
<£(2/i, 2/2)= ao'G/i - /?i?/2) O/i - ft?/2) - • . (?/i ~/?n?/2). Further, since xl - aix2 = ayi + by2 - «i(c?/i + d?/2)
a - * Bocher, Introduction to Higher Algebra, p. 260.
ELLIPTIC INTEGRALS IN GENERAL. 193
it is evident that one of the /?'s, say
and similarly
jj2=^d^bt etc a — a2c
From this it is clear that, if some of the a's are equal, some of the /?'s are also equal, and that there are just as many equal roots in the equation 0(2/i, 2/2) = 0 as there are in/(xi, x2) = 0.
ART. 175. The above correspondence gives rise to the following con sideration: Suppose we have given the quadratic form
a0z2 + 2aiz + a2. The roots of the quadratic equation
aQz2 + 2 diz + a2 = 0
are z = — ± — \ a\2 — a0a2.
If we write ai2 — a0a2 = £>(ao> ai> a2), 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(a0, ait a2, . . . , an) is an integral rational function of a0, 0,1, . . . , an and is homogeneous with respect to these quantities.
If next we form the discriminant D(a0', a/, a2, . . . , an') of the form
, 2/2) = fl(/2/in + niaifyin~li/2 + n2a2'ijin-2i/22 + • • • + an'?/2n,
then the vanishing of this discriminant is the condition that 0(?/i, 2/2) have equal roots /?. But we saw that 0(?/i, y2) had equal roots when the roots off(xi, x2) are equal. It follows that
D(a0',ai', . . . ,ow/)=CD(a0,ai, . . • , on), where C is a constant factor. This constant factor * is { ad — be } n(n~ x).
* 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 THEORY OF ELLIPTIC FUNCTIONS.
ART. 176. If the function f(xi,x2) = a0Xin+ n\aiXin- Ix2 + • • • + anx2n becomes through the substitutions
xi \\ayi + by2, x2 \\cyi + dij2,
</>(yi, 2/2) = a>o'yin + nialryin-ly2 + • • • + an'y2n, and if I is a function of the coefficients such that
/(«</, a>i, - • • , On') = (ad - bc)^I(a0)al} . . . , an),
where /JL is an integer, then / is called an invariant of the form/(xi, x2).
It may be shown * that, if / is an invariant, /* must be equal to £ up, where p is the degree of / with respect to the coefficients a0, «i, . . . , an. The quantity /JL is sometimes called the index of the invariant.
The following theorem is also truerf All the invariants of a binary formf(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,
there are only two fundamental invariants (cf. Sylvester, Phil. Mag., April, 1853).
The one of these is J
72= a0a4 — 4«ia3 4- 3a2.
If by the given transformations we bring f(x\, x2) to the form
0(2/i, 2/2) = «o'?/i4 + 4a1/?/13!/2 + - - - + a/2/24, then it is easy to show that
ao'a4' — 4 ai'oa' + 3 a22 = (a0«4 — 4 a^a^ + 3 a22)(ad — be)4.
In this case p = 2, w = 4, /JL = J n/o = 4. We thus have
72' = 72(ad
The other fundamental invariant § is
73 = a0a2a4 + 2a!a2a3 - a23 -
It is seen at once that
73' - 73(ad - 6c)«.
* Cf. Salmon, loc. cit., p. 130; Burnside and Panton, loc. cit., p. 376.
f Cf. Salmon, pp. Ill, 132, 175; Bocher, loc. cit., Chap. XVII, and Burnside and Panton, p. 405.
% 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, Crelle, Bd. 27, p. 81; Aronhold, Crelle, Bd. 39, p. 140.
ELLIPTIC INTEGRALS IN GENERAL.
195
ART. 177. The discriminant D of the binary form f(x\, x2) may be rationally expressed (cf. Salmon, loc. cit., p. 112) in terms of 72 and 73 in the form
D = 723 - 27 732. It is evident that
D' = 72'3 - 27 73'2 = D(ad- be)12.
ART. 178. The functional-determinant or Jacobian of the two forms L> X2), ^2(^1, xz) may be written
F =
dx2
6^2 dx2
If we make the substitution
*i = ^1(2/1,2/2), %2 = ^2(2/1,2/2),
>l! and /12 being functional signs, then ^i(xi, x2) becomes a function of ?/i, 2/2, which may be symbolically denoted by [^i(zi, x2)]s and ^2(xi, x2) becomes by the same substitution [^2(xi, x2)]s.
We form the functional determinant of these two forms
and we shall study the relation between F and <1>. It is evident, since
ch/r = 6^r 6X! 3
dz/i dxi d#i a that
L
'iOi,:r2)1 a^i . ^^1(^1^2)1 &k
I rl — ' ' I
dx\ js^l/2 dx2 J$a?/<
k
— L L
Suppose next that We then have
= AI(I/I, 2/2) = k(y\, 2/2)
- in
a, 6
a?/2
+ by
- 6c).
196
THEORY OF ELLIPTIC FUNCTIONS.
ART. 179. Let/(#i, x2) be a binary form of the nth degree. It is seen
that df(xij x<^ and dKXl> X2} are binary forms of the degree n - 1.
dxi dx2
The functional-determinant F of these two functions
axiax2
jy
ax22
= H(f), say,
is called the Hessian covariant * of the form /. Suppose that by the
substitution
Xl = ayi
+
the function /(zi, x2) becomes <f>(yit y2) and form the Hessian covariant for this latter function, viz.,
fyi \dyij dy2 \diji/
d_/d±\
)?/2 \dyal
We have
or
and similarly
When these values are substituted in the above determinant, it follows that a -
fyi
= (ad - be)
9^2
Further, since —
a + i — c, etc., we have
+ d
7,1
* Cf. Salmon, Zoc. ci«., p. 117.
ELLIPTIC INTEGRALS IX GENERAL. 197
It follows that
= (ad — be)2
and consequently
#(<£) = (ad-l
ART. 180. We may consider more closely the meaning of the covariant. Suppose we have a binary form f(x\, x2) of the nth degree. With its coefficients a0> ai, - • • > an and with x\, x2 we form an expression /"» ( )
Cj#o>al> • • • i ton') %1,%2\>
C denoting a functional sign which with respect to x\, x2 is of the vth degree, and in regard to the a's it is of the ^th order. Suppose further that by the substitution
xi \\ayi + by2, %2 \\cyi + dy2,
the function/(xi, x2) becomes </>(yi, y2}.
With the coefficients a0', a\ , . . . , an' of <f>(yi, y2) and with t/i, y2 we form the same function
/"»(// / . )
L {a0 , a\ , . . . , an , y\, y2j.
If then
,an';
wrhere ,« = ?(np — v),
we say that C is a covariant * of the binary form/(xi, x2).
ART. 181. In the theory of covariants it is shown that for every binary form f(xi, x2) there is a finite number of independent covariants, through which all the other covariants may be expressed.^
If f(xi, x2) is a binary form of the fourth degree, say
f(xi, x2) = a0xi4 -f 4ai^i3.r2 + 6a2Xi2x22 + 4a3Jix23 4- a4x24,
there are two fundamental covariants (Salmon, loc. cit., p. 192): The one is the Hessian, where
v = n-2 + n-2 = 2n-4, p = 2; and consequently
* Salmon, loc. cit., p. 135; Burnside and Panton, loc. cit., p. 376. t Salmon, loc. cit., pp. 132, 175, 176; and see also Clebsch, Theorie der binaren alge- braischen Formen, pp. 255 et seq.
198 THEORY OF ELLIPTIC FUNCTIONS.
This covariant is
2(0,10,4 — a2a3)
2ala3-3a22) — a32) x24.
The other fundamental covariant is the Jacobian of the quartic and its Hessian:
r-I
dx
dx2
dH(f
dx2
and therefore so that
For this covariant it is seen that
t " * t
T' = [T]s (ad - be)*.
ART. 182. Between the two co variants T and H(f) there exists the relation *
- T2 = 73/3 - I2f2H(f) + 4#(/)3-
This formula is given by Cay ley in Crelle's Journal (April 9, 1856, Bd. 50, p. 287). The formula, however, as stated by Cay ley, is due to a communication from Hermite.f
We have at once
or writing - = £, it is seen at once that
ART. 183. Consider next the determinant
H(f),f dH(f), df
dx
H(f), dx\ -
f
dx,
dxl
dx,
The functions/ and H(f) being homogeneous of the fourth degree in , x2) it follows that
dxi dx2
* Cf. Salmon, loc. cit., p. 195; Halphen, Fonctions Elliptiques, t. II, p. 362; Clebsch, loc. 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 INTEGRALS IN GENERAL. We therefore have
dJTi *L
dxl
199
dx2
«L
dx2
dx2
ax
*2
dx
= 2T(x2dxi-Xidx2).
On the other hand
A -
It follows that or
H(f), f dH(f), df
= H(f}df-fdH(f}
= -2 T(x2dxl -
— — 4 (x2dxi —
4
~ (- 2)* 4
-
(-2)*
(2/3 - 72C -f
From this it is evident that x2dxl
(-2)*
Since z = — , it follows that
where
We finall have
, x2) \ 5(
= a0z4 + 4 a^3 4- 6 a2z2 + 4a3z + a4.
r ^^ = (-2)* r
J \ R(z] - 4 J \
3 -
This is practically the transformation given by Cayley * in Crdle ~ Journal, Bd. 55, p. 23.
* See also Cayley, Elliptic Functions, p. 317; and Burnside and Pant on, 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 transformation.
200 THEORY OF ELLIPTIC FUNCTIONS.
The mode of procedure, however, as noted above, was suggested by Hermite (cf. Her mite 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 for £ in the above formula, it becomes
r dz = _ . r dt
J VR(z) J 2V4t* - I2t + /
ART. 184. Weierstrass employed a somewhat different notation. He put
^2 = 92, /3 = — 93,
and consequently introduced as his normal form of the elliptic integral of the first kind.
f:
He further wrote
4*3 -g2t-g3 = ±(t- ei) (t - e2) (t - e3)=S(t),
so that (cf. Art. 172) between the e's and the 0's we have the following relations:
ei + e2 + e3 = 0,
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
T)
write t = A + — , where A and B are constants. It is seen at once that z2
dt - Bdz
VS(t)
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) VB{(e3 - ei)z* + B} \ (e3 - e2)22 + B}
ELLIPTIC INTEGRALS IX GENERAL. 201
If further we give to B the value
B = ei - e3,
and put
e2 — e3 _ 7 2
- K ,
e1 - e3 wehave dt 1 dz
VS(t) Vei - e3 v(l-z2)(l-A:2z2) It has thus been shown that through the substitution
the Weierstrassidn normal form is changed into that of Legendre.
Other methods of effecting this transformation will be found in Volume II.
ART. 186. If we write * l
€l - e3 = -,
£
then is 1
ei = - + e3. (1)
£
Further, since k2
e2 - e3 = —,
we have 1.2
e2 = j + e3; (2)
and using the relation
ei + e2 + e3 = 0, we also have -, -, , T 2
1 1 ~r K
This value of e3 written in (1) and (2) gives
e2«_ (2*2-1). (5)
O £
From the equations eie2 + e2e3 + e3e\ = — \g2 and e\e2e3 = J g3 it follows with the use of (3), (4) and (5) that
- e202 = (2 - *2) (2 *2 - 1) - (2 - A- 2) (1 + A-2)- (2 A* 2 -!)(!+ A:2),
.
and from these two relations t
^23 108J1 -A-2 + A-4}3
* Cf. Halphen, Fonctions Elliptiques, t. I, p. 25.
t Cf. Felix Miiller, Schlomilch Zeit., Bd. 18, pp. 282-287.
202 THEORY OF ELLIPTIC FUNCTIONS.
We shall next show that the above expression is an absolute invariant* that is, it remains entirely unchanged by a linear substitution. We have
and we saw in Art. 176 that
/2/3 - /23M -
and 73/2 = /32 (ad - be)12.
It follows that 3 j 3 j /3
_ 2 72 J ' 2,' 93 +3 *3
From this it is seen that k is the root of an algebraic equation of the 12th
T 3
degree whose coefficients depend rationally upon the absolute invariant ^2--
ART. 187. Riemann's Normal Form.-\ If in Legendre's normal form
dz
V(l-z2)(l-/c2z2) we put z2 = t, k2 = X, it becomes
1 r dt
v t (1 - 0 (i - Jit) If in the latter integral we write
we have, neglecting a constant multiplier,
dr
Vt(l - pr + T2)
(Kronecker, BerZm £tte., July, 1886.)
In Volume II the transformation of the general integral into its normal forms will be resumed and the discussion for the most part will be restricted to real variables.
ART. 188. In connection with the realms of rationality we may consider more closely the integrals that have been introduced in this Chapter.
Let R denote any rational function of its arguments, and write the integral
where a — \/az + 6. If we put a = ^(2), where ^r is a rational function, then z — (j)(t) is a rational function. The above integral becomes
* Salmon, loc. dt., p. 111.
t Cf. Klein, Math. Ann., Bd. 14, p. 116, and Theorie der Elliptischen Modulfunc- tionen, Bd. I, p. 25.
ELLIPTIC INTEGRALS IX GENERAL. 203
where the integrand is a rational function of t. For example, put
,9 7
a = Vaz +b = t, then z = • In .this case the realm (z, a] is evi
ct
dently the same as the realm (t), since (z, t) is the same as (t), the pres ence of z within the realm adding nothing to it, as z is a rational function of t.
Consider next the integral
*, o)dz,
where a = \/(z — a\)(z — a2) = (z — a2) \ - - •
* z — 0,2
By writing t2 = z ~ Ol, it is seen that a = (z - a2)t and z = ^ .
z — a2 I - t2
We note that both a and z are rational functions of t and that t is a rational function of o and z. Hence every rational function of a and z is a rational function of t and any rational function of t may be expressed rationally through z and a. In this case we may say that the two realms (z, a) and (t) are equivalent and write
(z, o} ~ (0. In the case of the integral
J R(x, Vax2 + 2 bx + c)dx,
if we put y2 = ax2 -f 2 bx + c, we have the equation of a conic section. This conic section is cut by the line
y - r, = t(x - c),
where t is the tangent of the angle that the line makes with the x-axis, at the point c, y, say, and at another point
26 - 2rt + &2
t* - a
- 2a*t - 2bt
r — a Hence as above
(*, y} - (0.
In the case of the integral
where s is the square root of an expression of the third or fourth degree in z, 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. 166 it is seen that z and s may be rationally expressed through z and s = \/(l — z2)(l — /c2z2) 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 = \/4 t3 — g2t — g%, then
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 homographic transformation,
a . + /ft + yz + dtz = 0 for fc z = flj, z = a2, z = a3, z = a4,
let t = 0, t = 1, t = -, t = oo.
We thus have
a + yal = 0) a + /? + -ya2 + da2 = 0, «A + /?+ 7^3 + ^«3 = 0, The vanishing of the determinant of these equations gives
ai — 03 a2 ~ a4 Show that — - is thereby transformed into Riemann 's normal form.
2. In a similar manner transform — - into Legendre's normal form and from
the resulting determinant derive the 12 values of k given in Art. 171. [Thomae.] 3. Show that the substitutions
z~ai a*~a* .2 QS - <*4 "2 ~ «i
transform
into
±v^(o4-aa)(a1-as) / — = Jai \/(z-
- Z) (1 - k2£) JaiV(z- aj) (z- a2) (z- ag) (2- o4)
[Riemann-Stahl, ^/. ^wnd., p. 16.]
ELLIPTIC INTEGRALS IX GENERAL. 205
4. Show that the substitution
z - a2 ' 03 - transforms
/;
into ' *
VA(z - Ol)(2 - a2)(2 - as)(2 - a4) t/ >/4(U - a^^ -aj(t- aj(t - a4)
[Burkhardt, Ett. Fund.] Derive two other such substitutions.
5. Show that the substitution
t = e + ^ -gi)(g3-gi)
transforms Weierstrass's integral into itself.
6. If a is a root of az3 + 3 br + 3 cz + d = 0, by writing z - a = z2 transform
=1^=^= into Legendre's normal form. 3
7. If f(x) = x4 + 6 mx2 + 1, show that
4 /•-=* <* *
•/ vV +
where
x4+ 6mx2+ 1 /
[Appell et Lacour, Fonc. 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 elliptic integral of the first kind in Legendre's normal form,
'-^L, where Z =(l - z2)(l - /c2z2)= s2
VZ
has the branch-points + !,—!, + -> — -•
A/ /V
4-co
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
K Ki
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
b 8
The corresponding quantities here are, say,
Vz
and B(k)=2f+1-^.
J - i VZ
* Cf. Koenigsberger, Ellipt. 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 z0 = 0, SQ = 4- 1 ; that is, the origin in the upper leaf. We then have
«(z,.)-.| ^%inr.
*/o,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) u(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) — u(X) = B(k).
We form the integral between arbitrary limits, Z2, S2 and z\, s\, where the path of integration is free, that is, taken without regard to the canals a and b.
If the path of integration crosses the canal a (see Fig. 63) we have
/'Zj.Sj f>p ^»X /»Zi,S!
«/Z2,S2 t/Z2,S2 *J p */ A
the integrand for all these integrals being
- z*)(l - fc*z*)
Noting that the second integral on the right-hand side is indefinitely small in T, it is seen that
_^ = u(p) - i7(z2, 32) + u(zlf Si) - M(X) in T
VZ
= M(ZI, Si) — ^^(z2, s2) — A(k). If, however, the integration is taken in the opposite direction, we have
/*Z2,S2 ^^ _ _
< — = = u(zo. So) — u(zi, Si) + A.(k).
J*** \/z
We may form the following rule : // the path of integration for the integral
dz
- z2)(l - £
crosses the canal once in the direction from p to X, 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 oppos-ite 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'
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 - -, k
C~l dz
Jf , 7i
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
or
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)
-i VZ The same integral in the lower leaf and on the upper bank of the canal is
It follows, as above, that
Next forming the integral from + 1 to + - in the upper leaf and upper bank, we have
+1 vZ
and in the lower leaf, upper bank,
MODULI OF PERIODICITY. 209
We therefore have
We then form in the upper leaf, upper bank,
»+oo,+oo J7
k
and on the lower leaf, upper bank,
»+00, -00 fJ7
Adding these two integrals we have
(V) #(&) = 7Z(oc, + oc)+ ?Z(cc, — oc) — 2
ART. 191. If we form the integral
1 dz
V(l -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 + oo . At two corresponding points of the paths the abso-
*v K
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 — k2z2). The differential dz is the same along both the paths and positive, and consequently the elements of integration are equal in pairs and we have
- z2)(l - k2z2) «/i V(l - z2)(l -
In a similar manner we have
dz C+* dz
J
_ V(l - z2)(l - £2z2)
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 I — z taken over this path must be zero.
J V'(l - z2)(l - £2z2)
since the path of integration does not include a branch-point.
210
THEORY OF ELLIPTIC FUNCTIONS.
We therefore have .+ ?
/+ 1
upper bank
upper bank
We note that the two integrals
J.i Vz
+k
upper bank
upper lower bank bank
lower bank
VZ
lower bank
=o.
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
. i
dz
f ^
J+i VZ
and
+i Z
i VZ
are equal with opposite sign, since there is no canal between the two paths over which the integration is taken. f It follows that the sum of the above integrals reduces to
i VZ
k
-i VZ 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
-i VZ J+i VZ It follows at once that
or, owing to (III),
If we take the congruent path of integration in the lower leaf, we again have, since no canals are crossed,
or . - _
We have thus the formula
ART. 192. We compute the integral from 0 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
Jo
ro,i VZ Jo,-iVZ
upper leaf lower leaf
It follows that
-)+ i7(+ 1)- w(0, - 1)=0, or, since w(0, 1) = 0 from (I), we have (VIII) 2u(+ 1)- w(0, - l) = -
Further, it is seen that
r°-+1_rfz_ = /"+1 _^z_ */-i VZ Jo.+iVZ
upper bank upper bank
and consequently, multiplying by 2, we have
- = 2 -i v Z o,i \ Z
upper bank upper bank
212 THEOEY OF ELLIPTIC FUNCTIONS.
From this it follows that
u(+ 1)- Ti(- 1)+ B(k)= 2{u(+ 1) - u(Q, 1) - or, owing to (I) and (III),
We thus obtain (IX) -$B(k)=u(+l).
We have thus derived the following nine formulas :
(I)u(0,l)-0, (V)5(oo,
= B(K),
/TT\ — / -i\ ~~ / J-\ -^J- \ *v ) /TTT\ — / \
(II) u(- 1) - U [-- ) = —^r2' (VI) W(oo,+ oo)-
(VII)
From these formulas we have at once :
ART. 193. Legendre * and Jacobi f did not use the quantities A (k) and B(k) but instead two other quantities K and Kf. These quantities are connected with A(k) and B(k) as follows:
B(k)=2
_! V(1-Z2)(1-/C2Z2)
or, since
J%,.-2 I -^i (Art. 192). -i VZ JQt+iVZ
•£•
dz
* Legendre, Fonctions Elliptiques (1825), t. I, p. 90. t Jacobi, Werke, Bd. I, p. 82 (1829).
MODULI OF PERIODICITY. 213
If further we write
z= ~ 1 = , k'2=l-k2,
Vl - k'2v2
d -k'2vdv 2_ -k'2v2 2 2 _ k'2(l - v2)
^ ~ (1 - A;'2,2)* ~l-k'2v2 ' l-k'2v2
it is seen that i
A (k) = 2 Ck dz = =2i C1 dv = . [ JacobL]
Ji V(l-z2)(l-A'2z2) J0 V(l-v2)(l -k/2v2)
If then we write
Kf = r1 dv
JQ V(l - r2)(l - k"2v'2) we have A(k) = 2iK'.
The quantity /:' is called the complementary modulus. Since B(k) = 4K and A(k) = 2iK', the formulas of the preceding article become
M(+ 1)=-3JK:, u(- 1)=- K,
x) = -iKr, u(oo,-oo)^- 2K -iK',
, 1)=0, w(0,- 1) = - 2K.
Anticipating what follows, if we write
dz
- z2)(l - fc2z2) and if z considered as a function of M is written
z = sn u, we have from the above formulas
sn(-3K)=l, sn( - 3 K - iK') =L sn(- iK') = oc , etc.
n*
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/£-e* - et - e] J \ 5 '
where S(t) = S = 4(t - ej (t - e2) (t - e2).
214 THEORY OF ELLIPTIC FUNCTIONS.
In the Riemann surface T without the canals a and b let
and let u(t, Vs) denote the corresponding integral in T'.
"b
Fig. 65.
We may here write (cf. Art. 139)
u(fy— u(p} = A' on the canal a, and ^(p) _ u(X) = B' on the canal b.
The quantities u(ei), u(e2}, u(e3) may be computed as follows. In the figure we note that, when the integration is taken in the upper leaf,
C^_dt== PJL + re*_dt= = u(p)+u(el)-u(X)=u(el)-A'.
J* Vs J*Vs 'J* Vs
In the lower leaf along the congruent path of integration,
Through subtraction it follows that*
upper leaf
Similarly along the upper bank of the upper leaf of T , _^= p_d^+ Ce2J^ = u(p)-u(e1)+u(e2)-u(^=
Je, VS J+ VS J* VS
while for the congruent path in the lower bank,
* Vs
* Cf Riemann-Stahl, Ellipt. 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.
MODULI OF PEEIODICITY. 215
Hence through subtraction it is seen that in the upper leaf
J« VS
We may therefore- write
r*2 Hf f*£2 rit r*\ /it (II) 2 / -£L - 2 / -2L + 2 / **= — A' + ff.
»/oo VS Je,. VS «/oo VS
We further have in the upper leaf of T' ',
V S
= A' + u(e3)-u(e2); while in the lower leaf
* dt -x x -, \
— = = u(«8) - tt(«j).
Through subtraction we have in the upper leaf
9
It is also evident that
J«> V/S JK \fi&' Jez V/S
or
an) P4-f-
ART. 195. It follows at once from (I), (II) and (III) that in T
*, __:£ — „, say,
V S *
z dt -A' + B'
-, x C** dt B'
u(e-3}= I — =.= — = -a>'.
J* VS 2 From these definitions of w, a>f, to", it is seen that
co" = a) + a/. Again (cf. Art. 185), if we write *
- u =
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 Weierstrass. It is seen in Chapter XV that $u is an even function. It is called the Pe-function.
216 THEORY OF ELLIPTIC FUNCTIONS.
we have
(IV)
e2
ART. 196. In Art. 185 we derived the relation
Vs= x *vz
If we write then also
where t = £3 + — 2
dt F dt
— du = — =, or — u = I —=.,
Vs J» Vs.
du dz_} or JL= r _^L.
Ve VZ Vs Jo,iVZ
It follows that
\V e
and
1
= e3 + —
It is also evident that
K.fjfe 1 p* = -
Jo.lVZ V£ Joo V>S V£
or w = VeK, and similarly a/ = -V dK'.
ART. 197. T/ie conformal representation of the T' -surf ace. In Chapter VII we saw that if
r*>s dz
u = I . j
JZo>SoVR(z)
then z 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 z, s. The collectivity of these values was expressed by
u = u(z, s) + mA 4- 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 = (j>(u), then <j>(u) is a one- valued function of u. We
also saw that s= — = VR(z) is a one-valued function of u. Further,
du
in T' the quantity u is uniquely determined as soon as the upper limit z, s is known. Therefore for every value z, s in the surface T' we may
MODULI OF PERIODICITY. 217
compute the corresponding value of u and lay it off in the plane of the complex variable u. Since the integral u never becomes infinite (Art. 136), it follows that all the values of u which correspond to the collectivity of values z, s in the surface Tf ma}' be laid off within a finite portion of the u-plane.
It cannot happen that to two different values of 2, s on the surface T' there corresponds the same value u. For if this were possible, then re ciprocally to this value of u either there would correspond two different values of z in the T'-surface and z would not be a one-valued function of u, or there would correspond two different values of s, and then s would not be a one-valued function of 17. The points in the w-plane follow one another in a continuous manner and the region which they fill is simply covered. It follows that the portion of surface in the ?7-plane and the simply connected Riemann surface T' are conformal representations of each other, since to ever}' point of the one structure there corresponds one and only one point of the other structure and vice versa.
u — Plane
Fig. 66.
We may next investigate the form of the portion of surface in the 77- plane which is the ijnage of the surface included within the canals a and b. We compute the value of u for the point ft which is the intersection of the left bank of the canal a with the right bank of the canal b. The value of u at this point we also call fi and lay it off in the 77-plane. We compute for every point of the left bank of a the corresponding value of u and lay it off in the T7-plane. We thus have a curve an in the 77-plane which does not cross itself. Let the other end-point be denoted by d in the 77-plane, which point corresponds to the point d on the surface T'. If next starting from d we traverse the bank X of b and lay off the corresponding values in the plane 77, we have a curve b^ which ends at 7% say. Then starting from f in T' we go along the bank p of a and form in the 77-plane the corre sponding curve a p. Finally we return along the bank p of b back to /?, and the corresponding curve bp in the T-plane must lead back to the
218 THEORY OF ELLIPTIC FUNCTIONS.
initial point /?. The canals a and b are thus conformally represented on the tZ-plane.
Since the canals a and b are the boundaries of T' , the curve a^b^apbp must bound the surface which is the conformal representation of T' in the w-plane. The interior of the figure is this conformal representation, for u cannot be infinite for any value of z, s, which may be the case if the surface without the curve aA6Aap6p represented conformally T' '.
Remark. — The curves ax and ap are parallel curves, that is, to every point on a A there corresponds a point on ap, so that lines joining such pairs of points are equal and parallel. For if we take on the canal a in T' two points opposite each other on the left and the right banks respec tively, then we have
u(X} - u(p) = A.
Consequently the complex quantity A represents the length and the direction in the w-plane of the distance between two points lying on opposite banks of the canal a, which conformally in the w-plane lie on the curves a* and ap. Since A is a constant the two curves a^ and ap must be parallel.
Similarly b^ and bp are parallel curves and the distance between them is B.
If the variable crosses a canal a or b in T' ', we have values of u which lie in a period-parallelogram that is congruent to the first parallelogram, and by crossing the canals a and b arbitrarily often in either direction we have more and more parallelograms which completely fill out the w-plane.
ART. 198. The form of the two canals a and b was arbitrary. We shall show that they may be taken so that the corresponding parallel ogram in the w-plane is straight-lined. As a somewhat special case take Legendre's normal form and let the modulus k be real, positive and less than unity.
-i/k
C+i) +K,'
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, + 1, + -, as shown in the figure. k
We had in T'
- = rzs dz
t/o,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 PEEIODICITY.
219
For the bank / of a we have 1 — z2 > 0 for all points except z = 1, or z = — 1, and consequently also
1 - Fz2 > 0.
It follows that a 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 a^ coincides with the real axis.
On the bank X of b we have
1 - z2 < 0 and 1 - k'2z2 > 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 bji in the w-plane is a straight line that stands perpendicular to the axis of the real.
Since ap is parallel to a A and bp to b A, the conformal repre sentation of T' on the w-plane is a rectangle with the sides a*
dr = A(k)=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
1. Show that
2. Show that
3. Prove that
4. The substitution transforms
EXAMPLES
/»<»,- oo fJ7
/ -2== = K
Ji VZ
</ =
0) =
V40- ej (*- e2) (t-
into
r
J ۥ>
ds
V 4 (s - ej (s - e2) (s -
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
dt Cei dt — =+/ —;=
+ VS J^ VS
CHAPTER X THE JACOBI THETA-FUNCTIONS
ARTICLE 199. We saw in Chapter V that the 4>-f unctions of the second degree satisfied the two functional equations
$(u + a)= &(u),
If Q = e a, we saw in Art. 87 that
m = + oo 4 id
mu
We have now to write : "& = B (k) = 4
It follows that Q = e 2 K . If further we write
we have *
m — + oo Tti
-== mu
m=— oo TO = +oo
When K and K' are introduced into the functional equations, they are
= e~*(U ^
In Oi(w) the term which corresponds to m = 0 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, Cours redige en 1882 par M. Andoyer, p. 235 (Quatrieme Edition).
220
THE JACOBI THETA-FUNCTIONS. 221
tion the term which corresponds to + m with the term that corresponds to —m, we have
5) <r2U *
TTlfflM
~K~
m=l
or
HZ = 1
The terms in HI(W) may be combined as follows:
l
l,
It follows immediately, as we have already seen in Art. 148, that
ART. 200. \Ye introduce two new functions,* the first of which is defined by the relation
H(u)=Hi(M -K). We have at once
771=30 /2m+l\2
H(») = 2 X ,~ cos " ™ - (2 m + 1)
771 =0
But, since cos Li - (2 m + !)-=(- 1)TO sin A, it is seen that
m=x
« = 2 ( - l- sin
0
The second function is defined by the relation
e(«)-'ei(«-/io
and consequently we have
m= 1
It is seen at once that
0(- M)=0(M).
The functions 0i(w), HI(W), 0(^), H(M) are known in mathematics as the 0-functions. Excepting H(w) they are all even functions, and it is seen that they are more rapidly convergent than a geometrical progression.
* a. Jacobi, loc. tit., p. 235, and Werke, II, p. 293; see also Hermite, loc. tit., p. 235.
222 THEOEY OF ELLIPTIC FUNCTIONS.
ART. 201. From the equation
(U + K) = H(u + M + K) = - H(w).
0i
it follows that
and therefore
In a similar manner
We also have
9(*
and 0(tt
We thus have the four formulae
(I)
From these formulae we derive at once
(ID
From (I) and (II) we again have
2 K =
and finally
(III)
(IV)
0(ti
ART. 202. We shall next increase the argument of the 0-functions by iK'. We have
m= -oo
OT= +00
m= +00
*- X
m = -oo m=+oo
(TT)'-I.
1 Jrt'tt
m=-oo
THE JACOBI THETA-FUNCTIONa
If further we write *
JIT IU
we have
0i (z* + iK') = ^(M)Hi(u).
We may also note that
e
2m+l\ 2
- 1
le 2K e 2K
m=+x /2m+l\2 2m+l (2m +
— » ( — I + — JT-;
1 (m+l)rt«
'e Ke*K 2K,
= 2)
where m + I = mf. It is seen at once that
Since
we have 00
— - — (u-K) 4K 2K( W
In a similar manner it is seen that We may therefore write
(V)
It follows from (I) and (V) that (VI)
HI(M + K + iK') = -il( 0i(u + K + i,K') - U(u
* Hermite, loc. cit., p. 236.
223
224 THEORY OF ELLIPTIC FUNCTIONS.
It is clear that
HI(M + 2iK') = H,[(u + iK') + iK']
If we put
it follows that
We have the following formulae :
(VII) ®i(^ + 2iKf)
It is seen that H and 0 satisfy the functional equations &(u + 4K) = $(u), 3>(u + 2 t/g;') = -^(u) while HI and @j satisfy
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 $-f unctions 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(w)- was an odd function and therefore vanishes for u = 0. We also had
and consequently
H(2K) = -H(0) =0.
The points 0, 2 K are therefore the two incongruent zeros of this func tion; i.e., the function H(V) vanishes on all points of the form
2K + m24K +l22iK', where mi,m2,li,l2 are integers.
Hence all the points at which H(w) vanishes are had for the values of the argument
u = m2K + n2 iK' ,
where m and n are integers.
THE JACOBI THETA-FUNCTIONS. 226
Further, since
Q(u + iK')=M(u)H(u), when u = 0, we see that
and since
e( we also have
iK' + 2K)=Q. The zeros of 0(^) are consequently
m2K + (2n + \)i By definition we have
fi(M)-Hi(w~i
so that the zeros of HI(W) are
(2m + 1)K + Finallv, since
0(M) = 0i(u-
the zeros of 0i(w) are
(2m + 1)# +(2rc -f ART. 204. Write
where q = e K ;
and
m=-°o
where qQ = e
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
0 1 ( u • K', iK) and e ~ TKK'Ql (iu; K, iK') satisfy the same functional equations
The two functions have also the same zeros u =(2m + l)K' + (2n It follows that the ratio of the two functions is a constant.
226 THEORY OF ELLIPTIC FUNCTIONS.
We therefore have (cf. Jacobi's Werke, Bd. I, p. 214)
e 4KK'®i(iu; K, iK') = C8i(t*; Kl ', iK),
/Hi(ra; K, iK') = C®(u; K', iK),
e *KK"H.(iu; K, iK') = iCH(u; K', iK)}
e *KK'®(iu;K,iK') =
EXPRESSION OF THE THETA-FUNCTIONS IN THE FORM OF INFINITE PRODUCTS.
ART. 205. With Hermite * consider the two functions $(tO = <f>(u + iK') </>(u + 3i
and
It is seen at once that if (f>(u) has the period 2 K, then
It is also evident that
and consequently
If next we put
niu
(j>(u) = 1 + e K , we have
and also
<>- u
. K.
* See ./Vote swr la theorie des f auctions elliptiques placed at the end of Serret's Calcid Differentiel et Integral, pp. 753 et seq.; CEuvres, t. 2, pp. 123 et seq.
THE JACOBI THETA-FUNCTIONS. 227
It is thus seen that
(n = 1, 3, 5, . . . ) and
C7TtU\ 1 + e*]II(l + 2?n
(n = 2, 4, 6, • • • .)
These products are convergent (cf. Art. 17) if | q \ < 1 (see Art. 81).
ART. 206. The two functions &(u), <&i(u) both have the period 2 K and they satisfy the functional equations
Let us introduce a function W(u) denned by the equation
We have at once
si / iK'
$(M), ^(H + 2 K) = - V(u). It is evident from formulas (II) and (V) that we may write (cf. Art. 83)
where A is a constant. Noting also that
it is seen that
= 2 A t/sinul-22cos2u + *}l-2* cos2 u
where A is a constant.
228 THEORY 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) f(t) = (1 + qt) (1 + 230 . . . (1 + q2»-ty
This expression developed according to positive and negative powers of t is of the form
(2)
The following identity, which may be at once verified, f(q2t) (q2n + qt) = f(t) (I + q2n+lt), gives between two consecutive coefficients Ai and AI-I the relation
We thus have
4. -4.
1 -
If these equations are multiplied together, we find that A A ^ (1 -g^H1 n-
But it follows directly from (1) and (2) that
An = qn*. We therefore have
A° = (1 - q2} (1 - g4) ... (1 - 92 »)
When n becomes indefinitely large, it is seen that
\
° ~~ (1 - q2) (1 - q4) (1 - q6) . . .'
* Biehler, Crelle, Bd. 88, pp. 185-204; see also Hermite, loc. tit., 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 A.
it follows from the equation (2) that
(1 + qt)(l + <?30(1 + 550 . . . l + 1 4-
(1 - g2) (1 - q*) (1 - g6) ... Writing t = e2iu, this formula becomes
(1 + 2 q cos 2 u + q2) (1 + 2 g3 cos 2 u + q6) - • •
= 1 -f 2 q cos 2 H . -i- 2 g4 cos 4 K - • • •
(1 - 92) (1 - 54) (1 - Q6) ...
From this we conclude that the constant A of the previous Article is
A = (i - 92) (i - g4) (i - ?6) • • • ;
and at the same time it is shown that 0! as denned in the last Article as an infinite product is
Q f2J±!L\ = i + 2 q cos 2 u + 2 g4 cos 4 u + ' • • ,
or ®i(«)=
which is the original definition of this ©-function.
Example. — By means of the infinite products prove the formulas (I), (II), (III), (IV) and (V) of this Chapter, and therefrom derive the
. . i ., . f TT /2Ku\ u/2Ku\ ,r./2Ku\ expressions in infinite series of HI I — I H(— — — 1 and 0f - J-
THE SMALL THETA-FUNCTIOXS.
ART. 208. Jacobi (Werke, Bd. I, pp. 499 et seq.) introduced a notation similar to the following (see Art. 210):
m= -»-Qo
0(2 Ku) = &Q(U) = 5 (~ l)mgmV2 """'",
m= -
TO= — ao
m= 4-ao /27n.-t-lv-
-
— 30
-1-30
230 THEORY OF ELLIPTIC FUNCTIONS.
It follows at once [cf. formulas (I) and (V)] that
(T)
and if T = ,
(V)
J r) = i T) =
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 = <x> wi = oo
e<> = n a - 22m)> QI = n (! +
m = l
~m = <x>
m = l
m = oo
It follows at once that
- 2 g2™-i cos 2 ;m + q4m~
l
1 m = oo
4sin7rw JJ (1 - 2q2mcos2xu
m = l
1 TO = 00
4 cos TTW JJ (1 + 2^2w cos 2 TTU
If we write 2 = eiM7r, we have
COS 2 7m +
_ q2m+lz-2 I _
sin „ /?m.±I r _ M\ sin „ /i^L+j t + tt\
= \ ^ / »-iun \ 2 /
sn
. 2 m + 1
sm TTT
gittTT
THE JACOBI THETA-FUNCTIONS.
•
We therefore have
n/i
m=1l
m_
772 = 00
sin2 (m-r sin2 7:11 \
cos^
(mm)/
sm- TT
-w-
ART. 210. Jacobi's fundamental theorem. If we write r.u = x on the right-hand side of the equations above, the theta-functions as given by Jacobi*are m =+x
m= -x
= 1 - 2 g cos 2 x+ 2 g4 cos 4 x — 2 $9 cos 6 x +
771 = +00
fe q)= i2(-i)
sin z — 2 ^t sin 3 x + 2 q" sin 5 x — • • • ,
= 2 g* cos z + 2 ^ cos 3 x + 2 q3* cos 5 re + 2 3* cos 7 x +
m= + oo
#3(*, 9) = 5) ^^e2""*
m= — oo
= 1 + 2 q cos 2 x + 2 g4 cos 4 x + 2 g9 cos 6 x + • • • . We have at once
+ J log 5 • 0 = -iq-*exi$i(x) + l log g • 0 = -iq-*exi$0(x) + I log q • 0 - g-V%Or) + i log q • i) = q-
#0(x + log 9 • i) = - g-V^'
+ log 9 • i) =q- + log 5 • i) = g-^
+ ^ - + J log g • 0 =
&2(x + i TT + i log 9 • i) = iq-*exi$o(x) &3(x + \- + Hogg • 0 = - ^-V^i(
* Jacobi, Werke, I, pp. 497-538.
232
THEORY OF ELLIPTIC FUNCTIONS.
We next observe that if the quantities a, 6, c, d; a', b', c', d' are con nected by the equations
a' = %(a + b + c + d), b' = % (a + b - c - d), c' = i(o - b + c - d}, d' = 0 - 6 - c + d
(D
it follows that
(2)
c = i(a'- b'+ c'- d'},
and also that
(3) a2 + b2 + c2 + d2 = a/2 + b'2 + c'2 + d'2.
We shall next show that if a', ~bf, c'y d' are either all even integers or all odd integers, then also a, b, c, d are all 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
where p is an integer or zero.
For four even integers we may write
a = b = c = d =
4/9 4/9
4/?
4/9
4r
4r
4^ 4(5 4^ 4^
where the numbers in any column may be permuted among one another. If for brevity we put
a — ft + Y — d = f it follows from equations (1) that a'- 6'=
2 a'
2a>+ I 2a'+2 2a;+3
2p'
2r'-l
2d'
2^+1 2 d'
2d'-l 2 d'
* See Enneper, Elliptische Funktionen, p. 136.
THE JACOBI THETA-FUNCTIONS. 233
For four odd integers we may write
a = b = c = d =
4a + 1 4/?+ 1 4r + 1 4d + 1
4a + 1 4/?+ 1 4r + 1 40+3
4a + 1 4/? + 1 4r + 3 4£ + 3
4a + 1 4/9 + 3 4r + 3 4 d + 3
4a + 3 4,3 + 3 4r + 3 40 + 3
where the corresponding values of a', b', c', d' are, owing to equations (1), a' = 6'= c' = d' =
2a'+2 2a'+3
2a'+4 2/3'- 2 2 r'
2a'+ 5 2a'+6 If for example we write
a = I, b = 3, c = 5, d = 7,
we have a'= 8, b' = - 4, c' = - 2, dr = 0;
and reciprocally if a = 8, b = — 4, c = - 2, d = 0, we have a'= 1, £/= 3, cr= 5, d'= 7.
It follows that if for a, 6, c, d we write all possible combinations includ ing all systems of four even integers and all systems of four odd inte gers, the corresponding integers a', b' , c', d' will take the same systems of values in a different order and in such a way that none of the systems will be omitted or doubled.
Since x2 l
&3(X) = ^q^e2m,i=^em^ogq + 2mxi = ^^^^
and ^2W
it follows that a (u2+x2+yt + z2) L
1 \T
— (w2 + x2 + v2 + z2
and '
where
L = (2 v * Iog9 + u^')2 + (2 v'
+ (2 v" J log ? + i/i)2+ (2 i/" J
and M
ev"
the summation in the first equation to be taken over all positive and negative even integers 2 v, 2 !/, 2 v", 2 i/" and in the second equation over all positive and negative odd integers 2 v + 1, 2 i/+ 1, 2 i/'+ 1, 2 r/"+ 1.
234 THEORY OF ELLIPTIC FUNCTIONS.
Adding the two expressions we have
— — 2 + xz + vz + 2
(4) ^3 where
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, 6, c, d. . We note that N may be written in the form
(5) jv =\a + b + c + d lQg? + w + x + y + z H2
L 2 2 2 J
fa + b — c — d log q , w 4- x — y — z .I2
|_ 2 2 2 " J
fa — b + c — d log q . w — x + y — z .~j2
L 2 2 2 J
. [a - 6 - c + d log g , w — x — y + z -I2
[2 2 2 J '
We define w', xf, yf, z' through the equations
' = %(w + x + y + z), y' = $(w - x + y - z),
/ s*\
J x' = %(w + x — y — z), z' =%(w — x — y + z). It follows at once that
w'2 + x'2 + y'2 + z'2- =w2 + x2 + y2 + z2.
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
Jacobi (loc. cit.) made this formula the foundation of the theory of elliptic functions.
ART. 211. If for if we write w + n, we have
while at the same time w' , -x', y', z' are increased by J n so that $3(w' + becomes #0(^1) and$2(X+ i) becomes — &i(w'). The formula above becomes
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-FUNCTIOXS. . 235
produce corresponding changes of £ n and £ log q • i in w' xf, y', z' and secondly changes in w, x, y, z which cause changes of J n and \ log q • i in w', xf, y', z' .
The following eleven formulas belong to the first category, where for brevity we write
()^.p} for
and (Ipvp)' for
(A).
(1) (3333) -f (2222) = (3333)' + (2222)'
(2) (3333) - (2222) = (0000)' + (1111)'
(3) (0000) + (1111) = (3333)'- (2222)'
(4) (0000) -(11 11) = (0000)' -(11 11)'
(5) (0033) + (1122) = (0033)' + (1122)'
(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 wi 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 zf and x' becomes y'.
If we put w = x + y + 2,
it follows that w' = x + y + z, x' = x, y'= y, z' = z;
while if we write w = — (x + y + z) ,
we have w'=0, x'=-(y + z), \f=-(x + z), z'=-(x + y).
Equations (A) may then be combined into double equations. If for brevity we denote $0(0)#>iO/ + z)$ft(x + z)&v(x + y) by | Qtfjiv | and &x(x + y + z)^ft(x)^v(y)^p(z) by{^y(0}, the five most interesting of these double formulas are given in the following table.
236
THEORY OF ELLIPTIC FUNCTIONS.
(B).
| 0000 - {3333} - {2222} = {0000} + {llll } | 0033 | = {0033} - {1122} - {3300} + {2211 } | 0022 | - {0022} - { 1133}- {2200} + {3311 } | 0011 | = {3322} - {2233} = {0011 } + { 1100 } | 0123 = {3210} + {2301 } - { 1032} - {0123}
We may derive a more special system of formulas if in the
in table (A) we put
w = x, y = z,
w'=x + y, x'=x-yj ?/=0, z'=0;
or if we put
w = — x, y = - z,
ti>'=0, z'=0, y'=-(x-y), z' = - (x + Similar formulas, making in all thirty-six, are had by writing
w= y, x= z; w' w=-y, x=-z-, w' w= Zj x= y; w'
Using the notations*
formulas
y).
these thirty-six formulas are included in the following table.
(C).
(1) [3333] - (3333) + (1111) = (0000) + (2222)
(2) [3300] - (0033) + (2211) - (3300) + (1122)
(3) [3322] - (2233) - (0011) - (3322) - (1100)
(4) [3311] = (1133) - (3311) = (0022) - (2200)
(5) (6) (7) (8)
[0033] - (0033) - (1122) - (3300) - (2211) [0000] = (3333) - (2222) = (0000) - (1111) [0022] = (0022) - (1133) = (2200) - (3311) [0011] - (3322) - (2233) = (1100) - (0011) * Koenigsberger, Elliptische Functionen, p. 379.
THE JACOBI THETA-FUNCTIONS. 237
(9) [2233] - (3322) + (0011) = (2233) + (1100)
(10) [2200] = (0022) + (3311) = (1133) + (2200)
(11) [2222] = (2222) - (1111) - (3333) - (0000)
(12) [2211] = (1122) - (2211) = (0033) - (3300)
(13) [0202] = (0202) + (1313); [0220] = (0202) - (1313)
(14) [3232] = (3232) + (0101); [3223] = (3232) - (0101)
(15) [0303] = (0303) + (1212); [0330] = (0303) - (1212)
(16) [0213] = (1302) + (0213); [0231] = (1302) - (0213)
(17) [3210] = (0132) + (3201); [3201] - (0132) - (3201)
(18) [0312] = (1203) + (0312); [0321] = (1203) - (0312)
If in the above formulas we put x = y, we have from (1), (2) and (11) the following:
#3^3(2 x) = tV(*) + #i4 (*) - tVM + <VM
If we write y = 0 in the formulas (C), (1), (2) and (11), we have the formulas of the following table.
(D).
(i) tVWW - tfoWM + t^WC*)
d') #32<V« = tV#32(*) + TWO*)
(2) #3WW - #22#32(*) - tVVW
(3) ^2^12(^) - tWC*) ~ ^0^22(X)
If in equation (1) we put x = 0, we have
<v - <v + ^,
or
[1 + 2q + 2g4+ 2qg+ • • • ]4= [1 - 2 q + 2 ?4- 2 qg+ - - - ]4 + 16g[l +g1-2+g2'3+ q3A + - • • ]4.
ART. 212. We have denned and developed the theta-f unctions by means of infinite power series. These functions being integral transcend ents are susceptible of the treatment indicated hi 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 THEORY OF ELLIPTIC FUNCTIONS.
EXAMPLES 1. Show that
4KK'
[Jacobi, Werke I, p. 226.]
2. Derive the corresponding formulas for @t and H^
3. If
K' K
~*7T ~*~K'
q = e K,q0 = e K,
so that q, q0 are interchanged when K, K' change places, and if
@(w, q) = 1 - 2 5 cos 2 w + 2 g4 cos 4 w - 2 #9 cos 6 w + • • • , H(w, q) = 2 ^g"sin u - 2 ^9 sin 3 w + 2 "N/g^sin 5 w - • • • ,
prove that
[Jacobi, Werke, I, p. 264.] 4. Using the Jacobi notation show that
#0(u + mi log g) =(-l)w<?-"l2e2mMi^0(^),
i?2(M + mi log g) = q- $s(u + milogq) - q-
5. Show that, if n and m are integers,
2m ^ 1()g = Q^ ^(ns + mi log 9) =
CHAPTER XI THE FUNCTIONS snu, cnu, dnu
ARTICLE 213. It was shown in Art. 152 that z may be expressed as the quotient of two ^-functions in the form
z = where u = I* '
*J ZQ,S,
If we put
u=l
V(l-z2)(l-/;2z2)
and study a quotient of 4>-f unctions, it is seen that ^ must = 0, for
<&i(u) z = 0 in both the upper and the lower leaves of the Riemann surface;
and further for z = ao, we must have ^u' =oc in both leaves. It follows that ®i(u)
^ = 0 for z = 0, s = + 1 and for z = 0, s = — 1;
$fu)
and =QC for z = x , s = -f- oc and for z = oc , s = — oo.
In Art. 193 we saw that
77(0, + 1) = 0, 77(0, - 1)=-2K; and consequently
H[w(0,+ 1)] = 0, H[w(0,- 1)] = H(- 2A')=0.
Hence it is shown that H(w) becomes zero for 2 = 0. s = + 1 and for 2=0, s = — 1. We may therefore take H(w) as the numerator in the quotient of ^-functions.
On the other hand we have
and since
0(- iK') = 0, 0(- 2 K - iK') = 0,
we may use 0 (u) 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 OF ELLIPTIC FUNCTIONS.
the quotient 3-^ has the desired zeros and infinities, and has besides
9(tt)
no other such points. It follows that
where C is a constant.
To determine the constant C, write z = 1 and we have
=
But since (Art. 193) u(l) = -3 K, we have
rH(-3g)
°0(-3K)* In Art. 201 we saw that
Hi(N + 3£)=H(u), or H1(0)=H(-3K).
In a similar manner it may be shown that
0i(0) = e(-3K). We thus have
1 rHl(0) or C--®I<9). (i)
C' ~
It therefore follows that (M)
y <7r~2' ^•^
m = — GO
This transcendental expression, however, may be expressed algebraically
in terms of k.
i "HY?/^
If we write z = - in the formula z = C >
we have
1 _n I W
_n
*"
H[3
0[3 K + ^'] S[K + iK'] y(u)l,-oi0
It follows that
C_lHi(0). (n)
* 0i(0)
THE FUNCTIONS sn u, en u, dn u. 241
But from (i)
so that C2=i or C = -L
* Vk
where the sign is to be taken positive since it is definitely determined from the expression (M) above. We thus have
If in the integral of the first kind
dz
. - r
i/0,l
- z2)(l - k2z2)
we write z = sin
it becomes
Jacobi * wrote
(f> = am u (amplitude of u),
so that . ,
z = sin <j> = sm am u.
If the modulus k 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 modular sine and written
by Gudermannf
z = sn u.
ART. 214. Consider next the quotient
We have (cf. Art, 140)
HI(M) = HI(M(Z.S)+ m4K + n2iK'] Q(u) " 0[w(z, s)+m4X -h n2i'K/]'
Since HI(M) and 0(i*) have the period 4 K, it follows that
0(u) If we take n = 1, we have
* Jacobi, Werke, Bd. I, p. 81. Here Jacobi retained the word amplitude of Legendre [Fonct. Ellip., t. I, p. 14]
f Gudermann, 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
e<td,
a formula which is true for any value of n.
ART. 215. All the T he ta-f unctions have the property of becoming zero of the first order upon only two incongruent points. It follows that the quotient
0(M)
becomes zero of the second order upon two incongruent points, and upon two incongruent points it becomes infinite of the second order.
Since HI fa) = 0 for u = (2 m + l)K + n 2 iK',
it is seen that
Hifa) = 0 for u = - K and u = - 3 K;
and from above
0fa) = 0 for u = - iK' and u = - iK' - 2 K.
In Art. 193 it was found that
when u = — K, then z = — 1,
when u = — 3 K, then z = + 1,
when u = — iK', then z =00, s = oo,
when u = — iK' — 2 K, then z = oo , s = — oo .
It follows from Art. 150 that *v ' is a rational function of z. It be-
L©fa) 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 = - oo.
We note that the function z2 — 1 has the same properties. We may therefore write
©fa)
The function Vl - z2 is consequently like z a one-valued doubly periodic function of u. It has the period 4 K but -not the period 2 t/sT'; for when u is changed into u + 2 ^K', the above quotient changes sign. Hence the other period is 4 iK'.
We have
v 1 — z2 = vl — sn2?/ = cos am u = cnu,
or cnu-d^M).
©fa,)
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 = V 1 — sin2 u.
In order to determine the constant Ci, we may write z = 0, S = 0, so that
_ c Hi(0) c = 0(0) = l-2q + 2q4-2qg + - • • '
1 0(0) Hi(0) 2 t/q + 2 A/<p 4- • • •
Again, if we write z = - , then, since u(-r\ = — 3 K - iK', it follows that
/7~ ~T = c V A'2
0(- 3 K - IK') 0(3 K + iK')
' 0(0)
But, since Ci = " ^ \ , we see that
Hi(0)
#^2 = y/ -v* - " >, or Ci = -^-^
0(0)
the sign being definitely determined through C\ =
In the preceding Article we saw that Vk was definitely determined and consequently here Vk' is also definite. We may therefore write
Vk' Hiftt)
ART. 216. We saw in Art. 152 that -£• is a one-valued function of u and
from above it is seen that Vl — z2 is also one- valued. It therefore follows from the expression
that \/l - A-2z2 must be a one-valued function of u. This function is called the cte/ta amplitude u and written A am u, dn u or A<£.
Since — = — , it follows, since z = sin <£, that du = -& •
d* \/(l-z2)(l-A-2z2) A0
To investigate this function dn u, let us study the quotient
6(1*) J L0(")J
244 THEORY OF ELLIPTIC FUNCTIONS.
The zeros of the numerator are expressed through
u =(2m + l)K +(2n + l)iK'. We may therefore take as the two incongruent zeros the values
u = - 3 K - iK' and u = - K - iK'. In Art. 193 we saw that
u(z, s) = - 3 K - iK' for z = i,
k
and u(z, s) = - K - iK' for z = - i.
k
Hence the above quotient becomes zero for z = ± -, and it becomes
k infinite for z = oo, s = + QO and for z = oo, s = — <x>.
The function Vl — k2z2 has the same zeros and the same infinities. We may therefore write
Vl - k2z2 =
We shall choose the sign so that when z = 0 the root has the value + 1. Hence for z = 0 we have
r ®i(0) r @(0)
C2e(oy C2 = 0^o)'
If further we write z = 1, we have
) r @(0)
It follows that k' = C22 or C2 =Vkf, and consequently
~2q + 2 q±- 2 q» + •
@i(0) l+2g +
(Jacobi, Bd. I, p. 236.)
Finally we have
ART. 217. We may write * the three elliptic functions of u
(VIII)
snu = — -
Vk
* Cf. Jacobi, Werke, Bd. I, pp. 225, 256 and 512; Hermite, loc. cit., p. 794.
THE FUNCTIONS sn u, en u, dn u.
245
The first of these functions is odd, the other two are even. It follows at once that
sn 0 = 0,
(VIIIO en 0=1, dnO= 1.
The zeros of sn u are ...... . 2 mK + 2 niK',
the zeros of en u are .. . ; . ' . (2 m + 1)K + 2 niK',
the zeros of dn u are . .. . . . (2 m -f 1)K + (2 n + l)t'K';
tfi<? infinities of all three functions are . 2 mK + (2 n + l)iKf,
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
_ Vk QI(M) Vk
ecu)
dn u
or
/ , rr\ C
sn(u + K)=
dn u
We may consequently write
(IX)
f T^N en u
sn(u + K) = 1
dn u
v\ if sn u cn(u + K) = — K ;
dn u
dn(u + K} = -^- - dn u
(IX')
snK = l, en K = 0, dn K = kf.
When the argument u is increased by 2 K, it follows that
0(u)
We thus have
(X)
sn(w + 2 K) cn(w + 2 K)
dn(u + 2 K}
sn w, en u,
dnu.
246 THEOEY OF ELLIPTIC FUNCTIONS.
Noting that
/ , -j£f\ _ VA/ HI(W + iK') _ V A/ X(u Vk (d(u + iK) Vk i\(u
k snu
we may write
(XI)
and in a similar manner (XII) It is also seen that
We thus have
(XIII)
sn(u + tTT
1
k sn u
k snu
sn u
sn(u + 2 iK) = sn u, cn(u + 2 £./£') = — en u, dn(u + 2iK') = - dnu.
1
= dn u
k sn(u + K) k snu
, .™ dn u
cn(u
Ai. | I/A». y
A;cn w
fr'
K + ^KO = - ^
k cnu
en u
All three functions have the periods 4 K and 4 iK', so that
sn(u + 4 K)= snu,
(XIV)
and
cn(u + 4 K)= cnu, dn(u + 4 K)= dnu',
sn(w + 4 ijfiL7) = sn u,
(XV) cw(u + 4i7n=cntt,
c?n (u + 4 iKf) = dn u.
The periods of sn u are . . . . 4 K and 2 iK',
the periods of en u are . . . . 4 K and 2 K + 2 iK',
the periods of dn u are .... 2 K and 4 iK'.
ART. 219. The fundamental formulas connecting the elliptic functions follow at once from their definitions.
From the relations ^j
du = 23j > (f> = am u,
we have
du
THE FUNCTIONS sn u, en u, dn u.
247
It follows that
d ^^ = sn'u = en u dn u, du
cn'u = — snudnu, dn'u = — k2sn u en u.
The following two relations are also evident :
sn2u + cn2u = 1, dn2u + k2sn2u = 1.
Further, from the relations
— = V(l - z2)(l - du
we have and similarly
and z = snu,
- k2sn2u),
cn'2u = (I - cn2u) (1 - k2 + k2cn2u), dn'2u = (1 - dn?u) (dn2u - 1 + k2). ART. 220. Jacobi's imaginary transformation.* — If we put
sin (j> = i tan ty, it follows at once that
sin <j) = i tan t/r,
1
and also that
If next we write
cosy1
sin T/T = — i tan <£,
COS0
. d<f> = -i — ^i
COS0
1 - k2 sin2 v/1 - k'2 sin2
r» ^
Jo Vl - k'2s
= M*, say, t/u -v i — K"srn*<p *so vi— /j'^sin->/r
then >/r = am(w, A;') and $ = am(iu, k).
From the relations above we have
(XVI)
cn(iu} k) =
'ti l-'\ ,u> * ;
u,
sn(w, A;') = — i cn(u, k') =
cn(iu, k) 1
dn(u, k) =
cn(iu, k)
dnd'u. A:) cn(iu, k)
* Jacobi, Werke, Bd. I, p. 85.
248
THEORY OF ELLIPTIC FUNCTIONS.
ART. 221. As a definition Jacobi wrote
coam u = &m(K — u).
We have at once
(XVII)
sin coam u =
en u dnu
cos coam u =
A coam u =
dn u k' dnu
It also follows that
(XVIII)
sin coam (iu, k) =
I
cos
dn(u, k')
ikf
<MJ k) = — cos coam(u, k'), k
iu, k) = k' sin coam(w, k').
ART. 222. From the two preceding Articles it is seen that sn (u + iK') =
(XIX)
en (u + iK'} = -
I
k sn u
idn u
-ik'
k snu k cos coam u dn(u + iK') = — i cot am u ;
and also that
(XX)
sin coam(w, + iK')
\
k sin coam u
ik' k en u
A coam (u + iK') = ik' tan am u.
cos coam (u + iK') =
ART. 223. Linear transformations we put t = kz, we have
/*
dt
— If with Jacobi (loc. cit., p. 125)
dz
THE FUNCTIONS sn u, cnu, dnu. If further we write
- - f rfz .,
Jo \/(l -z2)(l -A--'z2)
249
we
have z = sn(w, A'). * = **(*"> 7h and consequently* if&tt, -J= A* sn(w, A;),
(XXI)
sni
en
We also have
dnlku, -]= cn(w, k).
sin coam (kit, -} = ,
\ kj sin coam (u, k)
cos coam fjfcu, -J = z'A;' tan am(w, A;),
A coam (ku, ±] = ^
\ k] k cos am (u, k)
(XXII)
Next put in in the place of u and observing that the complementary
modulus of - is y- , it is seen that
K K
(XXIII)
and
-
= cos coam (u, k'),
MX-
ik'\ ku,— \= sin coam(w, A;'),
(XXIV)
(ikf\ ku,-—\= cos am (u, kr),
Iku, -r"Jsaa sin am(?<, A*'),
cos coam
k
I ik'\ tan coamf &u} — - )
V A; /
cot am (it, A/).
* See also Hermite, CEuvres, t. II, p. 267.
250 THEOEY OF ELLIPTIC FUNCTIONS.
ART. 224. It follows from Art. 204 that
and
Oi(0; K,iK'
; K' , iK)
Hi(0; K,iK') 0(0; K',iK)
We have at once (cf. also Art. 220)
.sn(u\K'iK}
/)/>/• K 1 K M 1 X -
(XVI)
1
cn(u',K',iK)
cn(u] K', iK) ART. 225. Quadratic transformations. — If we write
we have
dz
Mdt
Z2)(l - k
1
where
Writing u = I —-=
Jo v(l
it follows that (1 + k)u = T
Jo and consequently
(XXV)
In a similar manner write
and M =
1
-f ksn2(u,k)
and we have
where
t - dz
(1 + k')z Vl - z2
Vl - k2z2
Mdt
- z2)( 1 - A;2z2)
1 _ j,r
1 + k'
and M =
- t2)( 1 -
k'
THE FUNCTIONS sn u, cnu, dnu.
It follows at once that
251
(XXVI)
en
(1 4- k')sn(u, k)cn(u. k} dn(u, k)
I -(1 +kf)sn2(u,k)
dn(u, k) 1 - (1 - k')sn2(u, k)
dn(u,k)
In formulas (XXVI) change A; to l/k and w to uk and observe formulas (XXI). It is seen that
(XXVII)
sn(k
+ tfc'J cn(u, k)
- ik'1 = I -(k + ik')ksn2(u,k) + ikr \ cn(u, k)
- &1 _ 1 ~ (k - ikf}k sn2(u, k)
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, 1, p. 32, London, 1780) and may be derived as follows:
Write
* sin (2<f> — <j>i)= ki sin 0i, (1)
where
, 1-k'
*1 = T— 77'
i < k,
Since
it is evident that
.sin (20 — 0i)< sin
(20-0!)<0!, 0< 01-
Solving (1) for 0, we have
sin220 =(1 +£1)2sin201
[~l -
sin* 0
]-
or, since —
it is seen that
We further have
A0
(1 4- A-'
! \/!-Jfc2sin20
252 THEOKY 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:
snu = u-(l + &2)^ o !
dnu =
U2n+l y2n
where the coefficient of any term, say - - — or - - — , is an integral
(2n + 1)! (2n)l
function of k2 with integral coefficients.
Following Hermite* we wish to determine these coefficients. From the formulas derived above
sn ( ku. -}= k sn(u. k), \ k/
en (ku, - } = dn(u, k), \ 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 •)• has shown that the coefficients of en u are'
1 + 4 k2,
1 + 44 k2 + 16 A;4,
1 + 408 k2 + 912 fc* + 64 k6,
1 + 3688 k2 + 30768 A;4 + 15808 k6 + 256 k8,
We note that if we put k = cos 0 and introduce the multiple arcs instead of the powers of the cosines, the above coefficients when multiplied by k may be written
k + 4 A;3 = 4 cos 6 + cos 3 0,
k 4. 44 yfc3 + 16 k5 = 44 cos 0 + 16 cos 3 0 + cos 5 6, k + 408 &3 + 912 k5 + 64 k7 = 912 cos 0 + 408 cos 3 0 + 64 cos 5 6 + cos 7 6,
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 rendus, t, LVII, 1863 (II), p. 613; or (Euvres, t. II, p. 264. t Gudermann, Crette, Bd. XIX, p. 80.
THE FUNCTIONS sn u, en u, dn u. 263
* u2n + 2
In general, if we denote the coefficient of — - —
\2i TL ~r~ — ) I
by
t = n
A0 + A, k2 + A2k* + - • - + Ank2n - ]£ Aik2* = cn<2"+2> (0, fc),
t = 0
we will have the relation
which may be demonstrated as follows: From formulas (XXVI) we have
|~,7 , ..,. k - ik*~\ I - (k + jkf)ksn2(u,k) cn\ (k + ik')u, - - — = - * - —*— — ^-^, L k + ik'] cn(u,k)
and changing i to — i it follows that
cn\(k _ *>, *±*H = i-(*-v*o*«»'(u.fe).
k — IK J cn(u, A;)
From these two formulas it follows at once that
(k + ik'}cn\(k - ik')u, L±jK\ + (k - ik')cn\ (k + ik')u, k ~ ik/ |_ fc — ik'_\ |_ k + ik'
= 2kcn(u,k). In this formula write k = cos 0, fc' = sin 0, and we have
e*cn(e-"w, e2i0) 4- e-i9cn(0*u, e~'2le] = 2 cos 0 cn(w, fc). Noting that
it is seen by equating the coefficients of - — on either side of this equation, when expanded by Maclaurin's Theorem, that 2AiC082t+lO = S^-cos (2n + 1 -- 4i)0.
From this formula the quantities J.0 = 1. AI, A-?, . • . , may be determined at once.
For example, let n = 4 and for brevity put At = 4l'az. If the multiple arcs are replaced by the powers of the cosine, we have
cos 0 + 4 a i cos30 + 16 a2 cos50 + 64 a3 cos"0 + 256 a4 cos90 = cos 0 + ai(cos 3 0 + 3 cos 0) + a2(cos 5 0 + 5 cos 3 0 + 10 cos 0) + a3(cos 7 0 + 7 cos 5 0 4- 21 cos 3 0 + 35 cos 0) + a4(cos 9 0 + 9 cos 7 0 + 36 cos 5 0 4- 84 cos 3 0 -4- 126 cos 0) = cos 9 0 + 4 a! cos 5 0 4- 16 a2 cos 0 + 64 a3 cos 3 0 + 256 a4 cos 7 0.
254 THEOEY OF ELLIPTIC FUNCTIONS.
We thus have among the a's the five equations *
1 - a4,
4«i = a2 + 7a3 + 36 a4, 16 a2 - 1 + 3 ai + 10 a2 + 35 a3 + 126 a4, 64 a3 = a! + 5 a2 + 21 a3 + 84 a4, 256 #4 = #3 + 9 a4.
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
«i= 922, a2= 1923, o3= 247, a4 = 1,
which agree with the above results of Gudermann. Since
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) = cn(u, k) dn(u, k).
[See Table of Formulas, LVIL]
DEVELOPMENT OF THE ELLIPTIC FUNCTIONS IN SIMPLE SERIES OF SINES AND COSINES. •
First Method. ART. 227. In Art. 206 we saw that
Noting that
A=l
^ t*
log(l +0- --2,(-V AT>
A=l X
and that
1 - 2qcos2u + g2 = (1 - qe2iu) (1 -qe~2iu),
it is seen that
- 1 , 9. 92 cos 4 w
— - log (1 — 2 g cos 2u + q2) = q cos 2 w + - — ^~-
2 2i
(f cos 6 u q4 cos 8 u ~~~ ~~~
THE FUNCTIONS sn u, cnu, dnu. 255
We therefore have
= const. - cos2u(q + q3 + <f + •••)
or 1 . - 2 #iA 9 cos 2 M. g2 cos 4 u
const. - <L__ - ___
3 cos 6 w 4 cos 8 u
1 . - /2 #iA - Iog0(— )
3(1 - ?6) 4(1 - g8)
The logarithms of the other Theta-f unctions 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 en u}
K STl U — • y
du
., _ d log (dn u + ik sn u)
IK CTl U — • i
du
idnu = d log (cn u + ?' sn u^ • du
We shall next derive the formulas
= 1-2 Vq cos u + q 1-2 \/g3 cos u + q3 1-2 Vq5 cos u + g5 1+2 Vg cos ?^ + 5 1+2 Vq3 cos u + q3 1+2 X/g5 cos w + q5
_ 1-2 V-qsin u-q 1-2V -q3 sinu — q3 l—2V — l+2\/-qsmu — q 1 + 2V — q3 sinu — q3
,0. 2 Ku , . 2 Kt*
(o) en -- h t s^i -
Jacobi [Werke, I, p. 143, formula (5)] has implicitly derived formulas (1) and (2) above, the first being had when in Jacobi 's formula u is changed
to ^ — u, and the second when — q is written for q.
256 THEOEY OF ELLIPTIC FUNCTIONS.
These two formulas may be derived directly in the manner which we now give for the formula (3) above, ^u
Write as in Art. 205, $(u)= 1 - e* "; the expression which we wish to demonstrate equal to
en u + i sn u will take the form
+ 3 iKf) <b ( — u + 5 iK') .
Multiplying numerator and denominator of this expression by
A<j>(- u + iK') </>(u + 3 iK')</)(- u + 5 iK') where A is a constant, and putting
xiu
<b(u) = Ae2K</>2(~ u + iKr)^(u + 3iK')cj>2(- u + 5iK') we have to demonstrate the formula
en u + i sn u
© (u) We further note that
$>(u + 2/0 = - $u + 4 i/T - $W
The same functional equations are satisfied by H(w) and HI(W). In Art. 90 it was shown that any three intermediary functions of the second order were connected by a linear relation, so that here we may write
Divide this expression byO(w), and we have
- u + 3 iK') (f>(u
= D en u + iB sn u.
Writing u = 0 and u = K respectively in this formula we have D = 1 and
B = 1, which we wished to demonstrate.
From the formulas (1), (2) and (3) we have (see Jacobi, Werke, II,
p. 296)
kK 2Ku _ Vq sin u Vq3 sin 3 u Vf sin 5 u 2*S> TT 1 -q 1 -<? 1 -36
kK 2 Ku _ \/qco$u Vq3 cos 3 u Vg5 cos 5 u ' —
— ri ^ ^U — I _i_ g CQS 2 u q2 cos 4 w q3 cos 6 w 2ffai „ - 4 + 1+52 j +g4 i+?6
THE FUNCTIONS sn u, en w, dn u. 257
Second Method.
ART. 229. Suppose with Briot and Bouquet (Fonct. Ellipt. , p. 286) that f(u) is a doubly periodic function of the 2 nth order with periods 4 K and 2iK' such that f(u + 2K)=-f(u)
and further suppose that/(w) has n infinities ah within (see Art. 91) the
period-parallelogram ABDC, where A is an arbitrary point u0, while B and
C are the two points u0+2K and w0+2tK'.
Form the parallelogram EFGH whose vertices E and
#are the points M.O - 2 ?rcW and u0 + 2 m'iK', while
F and (7 are the points u0 + 2 K - 2 m'iK' and
u0 + 2K + 2 m'iK'. The infinities of /(M) situated
within the parallelogram EFGH may be represented
by a = ah + 2 miK' , where w varies from — m' to
ifi'-J.
Let t be any point situated within this parallelo gram. The function
/(«)
has the period 2 #; its poles are the point * and the points a = ah + 2 miK'. It follows from Cauchy's Theorem that the definite integral
— du,
t)
j_ r /(«*)
••y **«••-
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 f(u) has a finite value and mod. - - tends towards zero * when
m' becomes very large. sm ^K ~ ^
Thus when m' becomes very large the definite integral tends towards zero and consequently the sum of the residues is zero.
The residue relative to t being —/(*), we have the equation
|
* In write u = x + iy and note that |
|||
|
sm u |
|||
|
1 |
2? |
||
|
sin M |
e^Tw _ e-ix±v |
= 0 f or v = oo .
268 THEOKY OF ELLIPTIC FUNCTIONS.
If f(u) has only simple infinities, which case alone is necessary for our investigation, the above equation becomes
Ah
sin — (t-ah-2 miK') 2 K
where A h 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 lines 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
= -00/1=1 sin—— (u — ah — 2 miK') 2 K
ART. 230. Consider next a doubly periodic function f(u) with periods 2 K and 2 iK' and having n infinities ah within the parallelogram ABDC of the preceding Article.
The function f(u)
admits the period 2 K, and the definite integral
du
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 u a point on the line A B, 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
(u-t-2 m'iK') tan^ (u - t + 2 2 K
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 (*un + 2K
M=~ I f(u)du
K Ju0
THE FUNCTIONS sn u, en u, dn u. 259
along the line AB. The residue of the function relative to the point t
9 TV- being f(t), we have, as in the preceding Article,
and consequently if the function has only simple infinities
m= + M h = n
Ah
m= -x h = i tan — — (
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 application of the results of the two preceding Articles consider the ratios of the four Theta-f unctions. Of these twelve ratios eight satisfy the relation f(u + 2 K) = — f(u) and four the relation
f(u + 2K) = f(u). Take the two functions ^v and ^-^ . Form a paral-
H(M) H(M)
lelogram EFGH with the origin as center and vertices ± K ± (2 m' + 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 " ^u' relative to the .infinity 2 miK' is , ' ; that of x^u^
s _
H'(0) We therefore have
m
v ;
* ei(0)mi
H(u) 2K H/(0)m=_oogm_._
2K
Replacing hi these two formulas u by the quantities u + K, u 4- + K + iKf we have six additional formulas including
* ©(O)"1
2K
m= 4-oc
- D
_Msi [u _ (2m _ x 2 A.
260 THEORY OF ELLIPTIC FUNCTIONS.
To develop the function 1* , 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 ra' + 1)2 iK', the function f(u) takes equal values with contrary signs; and the values of the tangent on these two sides being ^ i, the definite integral relative to these two sides is zero.
We therefore have
et(u) «• H^o
0(«) 2K H'(0)
2 K
Further, since
we have by differentiating sn u with regard to w, and then writing u = 0
« -; and since l « % , H'(0) V/c 0(0) v A;'
5lffli = JL and similarly M - -4=- Hr(0) x//cr Hr(0) V^A;'
It follows immediately from (3), (4) and (5) that
m— +00 IR\ ™ v - ^ ^
(0; sn u — ——
__OCgin_[M _(2m - l)iK'] 2 7£
""
2 K (8) *""-
If we group the terms two and two the equations (6) and (7) become
27iVo . nu
(9)
27:
do)
2K +q
*xu . ^m-2
2K a
THE FUNCTIONS sn u, en u, dn u. 261
The series (8) is not convergent in both directions; but if from dn 0 we subtract dn u, we have the convergent series
1 J_ n2m-l
(11)
_ r,
f\ TS-
Observing that
q Sl° 3 ' + I" sin 5 '
it is evident that (9) and (10) may be written
m= 1
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 !L as given in equation (1) and also the
H(uj
quotient — ^ , we may derive in a similar manner HI(U)
m = l
[See Jacobi, Werke, I, p. 157.] EXAMPLES
1. Prove that *n(iu + K} = ~^- •
2. Show that
ikf sin am u
in amf z'A/w. — ) \ Ar /
sm .
cos am u
I 1 \ A am u
cos ami ik'u, — } = ,
K I cos am u
A amhfc'w, — ) =
\ k'/ cos am u
262 THEORY OF ELLIPTIC FUNCTIONS.
3. Show that
am u
4. Prove that
(ik'\ tk sin am iku, — ) = — k I A am u
I i ik'\ ! cos am [ iku, — ] = ,
\ k I A am u
I ., ik'\ cos am u
A am ( iku, — ] = —
\ k J A am u
1
sn2(iu, k) sn2(u, k)
5. Derive the formulas
'
cn(l
dn(u,k)
Suggestion : apply formulas (XVI) to formulas (XXV). 6. Show that
~ (kf + ife) gn(M, fe) dn(u, fc)
cn\ (kf +ik)u, dn\ (kf + ik)u,
1 — (k — ik')k sn2(u, k) 2 Vikk' 1 cn(u, k)
i /j, i v^/^ i* o^>2^/ i-\
X yA/ I^ c/A/ ^ A/ o/t' ytC^ A/y
A;' + ik J = 1 - (k - ik') k sn2(u, k) '
7. Show that
sn\ (k -ik')u,
cn\(k-ik')u,
dn\(k- ik')u,
k + ik' k - ik' k + ik''
k + ik'' k - ikf
8. Show that
9. Show that
H'(it) =fc en
(k — ik'} sn(u, k)dn(u, k)
cn(u, k) l-(k- ik')k sn2(u, k)
cn(u, k) 1 -(k + ik')ksn2(u,k)
cn(u, k)
u} +Vk snu®'(u).
0(0)
0(10 '
THE FUNCTIONS sn u, en u, dn u.
10. Prove the following relations :
0(0,' k) VP6' 0(0, fc') '
H(m, k) . Ik _*& H(w, fc')
_ 0(0, A;) fc' 0(0,fc')'
K,fr) _ /fc" ^©(M +K', , Jk) VA;/e 0(0, A:')
0(0
11. Show that and that
—
du snu
w + iK'}-dn2u
d /snu dnu\ 2 2
— ( - = dn2u + dn2(iu. fc')— 1- rfw \ en M /
12. Prove that sn u dn"u — sn"u dnu = snudnu; and that
(sn w)2, sn u sn'u, (sn'w)2
(en u)2, en u cn'u, (cn'u)2 (dn u}2, dn u dn'u, (dn'u)'
kf snucnu dn u.
263
13. Show that 2kK
(G. B. Mathews.)
cos coam
R
14. Show that
2 k K 1 - 2X7* r
r
2 A"w 4 Vg sin w 4 \/q3 sin 3 M 4 \ cf sin 5 w « 1 +q 1 +53
4 q 4<f 4o3
1 — ^ cos 2 u + : cos 4 w — z cos 6 u + • • • .
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 dn u. 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 Her mite.*
The Germans use the word "Art" for the word "espece" which I translate by " sort " (see Art. 84 where the doubly periodic functions 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, Traite 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 denned through the functional equa tions
f(u + 2K)= vf(u),
f(u + 2iK')= v'/(iO,
where v and i/ are constants called factors or multipliers and are inde pendent of u. When v = 1 = i/, 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 sn u we have v = — 1, i/= 1; for en u we have v = _ 1; i/= — 1, while v = + 1, i/=— 1 for dn u. 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 theorie des fonctions elliptiques, 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, Ada Math,, Bd. 13, 1890; Picard, Comptes Rendus, t. 90, pp. 128-131 and 293-295; Picard, Crette, 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 doubly periodic functions of the second sort which have prescribed factors v and i/. — 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 v and i/, 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 u = 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 i/ as/(w). 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 k order at the points a.i (i = 1, 2, . . . , ri)j where the points 0.1, a2, . . . , an all lie within the period-parallelogram.
We shall show that F(u) may be expressed in terms otf(u).
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
where u is any point within the period-parallelogram, while £ is to be regarded as the independent variable.
Instead of c write £ + 2 K. It follows that
t(,- + 2 K) = F(£ + 2 K)f(u - c - 2 K). But we have f(u + 2 K)= vf(u),
If we put £ + 2 K for £ in this last formula, the result is
v
Also, since F(£ + 2 K) = vF(£),
it follows that
^(£ + 2/0=F(£)/
or +(* + 2K)= VT(£),
and similarly
It is thus seen that ^(£) is a doubly periodic function of the first sort. For such a function we have proved that
where the summation is to be taken over all the infinities within the period-parallelogram.
266 THEORY OF ELLIPTIC FUNCTIONS.
But ^r(£) becomes infinite on the points where F(£) is infinite and besides on the point u - £ = c, where f(u — £) is infinite. The points a i, a2, . . . , oin must be distinct from the point u — c = £.
The expansion of f(u — £) in the neighborhood of the point c is of the form
f(u-
- (u - 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(0 - F(u - c) + F'(u -C)[t-(u- and since
we have Res ^r(£) = - CF(^ - c).
In the neighborhood of the infinity ak, the expansion of F(£) is of the form (cf. Art. 98)
while the expansion of f(u — £) in the neighborhood of this point is f(u -f)-/(u -a.)-/-^T^(f- a^ + ^'^-^ff
Through the multiplication of these series it is seen that
Res TK£) - At, i/(« - «») - ^ /'(« - «*) + 4lr / e-ajk 2!
Since we have
DOUBLY PERIODIC FUNCTIONS (SECOND SOKT). 267 If next we write u + c in the place of u, it follows that
k = n
CF(u) = Ak, i/(u + c - a,) - 2*&f'(u + c - ak)
+ • • • ± ,' /(A*-1}(" + c - a (*k - 1)!
which is the expression of F(u) in terms of the fundamental function
/(«).
ART. 235. Formation of the fundamental function f(u) which has prescribed factors (or multipliers) v and v' ', where v and vf are any constants different from zero.
We had the formulas
2iK') = -
- ^(u+iK")
where JJL = ft(u) = e
If we write
0(u)=H(u + /?), it follows that
<j>(u + 2K) = H(u + ^ + 2K) = - H(u or,
<f>(u + 2K)=- <f)(u)t and similarly
_7nl
4>(u + 2iK') = - ae K >(M). Consider next the function
H(M) H(M) We have immediately
- V(u + 2iK')= V(u)e K .
The function ^(u) is therefore a doubly periodic function of the second
_^
sort having as factors +1 and e K . Suppose that v and i/ are the prescribed factors. To form a function having them, write
A«)-«flM*(«);
so that
f(u + 2K)= ea(»+2*) ¥(M + 2 K) = ea**f(u) and
_r^t
/(w + 2iK')=ea(u+2ijm&(u + 2iK')=ea2iKfe *f(u). Hence f(u) is a doubly periodic function of the second sort with the
a 2 j jf _ ^
factors ea2K and e K .
268 THEORY OF ELLIPTIC FUNCTIONS.
The arbitrary constants a and /? may be so chosen that (1)
(2) e
From (1) it follows that
and from (2)
or
= g'logy + g^logi/^
7T
The quantities a and /? being thus determined we have
The function f(u) is infinite of the first order for u = 0 (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 i/ have been so chosen that
/? = 2 mK + 2 niK',
where m and n are integers. We would then have
f(u) = eau K(u
HO)
(M + 2niK')
Further, since (cf. Art. 91)
H(u + n2iK')= ( - l)n e~ *'
it follows that
«H(1
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
/? = 2 mK + 2 niK', and consequently
2 mKn + 2 nK'in = K' log v + Ki log i/. Further, since log v = 2 Ka, it follows that
#i log *' = 2 mK- + 2 nK'ix - 2 KK'a,
or log i/ = - 2 ra-i + 2n — ~ + 2 K'cn.
/l
We thus have
v' = and
v = e2jfiro: = e If we put
mrt
« - ~K the above expressions become
We have the exceptional case* when v and i/ have this form. The quantity 7- is arbitrary; but if the factors v and i/ are given, then 7- is known.
We now write
where H'^) is the derivative of H(u). From the formulas
HCv)- H'(u)J-
2K)=- H(w), we have at once H'(w + 2K)=- H'(w), H'(w
It follows that
f(u + 2 We further have
H'(tt +2^0 771 H'(M)
H(w + 2iK;) " K H(w) ' so that
f(u + 2iK')=v'f(u)- vf7£ey».
K.
* First noted by Mittag-Leffler, Comptes Rendus, t. 90, p. 178; see also Halphen, Fonct. Ellipt., 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 e2Ky and e2K'iy, 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 and F(u + 2 iK') = v'F(u), v'
We shall express F(u) in terms of f(u) = W eyu.
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 ^(£) is here not a doubly periodic function of the first sort as was the case in Art. 234. From the formulas
it follows that -*!?(£ + 2 K) =
and further that
^ .Rl) _ ^ + K._
We again note that 2 iK' is not a period of
We compute next 2 Res ^(£) for the interior of the parallelogram whose sides are 2 K and 2 iK' . It is seen that $ = u is an infinity of -^(£); for H(0) = 0, and as H(w) is an odd function, its expansion is
H(w)= u(c0+ so that TT//,A i
where P(u) is a power series in positive integral powers of u, Similarly we have
Further, since e^ = 1 +
where P\(u — 0 denotes a power series in positive integral powers of
DOUBLY PERIODIC FUNCTIONS (SECOND SORT). 271 The expansion of F(£) in the neighborhood of £ = u is
We therefore have
As in Art. 234,
Res ^(0 - Aki if(u - ak) -
' *
(u - ak)
It follows that
k = nr 4
V Res +(*) = -F(u) + J) Ak,if(u-ak) - ™*f (« ~ «*)
(4 -I)'/
We cannot put S Res ^r(£) = 0, as in Art. 234; but after Cauchy's Theorem
where the integration is to be taken over the four sides of the parallel ogram in the figure.
We have as in Art. 92
or bv Art. 92,
= 2K (P + 2Kt)dt + 2iK'
iK'+ 2Kt)dt
But since ^(£) has the period 2 K, it follows that
2 7:1 Res V(c) - 2 K
2iK't)dt
or
= 2K
Res r(£)=-
+ 2 /ft) - ^(p + 2 tKr + 2 KO dt
F(p + 2 K«) ^ ^ dt;
re-(P+2K» F(p + 2 Kt)dt.
272 THEORY OF ELLIPTIC FUNCTIONS.
The definite integral is a quantity independent of u, which we may denote by A, 'so that therefore
Equating the two expressions that have been found for 2 Res ^(£), it is seen that
Further, since F(u + 2 iK') = v'F(u), we may write
k = n ,
F(u + 2iK') = v'Aey» + i/ V Ak,if(u - ak)-
k=i
(4-1)!
On the other hand if we put u + 2 iK' for u in the expression above, we have
v'ni
(4 - 1)! Comparing the two results just derived, it is seen that
This condition must be satisfied by the A1 8 in the formation of the function F(u).
Since 7- is an arbitrary quantity, it may be made equal to zero. We then have
But Akti is the residue of F(£) for £ = a&. We therefore have
SAfc,i = SResF(f);
and consequently
S Res F(£) = 0, when 7- = 0.
DOUBLY PERIODIC FUNCTIONS (SECOND SORT). 273
But if f = 0, then F(u + 2 K) = F(u) - '
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 ivithin 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 + (2 n + \)iKr.
It is seen that Q(u + v) vanishes on the point u + v = iK' or u = 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) { sn u — sn (iKf — 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) = sn(u + v} { snu - sn(iK' - v) }\snu - sn (iKf + 2 K - v} } .
This product remains finite for u = iK' — v and for u = iK' + 2 K — v and for all points congruent to these two points (modd. 2 K, 2 iKf). We have
sn(iK'- v) =
sn (— v) k sn v
It follows that
G(u) = sn(u + v} \>sn u H -- > ) sn u — — ^ — ', / ksnv) I ksnv)
= sn(u + v) \ sn2u — — — — I , ( k2sn2v )
or G(u)k2sn2v = sn(u + v}{k2sn2u sn2v — 1} = F(u), say.
* See Forsyth, Theory of Functions, p. 280.
t Hermite's " Cours " (Quatrieme 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 y = - 1 and
F(u + 2iK')= F(u), or i/^ 1.
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
sn h = h + c3h3 -f c5h5 + - • • ;
and consequently if we put
u = iK' + h or h = u — iK', we have
sn(iK'+h)= 1 l
k sn h kh 1 + e3h2 +
-— f 1 kh*
It follows at once that
and consequently
k2sn2usn2v — 1 = -sn2v + 2e2sn2v + • • • — 1.
(u — iK'}2
Noting that = en v dn v, it is seen that the expansion of sn(u + v)
in the neighborhood of u = h + iK' is
1
sn(u + v) = sn(v + iK' + h) =
(u -
k sn (v + h) which by Taylor's Theorem
1 _ 7 en v dn v , k sn v k sn2v
1 _ cnvdnv k sn v k sn2v We therefore have
•n, \ 1 sn v en v dn v 1 , n , . r,/x
F(u)=k(u-iK')*- —-^^ + P(U-*K)- Writing
_ en v dn v ^ * snv _ *
IV IV
we have
DOUBLY PERIODIC 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 K and 2 iK'.
We may consequently choose - for this fundamental function. We have
- = — h positive powers of u. snu u
Consequently we have Res/(w)= 1 = C (of Art. 234).
u = 0
Hence (see the formula at the end of Art. 234) it follows that F(u) = A0f(u - iK')- Ai/'(t* - iK').
We have further ff .v,^ 1 ,
f(u - iK') = — -- — — = k sn u, sn(u — iK)
and also f'(u — iK') = kcnu dn u, so that
rv \ cnv dnv j snv -,
F(u) = -- - - k snu -- k en u dn u.
K K
Equating the two values of F(u), it is seen that
sn(u + v) [k2sn2u sn2v — 1] = — sn u en v dn v — sn v en u dn u, or finally , >. sn u en v dn v + sn v en u dn u
87l(U -h V) —
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
«,<„+„)-_ dv du
1 — k2sn2u sn2v
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
Q(u + a) H(^ + a) QI(U + a) fii(u + a)
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 K
sin — (u + 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
.
eK - e
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
-e or e .
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'
either e - or e
If for the first a' + 2 K' > 0 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
sin — (u + 2 niK') 2 K
and noting that, since n varies from — oo to + oo , we may change n into n + 1, we have
(n-\-\}ifia . itina
K ^ rit
Sin JL. [u + 2 (n + l)iK'] sin -- [u + 2 iK' + 2 niK']
2 K 2 K
It follows at once that
or
* Hermite, Ann. de I'Ecole Norm. Super., 3e se>ie, t. II (1885); see also Hermite, Sur quelques applications desfonctions elliptiques, p. 35.
DOUBLY PERIODIC FUNCTIONS (SECOND SORT). 277 On the other hand we have immediately
so that $(u) is a doubly periodic function of the second sort with the
tVa
multipliers — 1 and e K .
The poles are obtained by writing
sin ^- (u + 2 niK') = 0,
from which we have
u = 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 u = 0, the corresponding residue being
9 jr
- -- We further note that the quantity
2K H'
n H(u)9(a)
has the same multipliers, the same pole, and the same residue. We may therefore write (see Art. 83)
idna
2K H'(0)0(M + a) = - H(w)0(a)
11 2^
If a and u are permuted in this equation, we have (1v 2K ^(0)0(1^ + a) _^ e^
We may deduce the others as follows: If we change a into a + iK', we have
2K
or
(2n + l)siu
u + a)
g
2 A.
278 THEORY OF ELLIPTIC FUNCTIONS.
If further a + K is written for a in (1) and (2), these formulas become
Ttinu
2K ~
(3)
2
a) _ • - 2«
~-^[«*<2«
If u + K is written for u in the four formulas above, we have the four following formulas, in which ®i(u) is found in the denominators:
Ttinu
tK\ 2K H'CO)©!^ + a) ^ (- l)ne^~
W ^ /.ATJ/.N ~2^~ ~~^~
sin (a + 2 niK')
2 A.
(2n + l)atu ;2n + l^ 2K
ART. 240. Hermite next formed a series entirely different from the one of the preceding Article which is represented as follows:
where n takes all even integral values from — oo to +00, while the quan tity e must be supposed zero for n = 0 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
jdna
cot || + cot || + %e 2X [cot ^ (« + niK') + t]
DOUBLY PERIODIC FUXCTIOXS (SECOND SORT). 279 which by an easy transformation becomes
(na+nitf +u)
cos -^7 (u + niK') 2 A
-—.(na-niK' + u)
cos — — (u — niK')
To prove the convergence of this series, note that for large values of n the two denominators
cos -^ (u + niK'} and cos — (u - niK') £ A 2 A
may be replaced by
\e 2X and ^ e2 K
the general terms becoming
If we put a = a + ta', we have for the limit of the nth root of their moduli as n becomes very large, the quantities
i 2
e and e2A
and consequently the conditions
a'+ 2K'>0, a'- 2 K' <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
¥(11) = cot —^ we have the relations
-ia
The second of these relations is evident from the expression of the product e K *&(u + 2 iK'), viz.,
T\GL ;rtct Trtfl T^ino.
~jr -jid ~jr ^-\ OK r 7T ^6 COl r€ 76" I COt
2 A ^ 2 A
280 THEOEY OF ELLIPTIC FUNCTIONS.
We have
and if we change, as is permissible, n to n — 2 in the general term, it becomes
-.
\ cot-2- (u + niK') + ei \ , L 2 K J
2 K. where now there is a modification regarding £.
The quantity e must be = 1 for n = 4, 6, 8, . . . , while e = 0 for n = 2 and e = — 1 for n = 0, — 2, — 4, • • • . We note that in adding
?rta
to the terms corresponding to n = 2 and n = 0 on the one hand ie K and
/ Trta \
on the other i, and consequently in causing the quantity i\eK + I/ to enter the summation, we find for e precisely the significance which was accorded it in the function W(u). We further note that within the rectangle of periods there exists the one pole u = 0, to which corresponds
9 /<"
the residue - . We may therefore represent the function W(u) by
2K H'0Htt + o
If we interchange u and a we have finally
where n represents all even integers and the unity e must be taken positive when n is positive and negative when n is negative. *iu
Next changing a to a + iHC', we have, after having multiplied by e2K, the formula
where m denotes the odd integer n + 1. Since
we have, if the term ie2Kis introduced under the summation sign,
where m represents all odd integers and s 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).
2 K H'
no,
IT
/1/n
Hl(M)e(a)
(16)
n
the quantities m, n, and e being denned as above.
EXAMPLES
1. If w = 1, 3, 5, . . . ; n = 2, 4, 6, . . . , show that 2tf
0(M)H(a)
2. Further if mf = 1, 3 5, . . . , prove that
0(n)0(a) 3. Show that
. HfttfHtM ^2K - L^1^' i; H(,)0i(a)a =cosec^-^2A[tan^(^^^)-^]-
0(M)Hl(a) ^ - —
4. Prove that
cos—
[Kronecker.]
CHAPTER XIII ELLIPTIC 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
- z2)(l -k2z2)
is finite and continuous on the finite portion of the Riemann surface. In the neighborhood of the point z = oo, we have
- z2)(l - & so that
- z + £i _L ^2 , . . .
— ^ T^ 1^ v I )
- z2)(l - &2z2) z z3
where the a's and 6'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
*£
'W
the expansion in the neighborhood of the point Z = 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
where C is a constant.
282
ELLIPTIC INTEGKALS OF THE SECOND KIND. 283
This is the simplest 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
At 4- B
f
(t - a
dt,
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 t = oo.
We shall so choose the constants A and 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
At+JS = Aa + B + _ _ (f ^ |
VS(t) VS(a) S(a)
It follows, if we put
(1) AVSM - I (Aa + B) -^ZL = J = 0,
2 VS(a)
that
(At + B)dt _ Aa + B log (t - a)
r (At +
J t- a2
(t- a)2VS(t) (t - a) VS(a) S(a]
p(t _ .
will not contain a logarithmic term in the expansion according to ascend ing powers of t — a. Further, since
Cdt C
(t - a)2 t - a
it is seen that the two integrals become infinite alike on the point
a,-VS(a), if
(2) -a
VS(a) It follows from equations (1) and (2) that
1 = 1 S'(a) ^ 2
B=-C S - Aa = C
VS(a)
* 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-a)2 (t-a)2VS(t)/
dt
J
is an integral of the second kind, which is infinite of the first order* at only the one position (a, \/S(a)). Write C = % and put
_ a)
2(t- a)2
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($, VjS(p) of the second kind which becomes algebraically infinite of the first order on the point a, \/$(a)? If such an integral exists, its development in the neighborhood of t = a is of the form
- a).
t — a
Writing E i (t, VS(f)) = E (t, VS(T) ) ,
it is seen that
E(t,VS(t))-E0(t,VS(t))
does not become infinite for any point on the Riemann surface. It is therefore an integral of the first kind, = F(t,\/S(t)), say. It follows that
E(t, VS(ij) - EoO, V^) + F(t, VSW>).
Hence, if we add to an integral of the second kind an integral of the first kind, we have an integral of the second kind which is infinite only at the point (a, VR(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 kind which are algebraically infinite of the first order on the one point (a,
* Cf. Koenigsberger, Elliptische Functionen, p. 250.
ELLIPTIC INTEGRALS OF THE SECOND KIND. 285
ART. 243. If we put
(t - a)
and then
da 2(t - a)2 We further write (see Art, 287)
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
fzs
u = I — =
«Au v(i
dz
we put z = sin </>, we have in Legendre's notation
Ffak) = r W == P4^-» where A0 = Vl - k2 sin26-
Jo Vl-A;2sin20 Jo A^>
The complete integrals of the first kind are therefore
Js
Vl -T'2 sin2<£
In Legendre's notation (Fonct. Elliptiques, t. I, p. 15) the integral of the second kind is
E(k,<f>)= fVl - k2sm2<i>d<f> =
J o «/o
286 THEOEY OF ELLIPTIC FUNCTIONS
The complete integrals are (see also Art. 249) :
= E,
E
JO \/l _ Z2 o YI _ Z2
If we put d<j) = dnu du, A^> = dnu, we have
E(k,<t>)= E(u)= CUdn2udu - P(l - Jo Jo
(Jacobi, Werke, I, p. 299.) ART. 245. To study the integral of the second kind
r «/o
- z2)(l - as a function of u, where
dz
r
Jo,i
u =
- z2)(l -
we may with Hermite * multiply this integral by k2 and put
'u) = I k2sn2u du.
f*
I
Jo
We note that the function sn2u has the periods 2K and 2iK'', and from the developments above it is seen that /2(^) is a one-valued function of z. But z considered as a function of 72 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, Ada Math., Bd. 8).
ART. 246. We saw (Art. 217) that snu became infinite on the points
2mK +(2n + l)iK' = a, say.
Writing u — a = h or u = a + h, we must develop sn2u = sn2 [2 mK + (2 n + l)iK' + h] in powers of h. Since sn2u has the periods 2 K and 2 iK', we have
1
sn2 [2mK +(2n + I)iK' + h] = sn2 (<*#' + h)
k2sn2h
* 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 INTEGKALS OF THE SECOND KIND. 287
so that
, or k2sn2u
, k2sn2h sn2h h2 + ch4 + •
It follows that
k2sn2u = - - ?— - 4- e0+ CI(M - (u - a)2
and consequently, since the integrand does not contain the term (u—a)~l, the integral
/2(M) _ CUk2sn2u du Jo
is a one-valued function of u.
ART. 247. The analytic expression for I2(u). — The function k2sn2u 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 k2sn2u as a doubly -periodic function of the second sort with the factors v = 1 and i/= 1; or v = e2^, i/ = where y = 0.
We have here the exceptional case of Art. 237 where
F(u) = Ce^ + 2) U»,i/[t« - a) - %V'(" -«
the function F(u) being fc2sn2M and f(u) = e>" being /(M)
_ tlvti]
since 7- = 0.
The development of k2sn2u in the neighborhood of the infinity iK' is
Hence in the formula above, Ak,i, the coefficient of (u — tK')-1, is
zero: and A^ 2, the coefficient of — (u — iK')'1, is — 1.
du
We consequently have
k2sn2u = C-//(^^ - iK'). It follows that
k2sn2u du=[Cu - f(u - iK')]
_
r/, r TL'(u-iK'
/2(w) = Cu ~
It is thus seen again that I2(u) is a one-valued function of u.
288 THEORY OF ELLIPTIC FUNCTIONS.
Since
we have
ic K' idu
H (u - iK') = -iel*+™:® (u)
-jjf(
= -ie 4X It follows that
We therefore have
W(u - iK') m jn_ K(u-iKf) 2K 0(w)
and W(- iK') =_ jn_ er(0)
H(-{Kr) 2K 6(0) since ©'(0) = 0.
It has thus been shown that*
To determine C, we have from above
Equating powers of u on either side of this equation, we have
r _Q"(0)
= w
It follows that
fUI2(u)du = %Cu2- Iog0(w) + C',
*J Q
where C' is the constant of integration. From this it is seen that
'0
or
Finally we may write f
0(tt)=C"'eiC"'--C7'(u)du, where C"= 0(0).
* Hermite, Serret's Calcul, t. 2, p. 829.
t Jacob! (Crelle, Bd. 26, pp. 86-88; Werke, II, pp. 161-170) defines the 0-function by this formula and therefrom derives directly the series through which this tran scendent may be expressed and its other characteristic properties.
ELLIPTIC INTEGRALS OF THE SECOND KIND. 289 ART. 248. We may next consider the integral of the second kind
0,1 V(l - z2)(l - k2z2)
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 Ti(z, s) is uniquely determined, and if u is known, then also /2(w) 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 /jt(z, s), that is, the elliptic integral of the second kind in I", 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 b.
For the elliptic integral of the first kind 77(z, s) we had
( "" M - u(p) = A (k) = 2 iK' on the canal «, I 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: *
( /2(/) — /2(/o) = 2iJ' on the canal a, ( Iz(p) — /2(/) = 4 J on the canal b.
We had (Art. 193)
K' =
r1 dz
Jo Vn - z2)(l - k2z2]
:>- /" , rfz - f
Jo V(l - z2) (I - k'2z2} Ji
1
dt
) Ji V(t2 - 1)(1 - k2t2)
In a corresponding manner we may write with Weierstrass (Werke, I, pp. 117, 118)
i
A;2*2^
= r1
Jo
- z2)(l - A;2z2)
Jf
'
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. cit., p. 828; Fuchs, Crelle, Bd. 83, pp. 13-38.
290 THEORY OF ELLIPTIC FUNCTIONS.
AKT. 249. We had above
If in this formula we write u = K, we have
From the formulas
0(t* + £)=©i(tO, &r(u + K)=Qlf(u),
it is seen that for u = 0
O'(K) =©!'(()) =0, and consequently I2(K)=CK.
To compute 72CK) we put u = K in z = snu, and if z0 is the value of z that corresponds to w = K, we have
z0= snK = 1 (Art. 218). It follows that
k2z2dz
r
t'O.l
= J.
V (1 - z2)(l - k2z2) We therefore have
J = CK, or C = j£;
and finally j @r( }
/2W- — ^ - 777^'
K 8(v)
We may next compute the constant C in a different manner. If in the equation
we write K + ^'K7 for u, it becomes
h(K + iK') = C(K + iX'
2K To compute 72(K + iK') we put w = K + t'K7 in sn u.
If Zx is the corresponding value of z, we have ~z.\ = —. Further, since
k i i
k
Z2)(l
-/
«/ 0,1
- z2)(l -
ELLIPTIC INTEGRALS OF THE SECOND KIND. 291
we have
iJ'=I2(K+iK')-J,
or I2(K + iK')=J + iJ';
and consequently
J + iJ'= C(K +iKf) + ~
2 A.
Eliminating C from this formula and the formula CK = J, it is seen that
J'K - K'J = |-
We note that _
r1 v/1 _ 1^-7-2
K-J=\ V X k z dz = E (Legendre); •/fl Vl - z2
and making the transformation
C1\-/-\ Z*'2,,,2
/ / VI -^ M
it is seen that
\ 1 - u It follows that
which is the celebrated formula of Legendre (Fonct. Ellipt., I, p. 60).
Remark. — The characteristic properties of /2(w) are expressed through the formulas
These formulas follow at once, when we note that
Change u to u + 2 K and z^ + 2 i'Kr respectively hi the equation
and use the relation j^j, _ jg/ = £.
ART. 250. We note that
J f« 9 , 0r(w)
-u- / k2sn2udu = — ^: K Jo B(u)
or
292 THEOEY OF ELLIPTIC FUNCTIONS.
With Jacob! (Werke, I, p. 189) we define the zeta-function by the relation
u)= (l - | ]u - rk2sn2u du,
which. is Jacobi's elliptic integral of the second kind. It follows * also that 0 (w)=0 (0)/« Z(u)du, where 0(0) = \/^^- (Art. 341).
* 7T
The ©-function may thus be considered as originating from the function 7i(u) [see Cayley, Elliptic Functions, p. 143]. From the formula
we have dn2w = — + 7t'(u) and consequently Z'(0) = 1 --- A. K
It follows at once that
k2sn2u = Zr(0)- Zr(^), and k2cn2u = k2 - Zr(0) + Z'(w) ; Zr(K) = Z'(0) - fc2
It is further seen, since
that
As ®i(tO is an even function, its derivative is odd, so that
Z(K)= 0.
ART. 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 a priori insight into the Weierstrassian functions. In Art. 220 we made the imaginary substitution
It follows at once that
* Jacobi, Werke, I, pp. 198, 224, 226, 231.
ELLIPTIC INTEGRALS OF THE SECOND KIND. 293
This expression, when integrated, becomes
or
(1)
It follows that (2)
L
From the formula (Art. 249)
FE(k') + F(k')E - FF(k') = ^
2i
we have at once
') = -£- [F(k')E(+, *') ~ E(V)FW, k')] r (K )
' 2F(k') Equation (2) becomes through this substitution
, k)
> iF F(kf)
2 FF(k') Using the Jacobi notation
$ = am iu, ^ = am (M, fcr), F(<£) - tu, F(^, A;') = u, we have
_ Z( ,,,.
and consequently from (3) we have
(4) iZ(iu, k)=- tn(u, k') dn(u, k') + -p- + Z(w, A;').
Z A A
Multiplying (4) by dw and integrating, this equation becomes
fUiZ(iu, k)du = log cn(u, k') + ~f + | Z(u, ^) Jo 4AA Jo
Further, since
it follows that
(ef.Art.204).
294 THEORY OF ELLIPTIC FUNCTIONS.
Formulas (4) and (5) reduce the functions Z(iu) and ®(iu) to real argu ments.
If in (5) we change u into u + 2 K', that formula becomes
In this formula change iu to u and we have
ir(K' - iu)
(6) ®(u + 2 iK') = -e K ®(u) (cf. Art. 202).
Again write u + K' for u in (5) and note that
cn(u + K', kf)=- k sn ^ k"> , dn(u,k')
Gi(», i IT i jpf\ dn(u, k ) s\, -.,%
Vk
It follows that
/-v / . 7r(M + A"')2
. @(0)
w(2 u + A^r
= — e 4K
,
0(0)
Write iu for w in this formula and it is seen that
7r(K' - 2 iu)
(7) ®(u + iK')=ie 4* Vksnu®(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(u + 2iK')=-^ + Z(u),
K
(9) Z(u + iK')=- ^- + cotnudnu + Z(u).
2 K.
Write u = 0 in formulas (6), (7), (8), (9) and we have
nK'
0 (2 iK') = - e K 0 (0) , 0 (iK') - 0 (cf . Art. 203),
H, K.
ELLIPTIC INTEGRALS OF THE SECOND KIND. 295 ART. 252. In Art. 227 we saw that = const. -
1 - q- 2(1- q4) q3 cos 6 u _ q4 cos 8 u 3(l-g«) 4 (1 - f) - From the relation
it follows that
- q2m
[Jacobi, Werke, I, p. 187.] We also have
(2) Z(tt)~e(to~i3:
" K
To be noted is the equality of the right-hand sides of (1) and (2). We further note that
2Ku -JK_ [£cos2M , 2 g2 cos 4 ?*
= -
ART. 253. Thomae * introduced the notation
Differentiate logarithmically
and we have 2 .2. „
rj t,.\ rj f..\ k^snucnu
Similarly we have
dn u [Jacobi, W^erke, I, p. 188, formula (6).]
/ x _ en u dn u
sn u
/ v = _ sn u dn u 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
K
it is seen that
J_
K
Further, since
it follows that
- log ©!(«)= - *»«i*(« + K) = - fc2
Similar results may be derived for H(u) and HI(W).
The functions @(w), ®iCw), etc., when for u is written the integral of the first kind u(z, s), are functions of z, s, but not one-valued, since u(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 dn2(u, k)du,
2
o E'= C* dn2(u,k')du.
2. Through the definitions of the zeta-functions of Art. 253 derive independently the formulas given in Chapter X for &i(it)j Hj(w) and H(tt).
3. Prove that iZw(iu, k) = Z00O, fc') + and iZM (iu, k} = Z1n(tt, k') +
2 TT
4. Prove that Z^ = — K
2 TT
K l + 2?cos cosgcos. - •
K K K
Derive similar expressions for Z10(w) and Zu(w).
(Thomae, Sammlung, etc., p. 16.)
|
( \m • m7tu |
|
|
oo ( q) sin — |
|
|
r i-g2w |
|
|
/V j?V |
I sin ^ |
ELLIPTIC INTEGRALS OF THE SECOND KIND. 297
5. Verify the results indicated in the table :
iK'
|
zw + ^ |
0 |
oc |
|
? 77 |
X |
0 |
|
Z10 + .^ |
0 |
0 |
|
Z» + ^ |
0 |
0 |
|
Zoo |
0 |
0 |
|
Z01 |
0 |
0 |
|
Z10 |
0 |
OC |
|
Z« |
CiC |
0 |
6. Show that
*<*""*
V7! -
7. Prove that
Roberts (Liouvitte's Journ. (1), Vol. 19), Wangerin (ScMomtich's Zeit., Bd. 34, p. 119).
8. Complete the table of Ex. 5 by letting u take values i K, K + J i'A', 4 iA', ^ A + i iAr, | A + | iK', etc.
CHAPTER XIV INTRODUCTION TO WEIERSTRASS'S THEORY
ARTICLE 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,
V4(t- ei)(t- e2)(t- e3)
where V±(t-ei)(t- e2)(t- c3) = V/S(0 (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 u = 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)
it follows immediately that
du
In Art. 185 we saw that the transformation of Weierstrass 's normal form to that of Legendre is effected through the substitution
t = e3 H > where e =
el - e3
We therefore have 1
<$u = e3 H -- - --
Since sn I -^= ) is a one-valued function, the function <@u must also be one- Vv«/
_
valued; and since sn2(u \/ei— e3) has the periods 2 \/eK and 2 these are also the periods of <@u. We put (Art. 196)
2 VI K = 2a), 2 V~eiK' = 2 co',
so that the function $>u has the periods 2 co and 2 a>'. We further note that sn2u being an even function, the same is true also of <@u.
298
INTRODUCTION TO WEIERSTRASS'S THEORY. 299
ART. 257. As we have introduced the new function $>u 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
we put u + iKf in the place of u, we have
= ~ log H(tt)-
sn2u K Since p(v\/e)= e3H -- — >
£snv
it follows that f(v VI) = ea + -i - - d*
K e dv2 Noting the identity
or
it is clear that
Writing v \/£ = u, this formula becomes
which is a one- valued function of z (see Art. 254). We thus have
or, if we put
ou = fie 2 ^ £ ]
where /9 is a constant, then is
- &u = — - log ou. du2
The arbitrary constant /? we may so choose that in the development of ou, the coefficient of the first power of u is unity.
300 THEORY OF ELLIPTIC FUNCTIONS.
By Maclaurin's Theorem this development is
ou - <r(0) + ua'(Q) + • • - . Since H(0)= 0, we also have <r(0)= 0; and noting that
A/e
we have o
</(()) = 1 = JL-H'(0).
It is thus shown that , /-
H'(0) and consequently /? _ 1 / . I -
If we differentiate the expression
1
wehave 1
cnudnu = —^
Vk Writing u = 0, it is seen that
or
\//c 02(0) Hr(0)
It follows from above that
ART. 258. The expression
d2 log av
becomes when integrated
where the lower limit w and the constant y are connected as follows : If we define the small zeta-function by
rv^— (see Art. 277), av
we may write jv
av
INTRODUCTION TO WEIERSTRASS'S THEORY. 301
Putting v = a) in this formula, we have at once
'cu = — (at) = rj = - I $(v)dv + TJ. o J \>
We may similarly introduce the new quantities
If we put (see Arts. 195 and 256)
pv it follows that
= t, dv = —=, &w = ei, #w" = e2, &*>'= e3,
VS(t)
and
or
tdt
In a similar manner as in Art. 194, it is seen that
9 I — R along the upper bank of the canal c^e?
J V in the upper leaf; and
tdt
- = A (upper leaf) , VS(t)
where B denotes the difference in the values of the integral
on the right and the left bank of the canal b, and A the corresponding difference on the left and the right bank of the canal a. If any arbitrary path of integration is taken, we have *
rtdt B . V^m 2
= + m'A + I'B, * VS(t) 2
where m, /, m' ', /; are integers.
* See Bruns, t/e&er die Perioden der elliptischen Integrals erster und zweiter Gattung, Math. Ann., Bd. 27, p. 234.
302 THEORY OF ELLIPTIC FUNCTIONS.
It follows from above that
and
the congruences being taken with regard to integral multiples of A and B. ART. 259. By definition of Art. 257 we have
It follows that
From the formulas H(w + K) = HI(W)
and H'(tt + X)=Hi'(M) we have at once H(K) = Hi(0)
and H'(K)=H/(0)=0.
It is seen that
Further, since J = K - E and o> =
we may write *
Further, since \/eiKf= to',
, ofajf I 1 J\ , , 1
-- -
, ofajf I 1 , ,
we have if = - = - ( e3 H --- w H -- -=
aw7 V t K/ V^
or, since (see Art. 247)
we
(2) or, (20
* See Schwarz, Zoc. cif., p. 34.
INTRODUCTION TO WEIERSTRASS'S THEORY. 303
It follows at once from (2') and (!') that
From the formulas
we have
1 W(K + iK')
x/7 H(K 4- iK') Further, since (Art. 247)
it follows that (3)
From the formulas (1), (2) and (3) it is evident that
f + t'- *"•
It is seen from the preceding article that
and since — -r
, Jo A
l-^-J.— 2
we further have —
'S2
and
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,
yo)'- TI'UJ = ~
We may note that
£(u + 2 co) = £u + 2 i),
£(u + 2o>')= C" + 2)?/;
for pu being an even function, its integral £u is odd, and writing u = — aj and — a>f respectively in the two formulas just written, we establish their existence.
304 THEORY OF ELLIPTIC FUNCTIONS.
ART. 260. We have already derived the formulas
and T , 1 J
If we put
then is ou
from which it is seen that ou is an odd function, the function H being odd. It follows immediately that
o(u + a>) = [3e2riw(v+®2 H(2 Kv + K) The following new notation is suggested:
where /?i, /?2 and /?3 are constants.*
It is seen that o\u, a2u and o3u are even functions. We shall so deter mine /?i that for 2 cov = u = 0 we have <7i(0) = 1. We thus have
i-AHiW, or 0!
and similarly
- and
ART. 261. It is evident from the previous Article that a(u + w)
where Ci is a constant. For v = 0, it is seen that C\= oat, and conse quently
0(J)
We further have
2 ( - - ~
o(u + at') = (3e 2 H(2 K
or
These constants are expressed through Weierstrassian transcendents in Art. 345.
INTRODUCTION TO WEIERSTRASS'S THEORY. 305
Writing u = 0 = v and noting that 2 r)a>' — -i = 2 r)'a>, it is seen that
It also follows without difficulty that
00)
The functions oiu, 0*2,11, o%u are like ou, one- valued functions of u, that have everywhere in the finite portion of the plane the character of integral functions.
ART. 262. From the formulas above we have
or
0(2K
^cn'22Kv.
and similarly
fef)2=a
(02U\2 = b,dn22KVf
\auj
where a, a', bf, c' are constants.
Since
9U — e'.
it is evident that
where c3 is a constant. If we put
we have so that
— 63
sn22Kv
e1-
or
<tu
c\(®u — e\), where GI is a constant.
In the same manner we have
- k2 =
dn22Kv
\au/
_ 62^ where c2 is a constant.
306 THEOEY OF ELLIPTIC FUNCTIONS.
We have accordingly
v @u — e\ = cti
au
au
au where d\, d2, and d3 are constants.
To determine the constants we note that au may be developed in the form
au = u + b3u3 + b5u5+ • • • , and also that
oku =1 + b2,ku2+ 64,* u*+ ... (k = I, 2, 3),
where the b's are definite constants. We therefore have
OkU 1 1 -f- b2tku2-\- • • - 1
au u 1 + b-)U2+ ... u
In the neighborhood of the point u = 0 we also have sn v = v 4- esv3 + « • • ,
sn^==^=
so that
and
Since
it follows that
1
I
$>u — e\ = — - + HQ -\- 63 — uz
On the other hand we had
+dltku + . - . (jfc = 1,2,3). u \
It follows that rffc2= 1 or dfc = ± 1, and consequently
INTRODUCTION TO WEIERSTRASS'S THEORY. 307
Since the quotient ^^ is a one-valued function, we may take the positive
au sign (see Schwarz, loc. cit., p. 21).
We further have _
Cu \ \/e\ — e3 1 au — -]= — ===== — - = -- V«/ Vf?tt - 63 V £ ^3^
Similarly it is seen that
or
and also that
ART. 263. It follows from the formulas
that
^2(^7^3" • w) = 1 Further, since
_,.
azu = e -i u — - - — " »
Oil)
we have
72 J2
-— log a3u = -— log a(u + a}') = - %>(u + &') = - p(u - a)'). du2 du2
Admitting the relation (see Art. 316)
we have
d/asu + \ au\ cr3u /
Since
it is seen that *
E(u)= C Jo
u) = -±= l^L + eiu\
Vel-e3\(>3u I
* See Schwarz, loc. cit., p. 52.
308 THEORY OF ELLIPTIC FUNCTIONS.
Further, since (Art. 259)
E = ZlZ. and K = ei- e3 • w,
V6!- 63
it follows from
Z(u) = E(tO-uf K
that
V ei - e3 V ff3tt / V el - e3 W
1 /fr3'^ _ 7£W\
Vei- e3 \ <*3^ w/ The last formula may be written *
+aj')-riu- /I. u J
- e3
EXAMPLES
1. Jacobi, Werke, I, p. 527, wrote
f
^> 2
9 = am
2 K show that x?(Q)— £(x) =
71
2. Prove that
3. Let
Show that
Px x 1 , l 2 , 4,
P(u} = -- 1 -- u2 H -- u4 + u2 15 189
4. If F(k2) is the coefficient of u2n~2 in the preceding example, show that
5. Prove that the function P(u} of Example 3 satisfies the relation
P'(w)2 = 4 p(tt)3 _ 4 (j _ p + j.4) p(u) - /T (1 + A;2) (1-2 fc2) (2 - /c2) ;
or P'(u)2=4P(u)*-g2P(u)-g3.
(Hermite, Serret's Calcul, t. II, p. 856.)
* See Enneper, Elliptische Functionen, p. 221.
CHAPTER XV THE WEIERSTRASSIAN FUNCTIONS $u, %u, an
ARTICLE 264. We saw in Chapter V that the doubly periodic functions of the second order or degree are the simplest doubly periodic functions. These functions are either infinite of the first order at two distinct points of the period-parallelogram, or they are infinite of the second order at one point of the period-parallelogram. The first case has been considered in Chapter XI. We shall now consider the latter case. Among this group of functions we shall take the simplest, viz., those which become infinite of the second order at the origin.
Such a function may be expressed in the form
where 6^0, and where P(u) is a power series in integral ascending powers of -M.
It is shown below that the constant a = 0. We therefore have
<£(") _ L | p(") 6 u2 b
The constant term that occurs hi 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-f unction and denoted by
g>(w) or more simply gra. This function is of the form
&u = -5 + * +(("))• u2
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 ((u)) denotes that all the following terms are infinitesimally small when u 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 u — v.
309
310 THEOKY OF ELLIPTIC FUNCTIONS.
We may show as follows that the constant a is zero: We had
4>M = 4 + - + c + ciu + c2u2+ c3u3 + .... u2 u
Consider also the function <j>(— u). It is doubly periodic, having the same pair of primitive periods as has (f)(u), and consequently like (j>(u) is infinite of the second order on all points congruent to the origin. It may be
written h
$(- u) = JL_ « + c _ ClU + c2u2- ....
u2 u We therefore have
It follows also that (j>(u)— <j>(— u) is a doubly periodic function with the same pair of primitive periods as <j>(u), and consequently can become infinite only where (f>(u) and $(— u) become infinite and therefore only on the points congruent to the origin. But, as seen from the last equation, <j)(u) — </>(— u) 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
</>(u)— </>(— u)= 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 4>(u)— <£(— u), we conclude that
<£(u)-<£(-W)=0,
or <j>(u)= </>(— u).
It is thus seen that the elliptic function of the second degree which becomes infinite of the second order at only one point of the period- parallelogram must be an even function. It follows that
or (W)_C= +M+
b u2 b b
This function we denote by ®u and we require that pu be a one-valued doubly periodic function of the unrestricted variable u which has the char acter of an integral rational function at all points that are not congruent to the origin. At the origin and the congruent points $w must be infinite of the second order and is to be an even function.
THE WEIEKSTRASSIAN FUNCTIONS ffu, &, <ru-
311
ART. 265. We may next show that in reality there exists a function which has the properties required of <@u.
Let w = 2fiw -f 2(«V,
where // = 0, ± 1, ± 2, . . . ; a' = 0, ± 1, ± 2, . . . ; w = 0 excluded. Form the function
1 , ^ 1
This function does not have the properties desired of pu, since the series
V is not convergent. For if we give to u the value zero, we have
T (u ~ ™)2
V — , which is not convergent (see next Article). ^fw2
But if we form the series
(u
J_ .J_J
- w)2 w2 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 w = 0 is excluded from the summation, we may write
^ = L+vf\—L- ±1.
u2 ** l(u-w)2 w2) ART. 266. We must show that the series
6u>'
(u- w)2 w2
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 a>, — 2 co, 2 w', — 2 a/ be d\, and let d2 be the longest distance from the origin to any point on the periphery.
Fig. 70.
On the periphery of this parallelogram there He 8 = 32 - I2 period- points. For these points we have
312
THEORY OF ELLIPTIC FUNCTIONS.
On the second parallelogram, passing through the points 4 co, — 4 a), 4 w'f _ 4 &' there are 52 — 32 = 8 • 2 period-points, and for these we have
2di ~ \ w\ ^ 2d2.
On the third parallelogram indicated in the figure there are 72 - 52 = 8 • 3 period-points, and for them there exists the inequality 3 di ^ \w\ ^ 3 d2j and for the n + 1st parallelogram there are (2 n + 3)2 — (2 n + I)2 = 8(n + 1) period-points, and for them we have
(n + l)di ~ | w | ^ (n + I)d2.
In the first parallelogram we have in the second parallelogram we have in the third parallelogram we have
J_
w2
1
It follows that
J_
w2
8*1
(2
for the first parallelogram,
J-- for the second parallelogram,
8 • 3
• = — — — for the third parallelogram,
and consequently
1 2 ) 12
32
The series on the right is the well-known divergent harmonic series. We have further
01
= — '— for the first parallelogram,
8-2
w
8-3 (3di):
for the second parallelogram,
for the third parallelogram,
and consequently
3< jMJL ,1 , _3_
di3n3 23 33
which is absolutely convergent.*
* Eisenstein, Genaue Untersuchung, etc., Crelle, Bd. 35, p. 156; Vivanti-Gutzmer, Eindeutige Analytische Functionen, pp. 168 et seq.; Osgood, Lehrbuch der Funktionen- theorie, p. 444.
THE WEIERSTRASSIAN FUNCTIONS pu, &>, all. 313
ART. 267. We may next show that
(u - w)2 is absolutely convergent.
We limit u to the interior of a circle with radius R, where R is arbi trarily large, but finite. With 2 # as a radius a circle is described about the origin. Within this circle there is only a finite number of points w. Any of these quantities w situated within or on the circumference of the circle with radius 2 R is denoted by w', so that
We denote any of the points w without the circle by w" so that
I -../' I \ o P I W I > Z K.
It is clear that
w w' u/
The series
(M - w)2 w2
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
ur
and limit u to the interior of the circle with radius R about the origin as center. We then have u \
rf~' < 2 We also have
1 ( 1
(v/>-u)2 rfvj, ^y
(\ u/') \
and since u I
the expression may be developed in the series
(w"~ or
314 THEORY OF ELLIPTIC FUNCTIONS.
By reducing all the terms to their absolute values we have
1
R
1
(w"-u)2 w"2
The expression in the braces converges towards a definite limit, G, say. It follows that
w
w"
which we saw above is an absolutely convergent series. It follows that
w"
(u - w"}2 w"2
is a finite quantity, and since
r
w'
\
(u - w'Y wft-
is a finite quantity, it is seen that
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
u
(u - w)
w
(w = 2uw + 2/*V:M = 0, ±1. ±2, . . . ; w = 0 excluded) \ jf* I
has only at the points u — w (including w = 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.
ART. 268. In order to show that the function
1 1 )
corresponds completely with the function g?w defined in Art. 264 we must first show that it is doubly periodic.
THE WEIERSTRASSIAN FUNCTIONS W, fu, <m. 315
Since the expression is uniformly convergent,* we may differentiate term by term and have
fu = - - - 2 £' l - 2 Y_J_,
u3 7 (u - w)3 yKu- w)*'
or
<&'u = - 2 V (w = 0 inclusive).
±T (u - w)3
It follows that
From this it is seen that the totality of values on the right-hand side is not altered provided the series is absolutely convergent, and consequently
jp'(M + 2w) = tfu. In a similar manner we have
We have thus shown that the function p'u is a doubly periodic function which is infinite of the third order for u = 0 and for the congruent points. For all other points this function has the character of an integral function. We may prove that the series ]jT - - — is absolutely convergent as follows: As above write
(u — w)3 y (u — w')3 y (u — w")3
The series ^ — — has a finite value if u does not take one of the
values w'. To show that 2) 77-^ is convergent, we note that
w" fa ~" W )
- 1 = 1 1 (u-w)3 ^/^y'
\ wl and since
we have
<•
1
(u - w")
<
8
n" I 3
also, since 5^ - -, as shown above, is convergent for all values of w"
"3
)
except w" = 0, it follows that
*
is absolutely convergent.
(u - w")3
Osgood, Lehrbuch der Funktionentheorie, pp. 83, 258.
316 THEORY OF ELLIPTIC FUNCTIONS.
ART. 269. We have at once from the formulas above
$>'(u + 2co)du = p'udu,
and consequently also
p(u + 2 co) = <@u + c.
Similarly it is seen that
p(u + 2 to') = pu + cf.
Since @u is an even function, its first derivative <@'u is necessarily odd, so that
If then we write — co for u in the formula * above, we have $>co = §?(— co) + c, so that c = 0.
Similary it is seen that c' = 0.
ART. 270. We may derive as follows another proof that <@u is doubly periodic without making use of its first derivative.
The formula
1 , v*'(
1 \_\
w)2 w2 }
becomes, if w is changed into — w,
1
The term which corresponds to w = — 2 co is taken without the summation sign. The sum taken over all the values of w except w = 0 and w = — 2 co is denoted by 2 *. We thus have
1 i 1 1
u2 ( (u - 2co)2 (2co)2) % ((u + w)2 w2
The totality of the values of w under the summation sign is not changed if we write — w — 2 co instead of w. It follows then that
1 1 )
(u-w-2co)2 (w + 2co)2 Adding these two expressions and dividing by 2, we have
i ?
2 co)2 )
2 (u + w}2 (u - w - 2 co)2 w2 (w +
* See Osgood, loc. cit., p. 444; Humbert, Cours d' Analyse, t. II, p. 194.
THE WEIERSTKASSIAN FUNCTIONS ?M, ?M, <ru. 317
In this formula write — u for tt; then since pu is an even function, it is seen that
+ ; i
u2
i 111
(u - w)2 (u + w Finally, changing w into u — 2 w, we have
(ii)
1
((u- w -2w}2 (u + w}2 w2 (w + 2 to)2 Comparing the formulas (II) and (I), it follows that
p(u - 2 w) = $u, or writing u + 2 a> for u,
In a similar manner it may be shown that
pu = p(u + 2o>')-
ART. 271. It is evident from the formulas above that 2a>, 2 a)' form, a primitive pair of periods of the argument of the function $u. The parallel ogram with the vertices 0, 2 at, 2 a>', 2 CD + 2 a/ is free from periods, since all the quantities -a? represent points that are congruent to these four points. If we select the pair of periods 2 M, 2 a}', we may bring them into promi nence by writing pu in the form
If a transition is made to an equivalent pair of periods, we write 2 5 = 2 pw + 2 qw', 2a) = 2p'u) + 2 g'o/,
where pqf — qp' ~~ 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
It is thus seen that jpu remains unchanged by a transition to an equivalent pair of primitive periods.
318 THEORY OF ELLIPTIC FUNCTIONS.
THE SlGMA-FUNCTION.
ART. 272. By integrating twice the gw-function we may derive another important function. It is clear that
or - Cpu du = - + V ' \ — — + ~ I + Constant.
J u ** (u — w w2)
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
w
we shall show that this expression is absolutely convergent and becomes infinite of the first order only at the points u = 0 and u = w. It is seen that
u — w w
w
w
u — w w w2 w3 w w2
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 — / @u du cannot
be doubly periodic.
Integrating again the above expression we have
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 &U, &, <ru. 319
The quantities w we again, Art. 267, distribute into two groups, so that
We then have
2R,
1C'
>2R,
<l
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
1/1 u\ u 1 1 u\2 l I u \3 10g( l-^)=-^-^l~^)~
it is seen that
which is an absolutely convergent series (Art. 268). It follows that
— I du I pu du
is absolutely convergent for all values of u other than u = 0 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
-fdufpudu
au = e
This sigma-f unction 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 w = 0 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
The function ou is not doubly periodic. It has like the theta-f unctions for all finite values of u the character of an integral function and may be expressed as an absolutely convergent power-series with integral positive exponents (Arts. 276, 336). Like the function pu it is not changed when a transition is made from one pair of primitive periods of the function <@u to an equivalent pair.
ART. 273. Historical. — Eisenstein (in Crelle's Journal, Bd. 27, p. 285, 1844) formed the product
where A and A' are quantities such that
4; = a + ip (/? ^ 0),
-A
while n and /*' take all values ± 1, ± 3, ± 5, • • • ; and on page 287 he formed the products
(>U')= ±2, ±4, ±6, • • . , P = ± 1, ± 3, ± 5, - - - .
On page 288 Eisenstein 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 ubrigens so elementar Natur, dass sie sich wohl eignen mochte, den Anfanger in die Theorie der elliptischen Functionen einzufuhren . ' '
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, Camb. and Dublin Math. Journ., 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 WEIERSTBASSIAN FUNCTIONS VU, Jfc, ru. 321
It may be observed that the above remarks are applicable also to the infinite products of Abel (Recherches sur les fonctwns elliptiques, Crelle, Bd. 2, p 154; (Euvres, t. I, p. 226) and of Jacobi, Fund, nova, § 35; Werke, I, p. 141.
Professor Klein, Theorw der elliptischen Modulfunctionen, Bd. I, p. 150, calls attention to the fact that the quantities pu, p'u, g2, #3, e\, e2, e$ are defined by Eisenstein, Genaue Untersuchung der unendlichen Doppel- produkte, aus welchen die elliptischen Funktionen als Quotienten zusammen gesetzt sind, Crelle, Bd. 35 (1847), pp. 153-274, and Mathematische Ab- handlungen, pp. 213-334.
We also note that the relation
is the identical relation given by Eisenstein, Crelle, 35, p. 225, formula (5). On page 226, Eisenstein derives the normal integrals of the first and second kinds in the forms
f d" and - C
J 2 V(u- a\(n - a'Mu - a"} J '
2 V(ij- a)(y - a')0/ - a") ^ 2 V(y - a)(y - a')(y - a"}
It also appears from this paper that Eisenstein had some idea of the nature of the quantities g2 and #3 whose invariantive properties were discovered by Cayley and Boole in 1845.
Weierstrass, recognizing the true nature of these invariants, was the first (cf. Klein, loc. cit., p. 24) to make the Theory of Elliptic Functions from the standpoint of the infinite products and series as given in this Chapter (and developed by him) of consequence, and so he is to be considered the founder of this theory.
In his last lectures Professor Kronecker, Theorie der elliptischen Func- tionen zweier Paare reeller Arguinente (W. S., 1891), especially empha sized the Eisenstein theory and made paramount a certain function En (denoting Eisenstein's name) which is a generalized ^-function.
ART. 274. \Ye saw in Chapter I that the infinite product
(1 + av) is absolutely convergent if
v= 1 v=l
a,
is absolutely convergent.
To prove the absolute convergence of the infinite product through which the sigma-f unction is expressed let \u < R, \ w' \ = 2 R, \w" \ > 2 R as above. We omit from the infinite product all those factors which correspond to the quantities w'. Such factors being finite in number exercise no influence upon the question of convergence.
322
THEORY OF ELLIPTIC FUNCTIONS.
The factors remaining in the product are of the form
U I V?
1 U2
Since
or finally
< 1, we may develop the logarithm in a power series and have
z
_jU__ljM2__ljW3__ ... +^-+- "2
1 V3 ii 3ji_4.3_^- +
~ 3 w773 4 w" 5 w"2
or
Since
-, this expression is
< e'
w"
and consequently
«;
1.2
w"
It is thus seen that the quantities in the sigma-function corresponding to aw above are such that
or finally
It follows that
'
16 16~2
.16
<15
to"
t
" 16
which we saw above was absolutely convergent. To the S |av| we must add the quantities |ov| which correspond to the quantities «/; but the convergence is unchanged by the addition of these terms. It follows 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 u = 0 and u = w and at these points au is zero of the first order.
THE WEIERSTKASSIAN FUNCTIONS &u, &, <ni. 323
ART. 275. Other properties of the sigma-f unction may be developed as follows:
We have i
w
If w is changed into — w the product is not altered, and we have
It follows that
a(— u) = — au,
and consequently the function au is an odd function.
ART. 276. We shall consider next more closely the form of the develop ment of au. In the product
U III*
we join any two factors that correspond to opposite values of w and thus have *
the star denoting that of every pair of values w and — w only one value is to be taken. It follows that
If u is chosen smaller than any of the values w, we may write
1 U^ 1 U^
and consequently
or
* Cf. Daniels, Amer. Journ. Math., Vol. 6, p. 178.
324 THEORY OF ELLIPTIC FUNCTIONS.
We may write 02 Q c V *
• • 5 2, - 22,
where, as will be evident from the sequel, the quantities g2, #3 are the invariants introduced in Art. 184. It is also evident that g2 and g3 remain unaltered when we pass from one pair of equivalent primitive periods to another pair. It is seen that
the star indicating that the term with u3 is wanting. The function ou 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 ^-FUNCTION. ART. 277. From the formula just written it follows that
log an = log j u - ^75 "'" 2» . 3^ 5 . 7 "' " ' ' ' j
It is evident from the consideration of the product through which ou 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
^ = I + * _ , _ 22—^3 __ 23 _ U5_ .... au u 22 • 3 • 5 22 • 5 • 7
The quotient ^^ is often denoted by £u (Art. 258, see Halphen, Fond. au
Elliptiques, t. I, Chap. V).
Differentiating this expression again and multiplying by — 1, we have
„.-*+. + *.
The series through which pu, <@'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 &U, fy, <ru. 325
The functions %>u and p'u are, as we have already seen, doubly periodic, yu being an even and <#u' an odd function. The function (p'u)2 is an even doubly periodic function of the sixth degree and is infinite of the sixth order at the origin and all congruent points.
ART. 278. We may next prove that %>u satisfies the differential equa tion of the first order *
(g/w)2 = 4(&m)3- g2 $>u - gs- We have
5"u' 7'3 + (("2))
and
It follows that
and also that
- g3
We note that the left-hand side of this expression is doubly periodic, while the right-hand side has everywhere the character of an integral function. By the theorem of Art. 83, such a doubly periodic function must be a constant, and as there is present no constant term, the right-hand side is zero. We therefore have as our eliminant equation
(p'u)2 = 4 yPu - g2 pu - g3.
ART. 279. If in the above equation we use Weierstrass's notation and
put pu = s, and $>'u = — ^-, Art. 256, we have
du
or
u= ± I
v 4 s3 — g2s - g3
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 = oc for u — 0, we may write this integral in the form
JBL..JSL
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 ^ we mav expan(i the integrand in a power series and
4s3 then integrate term by term. We thus have
or u = —
v It follows that
All the coefficients of this power series are clearly functions of g2 and g3 with rational numerical coefficients.
When this series is reverted, it is seen that — may in the neighborhood
of the origin be expanded in powers of u\ and it is also evident that s = <@u may be expanded in the neighborhood of the origin in a power- series whose coefficients are integral functions of g2 and g3 with rational
numerical coefficients. The functions ra = , ' and loge ou have the
o(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 ou, so that the development of ou in the neighborhood of the origin is such that all the coefficients are integral functions of g2 and g% with rational numerical coefficients. The sigma-function is therefore a function of u, g2, g%. A method of determining the coefficients of ou by means of a partial differ ential equation is found in Art. 336.
ART. 280. It follows from the equation above that
or ,.
a2 r 20 «2 28
Hence as an approximation (up to terms of the order ue) we have
If then on the right-hand side of the last equation we write — for s, we
have
Writing <@u = - + * + c2u2 + c3u* + c4u6 + • • • + ctu2*~2+ - • - , it follows that c2=A02 and c3= &g3.
We shall express the other constants c4, c5, . . . through these two quan tities.
THE WEIERSTRASSIAN FUNCTIONS &u, &, *u. 327
From the relation
(#>'u)2 = 4 $*u - g2&u - g3
we have through differentiation
2 tfu tf'u =12 p-u <$'u — g2&'u, or, if we give to u such values that $'u ^ 0,
9»u = 6 $2u - & • (Eisenstein, Crelle, Bd. 35, p. 195.)
Multiplying through by u4 we have
(A) u*v"u = 6 u*j?u - J sr2M4.
From the equation
-. 4- ii
it follows that &'u = - 2- + * + 2c2u + 4c3u3+ • • • +(2 J - 2
or
We also have
W2§?U = 1 + *
or u2$w — .-.. and consequently
v=
where we have written down only the terms that contain u2*.
Writing these values in the equation (A) above and equating the coefficients of u2*, we have *
v=2
v = A-2
(2 J + 1XJ - 3)
This is a recursion formula by means of which each of the coefficients ca in the development of %>u 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 FUNCTIONS
We have, for example,
or, since C2 = ^V 92, it follows that
1
24-3.52l/; and similarly
c = 3 ff2!73 5 24- 5- 7-11'
Cft= -J_/22! + ^ \ 6 24-13V7 2.3-5V'
We may therefore write
GU = u + * _ _^|_ _ _^_ ^7-
ART. 281. We saw in Art. 268 that
If we make the condition that \u\ < w, we may write
w1
This equation differentiated with respect to u becomes
_L 4. ?Jf , . . nun~l
(w - u)2 w2 u It follows at once that
"
We note that all terms in which it? appears with an odd exponent vanish, since a value — w belongs to every value 4- w.
THE WEIEKSTRASSIAN FUNCTIONS yu9 £u, an. 329
If then we write n — 1 = 2^ — 2, or n = 2 / — 1, and compare the above expression with
it is seen that
It follows from the results of the preceding Article that Jj — r; may be
integrally expressed in terms of <?2 and gr3. This is a very remarkable fact (cf. Halphen, Fonct. Ellip., t. I, p. 366). In Art. 272 we saw that
— _ ^2 log ou du2
= ±(- 2^\
du \ ou /
or OILO"U - (o'li)2
(ou)2
The function ou is uniformly convergent for all values of u in the finite portion of the plane. The same is true of o'u and o"u. Hence it is seen that %ni may be expressed as the quotient of two power-series that are uniformly convergent for all values of u in the finite portion of the plane. We saw in Chapter XI that the functions sn u, cnu, dnu have the same property. In Arts. 262, 324-326 we consider the analogues of these three functions in Weierstrass's Theory.
ART. 282. Another expression for the function pu. — We write (cf . Art. 60)
and we shall first derive a function of t which behaves at the origin in the same manner as pu. The development of t in the neighborhood of u = 0 is
co 1 '2\co ,
or
1>2 o> 1 -2 -3
We note that t — 1 becomes zero of the first order at the point u = 0 and at all other points where t has the value 1. The totality of all these points is expressed through
u = 2[j.uj(u. = 0, ± 1, ± 2, • • • ). The function
co 12\ co I becomes zero of the second order at all the points u = 2
330 THEORY OF ELLIPTIC FUNCTIONS.
Let g(t) be an integral function of t which does not vanish for t — 1. The function
(t - I)2
will therefore be infinite of the second order for the value u = 0 and for all the values u = 2 /*to. Hence this function behaves at these points in the same way as does the function yu.
We may write g(t) = a + bt + ct2, a, b and c being constants. It follows that
g(t) = a + bt + ct2
(t - 1)2 (t - 1)2
Uiri
Since t = e " , it is seen that t2 may be derived from t by writing 2 u in the place of u in the expansion of t. Accordingly we have
(t - 1)2
+^+ W^\2+. . .]+c[i+2tm + J_^rmV+ , . 1 u) I »2\ co ] J |_ a} 1 »2\ a) ] J
_ ^27r2["-i , uni , 1 /U7ii\2 . ]
~^t "IT auy " J
We wish that the following conditions be satisfied : First. The term which becomes infinite of the second order must be of
the form— -• u2
Second. The term which becomes infinite of the first order must not be present.
Third. The constant term in the development of the function in powers of u must be zero.
To fulfill the first condition we must have
_ ^2 a + 5 + c = j^ ^
2 2
U U
for the second condition, we must put
OJ2[, xi 1 , 2cxi I f . , N id 1") n — — o -- -+. - ---- (a + 6 -f t;) -- = 0, ^2 [_ w ^ a; w w wj
or c - a = 0.
THE WEIERSTRASSIAN FUNCTIONS yu, ty, au. 331
From the first condition it follows that
/t\ These values substituted in — ^ ' cause this function to become
(t - I)2 W2 (t - I)2
a.
The constant a must be so chosen that the third condition above may be satisfied.
We note that
24 and since
it follows that
12
and consequently that the third condition may be satisfied, we must have
Noting that ^ _ t-±
2i it is seen that _2
12 0,2
= sin
=fl_
We have thus shown that the function
LVr_J_ .11
Sin2ȣ 3 2w
corresponds in its initial terms with the development of pu, so that it differs from $>u only in quantities which become infinitesimally small of the first order when u becomes indefinitely small.
332 THEORY OF ELLIPTIC FUNCTIONS.
ART. 283. We had
1 ]
The quantities w may be distributed into two groups. The first group contains all values w for which // = 0, so that w = 2 /j. a). The second group contains those w's for which // S 0, so that w = 2 p oj + 2 /*'o/. If then we let a>' become infinite, the values w of the second group become infinite, and we have
It is seen from Art. 22 that this expression is none other than the function
(t - I)2' If then the period 2 a/ becomes infinite, the function $ru is represented by
1
"3
AKT. 284. We shall next write (cf. Eisenstein, loc. tit., p. 216)
F(t) = - — _ - _
OJ2 (t - I)2
and we shall seek to express * %>u through t even when the second period
Ttiu
2oj' is finite. F(t) being a rational function of t = e "• remains un changed when u is increased by u + 2 co; but when u is increased by 2 a/
then e w = e w <. Weierstrass used the letter h to denote the quan-
tity e w , which Jacobi denoted by q. In Art. 86 we wrote — = a + i/3,
0)
where ? > 0. From this it is seen that
q = = and consequently | ^ | = e-0*m
Noting Art. 81, it is evident that we may always choose a pair of primitive periods so that \h\ < 1
Since t becomes h2t when u is increased by 2 a/, it follows that when u becomes u + 2 a/ F(Q becomes
becomes becomes
See also Halphen, Fonct. Ellip., t. I, Chap. XIII.
THE WEIEKSTKASSIAN FUNCTIONS <WL, &t, <M. 333
If we consider the infinite series
(Sr) F(t)+ F(h2t)+F(h*t) + .. • - + F(h2nt) +
then, if u is increased by 2 a/, each term becomes the following term. Hence the series
+ F(h2t) + F(h*t)+ •'... + F(h2nt)+ • . .
is a doubly periodic function having the two periods 2 a), 2 CD'. At the point u = 0 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 u 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
*v = efev, it is seen that
1 1 1 = e",
so that | 1 1 becomes a maximum with //, that is, with R(— j- If we put u = to', then is \ w /
\ w
If M is the greatest value that R() can take for values of u within
OJ
the region in question and m the smallest, it is always possible to find an integer no, say, such that
— no ,3/r < in
and M < n0 /?-.
Hence for this region there exists the inequality
Q / n/u~i\^ o — no-j" <•» i«j — j <, noJ/T,
\ OJ /
and consequently, since | h = e~^, it foUows that
—
Since F(t)= — — • -, it is seen that in the first term of the in-
oj- (t - 1)- finite series (S') there appears (1 — t)2 in the denominator; in the second
334 THEORY OF ELLIPTIC FUNCTIONS.
term there appears (1 — h2t)2 in the denominator; in the third term there appears (1 — h4t)2 in the denominator; • • • .
The greatest absolute value that t can take within the fixed region being < |^~"o|, the greatest absolute value that h2nt can take in the same region is < | h2n-^ \ . If then we choose 2 n = n0, then is | h2nt \ < 1. In the series (SO we separate from the remaining series those terms (finite in number) in which h occurs to a power less than UQ.
The denominator in any of the remaining terms is
where
and consequently
We therefore diminish the denominator of the terms in question if instead of (1 — h2*t)2 we write (1 — | hn° |)2, and consequently we increase the value of the term F(h2H).
The numerators of the terms which have been separated from the first UQ terms are
h2n<>t, h2n»+2t,
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
F(t)+ F(h2t) + F(h*t) +
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") F(t)+ F(h-2t)
We may write
- h2t~1)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 &u, &, all. 335
It follows that the series
+ F(h2t}- (S) F(t)
+ F(h~2t) +
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 doublt/ 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 = 0 o/nd the congruent points this func tion becomes infinite of the second order.
ART. 286. We note that F(0)= F(*)= 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 — -*-•
Further, put z2 for /, retaining the notation of Weierstrass, as no confu sion can arise between the z used here and the z formerly employed.
MTTt
It follows,* since z = e2(a, that
*> = Y\ 17 = 3f»
h2nz~2
aj aj2((z - z-1)2 mtt (1 - h'2nz~2}2 £( (1 - h2nz2}2
where h = e ™ = q.
In order to determine the constant y, it follows, when we expand
urt
z = e*" and
that
and consequently
3-4
We note that
u2 3 • 4 a>2
* See Schwarz, Formeln und Lehrsdtze, etc., p. 10.
336 THEORY OF ELLIPTIC FUNCTIONS.
If we write this value in the above expression for @u, we have
a) u2 12 w2
It follows that *
! ,2 ^"g 2fr2n _
w 12 w2 w2nf?[(l -/i2")2
or
The above expression for §?w is not unique, since the period 2 a> 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 — du, it follows through integration that
£u = -(u)=±-+ * a u
, 2h2nz~2 2h2n
-h2nz~2 1 -h2n
y ^ 2h2nz2 2h2n n
3i ll -h2nz2 1 -h2n\]
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
, 2n~2
tt + -_— +
If with Eisenstein (loc. tit., p. 215) we note that -2 2h2nz~2 zh
- h2nz~2 l-h2nz~2 zh~n - z~lhn
* Schwarz, loc. cit., p. 8. t Schwarz, loc. cit., p. 10.
THE WEIERSTKASSIAN FUNCTIONS f>u, &, <ru. 337
and further that
ni— £
= e », zh~n = e2
7T Z — Z~l
u' 2'
it is seen that the above expression may be written *
' r n=x (
£U = ^(U)=1U+ JL cot-^i + V ]cot^-(u- 2nu)')- a w 2aj\_ 2a> £?1 ( 2a>
n = x ,
f+ T ]cot-^-(u + 2 ncof) + n=l ' 2 ^
It is evident that the constant term of the series is zero; for if u is changed into — u, the right-hand side of the series takes its opposite value and is consequently an odd function of u. If u is increased by 2 a>, the quantity z becomes — z, for
e It follows at once that
-(u + 2aj)= 2 r) + - (u), a a
or £(u + 2w)= ^u + 2 fj.
Writing u = — w in this formula we have (cf . Art. 258)
tu = y,
where T? is finite since — (a>) is finite. a
We saw that
p(-u + 2,0)')= <$u.
Multiply both sides of this formula by — du and integrate. It follows that
-(u + 2co')= ^.(u)+2r)f, a a
or £(u + 2aj')= £u + 2 if,
where r[ is the constant of integration. Again writing u = — a>f, we have
By interchanging w and a)' in the preceding Article, it may be shown that
where h0= e u/.
* Schwarz, loc. cit., p. 10; see also Halphen, Fonct. Ellipt., t. I, p. 425; Tannery et Molk, Fonct. Ellipt., t. II, p. 237.
338 THEORY OF ELLIPTIC FUNCTIONS.
From the formula
w=--lgau,= --f £(U) du G
it follows that (cf. Art. 258)
^u = °~(u)=-f\udu= r**»,
a J J VS
where %>u = s and du = -- ^4- •
VS
The constant of integration on the right-hand side is so chosen that for sufficiently large values of s the series on the right-han%ide is (cf. Art. 279)
p^
J VS
VS L 24 s2 40 s3
uiti
ART. 288. If u is increased by 2 a/, then 2 = e2"" becomes z*h. We consequently have
c(* + 2o/) = c* + 2 ?'= att + *u 21 ^"^" : +
o» 2a)(hz — h lz n=1 x - ,«, -* -
i 2 rX
-(- — « — .
Comparing this formula with the one given above for £u, we note that here under the first summation the new initial term is
{-* Q -
- — -, which may be written = ^ 1,
1 — z~2 z — z-1
and consequently the first summation is transformed into
z + z-1 _ 1 z — z~l
while the second summation
becomes
We further note that
hz + h-^z~l h2z2+ 1 h2z2+
hz-h-tz-1 h2z2-l l-h2z2
It follows at once that
C(u + 2cw)= C^ + 2^= ^ + ^^ + -'i - ^L+i- 1 + -^
w 2 ( 1 - h2z2 1 -
THE WEIERSTRASSIAN FUNCTIONS VU, &, au. 339
so that
or finally (cf . Art. 259)
We have assumed always (Art. 86) that R ( — ) > 0.
\<Mj
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-parallelogram with vertices 0, 2 a>, 2w', 2uj" = 2 oj + 2 a/.
By sliding this parallelogram parallel with itself, it may be caused to take u0^^'=u3 / u0^ul=us
a position such that for all points on / / /
its boundarv and within the interior * *— — ^ —
/ 0 /
(except the point u = 0) the function / /
£u has the character of an integral u0 w0+2
function, being of the form Fig. 71.
It follows that
lu = 27ri,
where the integration has been taken over a small circle about u = 0. Since this integral is the same as that taken over the parallelogram , we have
or
2m = ^u - £(u + 2a)')}du
- 2 T'di^ + 32 7 <fa = - 4 T'
'udu+ l£u<fuj
*/ Us
ART. 290. If we multiply by dw the expression — (u + 2w)=-(u)+27/,
C7 (7
we have through integration
log <J(M + 2 o>) = log <TU + 2 TJW + c, or
340 THEORY OF ELLIPTIC FUNCTIONS.
If the value -co is given to u, it is seen that
gC_ _ 'e2^.
We consequently have
o(u + 2a>) = — e2^u+^a(u).
If — u is written for u in this formula, we have
o(u - 2w)= - e~2^u-^ou. Combining these formulas into one formula, we may write
(A) a(u±2a>) = - e±2*(t«±«> a(u). In a similar manner it may be shown that
(B) o(u ± 2 a>') = - e
Further, if 2 to = 2 pco + 2 qa*' ', where p and q are positive or negative integers (including zero), it is seen that
a(u + 2w) = <7(u + 2paj + 2 Writing
2py + 2qi)'= 2 •it follows that
a(u + 2 a>)
To determine the constant C, write u = — co + v, where v is a very small quantity. It follows that
o(to + v) = — Ce-27'"+2w<7(a) — v). If we develop by Taylor's Theorem, it is seen that (C) o(w + v)= a(oj) + va'(a>)+ - • - - - Ce-2w+2*vo
Two cases are possible:
(1) either | a(a>) > 0, or
(2) ,7(5) | - 0.
In the first case we have by writing v = 0,
<r(5)=- Ce~2^a(a)). It follows that C = - e2**,
and consequently
a(u + 2 w) = - e2~r<(u+^o(u}.
In the second case we have by developing both sides of (C)
<?'(£)+ (M) - C or by making v = 0,
C = It follows that
<r(w + 2 5) = ±
according as we have case (2) or case (1) respectively.
THE WEIERSTRASSIAN FUNCTIONS yu, &, <rii. 341
The quantity a (at) vanishes when p and q are even integers. We may therefore write the general formula
a(u + 2pa} + 2qaj') = (- ART. 291. We derived in Art. 287 the formula
which is uniformly convergent within the period-parallelogram (vertices excluded). If this series is integrated term by term, it follows that
log „
When u = 0, we have z = 1, so that
[log <™]u=0= C + log sin ^-
L 2 wju=
|f^ +((«=>))
L2^ Ju=o
It follows that *
and
where ?i = 2 wv.
Writing — = 7, it is seen that
CO
1 — h2nz~2 = sin[(r — m)-] _l = sm[(rn — r)~] _1 I _ h'2n h~n — hn sin TIT- ~Z '
2i
with a similar formula for -^--
It follows that f
(2) ou - ^-' ^ sm v* n sin[(nr~ rkig-^- n ^i^±_
- „ sinrir^- „ sinnrr
or
n=i
* Compare this function with Eisenstein's x-function, loc. cit., p. 216. t Schwarz, loc. cit., p. 8. Formulas (2) and (3) are precisely the same as those derived by Jacobi for H(u) [Werke, I, pp. 141-142].
342 THEORY OF ELLIPTIC FUNCTIONS.
The formula (2) may be written (3) ~-
Since 2 aj 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 n=00
TT 2r h2nsm 2 vn
10 , u = — cot vn + 2 vv H --
2 a) co nr{ I - 2 h2ncos 2 wr + h4"
Noting that
- L_ = 1 _|_ u 4- U2+ . . . + Um+ - • :. ( I U I < 1),
it is evident, if u = r(cos 6 + i sin 6), that
- Srsintf - _ mv 2 rm sin m^ 1 - 2 r cos ^ + r2 ^
an identity which is true for complex as well as for real values of r. If we put r = h2n, we have
ii--'i 2
and consequently
If we differentiate with regard to u, we have (A) p
4 OJ2 6t» 6t>
The right-hand side of this equation is
1 . (7o o | ^7j
g?tt = — + **• U H~
while the expansion of cosec2 t is
By equating like powers of u on either side of (A), we have *
* Harkness and Morley, Theory of Functions, p. 321; Halphen, Fonct. Ellipt., t. I, Chap. 13
THE WEIERSTRASSIAN FUNCTIONS <?u, &, <ru. 343
ART. 292. Homogeneity. — Write the functions an, £u, pu in the forms o» - o(u', co, a/) = a(u; g2, g3),
It follows at once from the infinite product through which the function au is defined (Art. 272) that
where A is any quantity real or imaginary. We also have
and consequently
p(/w; ko, )*}')= — In the formulas
when w and o>' are replaced by Xco and Aw', w becomes Xwt so that g^ and are transformed into
2| and 23. x4 /6
It is also seen that
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 a/= oo, we saw in Art. 283 that
We further have
From Chapter I we have
32 -o m6 33 .5.7
344 THEORY OF ELLIPTIC FUNCTIONS.
It follows that
= i/jELy = i / 7T2 \3
and consequently . 3 _ 2_
The discriminant being zero, the roots of the polynomial
4s3— g2s — g3= 0 = 4(s — ei)(s — 62) (s — 63) are equal. Further, since
61+62+63=0 and ei>62>e3,
the quantity e\ must be positive and 63 negative.
Two cases are possible: either e2 coincides with 63, or e2 coincides with e\. In the first case: 62= 63= — \e\\ g2=3ei2, g3= 6i3, g3> 0;
2
We also have
o'u x , nu .If x \2
u = — = — cot -- — ( — j u,
on 2co 2aj 3\2ajj
l/Jrt*\2
6\2^j 2<y . TTU
GU = 6bV ' Sin
In the second case: e2 = e\= — \e^\ g%<. 0,
92= 3e32, ^3= 633; k = I, snu = ~_u, K =00, w = oo.
6 ~t~ e
9 2, where , = t«
, where
, , T?
V--,
in 2
When the roots of the polynomial
4s3- g2s - g3= 0
are equal, it may be shown directly that the values of s = @u derived from the integral
a) u
V4s3- g2s- g3 agree with the results above-
THE WEIEKSTRASSIAN FUNCTIONS &u, &, <m. 345
When both periods are infinite, then g2 = 0 = g$ and e\ = 0 = e<i = €3. The integral (1) becomes
u= -=, or 5 = =
- r d±_
Js vTs*
- , ou = u. u
EXAMPLES
1. By making a/ = oo in the formula
~ + 2
derive the results of Arts. 283 et seq. (Halphen, loc. tit., Chap. 13).
*. K /- f ds ,
Jo V± s3 - a,s
show that
3. If F(f) is any rational function of t = c ™ , such that F(0) = 0 = F(oc), show that
n=l 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 <&(u) 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
(I)
2iK') = e K^
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
@2 (w), H2(w) and ®(u - v)@(u + v),
where v is an arbitrary parameter. It follows that
C®(u + v)®(u -v) + Ci02(w)+ C2H2(w)= 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
5^ = Constant,
8(«)
which is not true.
* See Hermite in Serret's Calcul, t. II, p. 797; and Koenigsberger, Elliptische Functionen, p. 368.
346
THE ADDITION-THEOREMS. 347
Writing
we have
B(u + v)®(u ~v) = f(v) ®2(u) + g(v) H2(u).
If we consider f(v) G2 (u) + g (v) H2 (u) as a function of v, say ¥(v), we have
It follows that
¥(i? + 2K)= ¥(r) and
2;rt
-rir* *V(i'),
from which it is seen that "^(v) satisfies the functional equations (I). If we write v + 2 K in the equation
(II) 0(W + V) 0(u - v) = f(v) 02(w) + gr(v) H2(u),
we have
0(t* + v)Q(u - v) = f(v + 2K)02(^)+ g(v + 2K)H2(u); and consequently through subtraction it follows that
[f(v + 2 K) - f(v)] 02(M) + [gr(y + 2 K) - </(iO] H2(M) - 0. As this relation is true for all values of u, we must have
/(• + 2JO-M
0(0 + 2/0=0(1;).
On the other hand, if in the equation (II) we write v + 2 i7£' for v, we have hi a similar manner
g(v + 2 iK')=e~*
It follows that /(v) and g(v) satisfy the functional equations (I) that were satisfied by 02(u) and H2(w). We thus have the following relations:
g(v)
where a, /?, 7-, ^ are constants.
When these relations are written in the equation above, we have
(1) 0(w + v)B(u - r)=
348 THEOKY OF ELLIPTIC FUNCTIONS.
To determine the constants a, ft, ?, d, write v = 0. We then have
G2o)[i - /?e2(o)]= £02(o) n2(u),
a relation which can exist only if
1 - /?@2(0) =0 and £B2(0) = 0.
We thus have
1
and *-
If next we write w = 0 in the above equation, we have a = 0. To deter mine 7-, we write the values of a, ft, d just found, in (1), then write u = v + iK' and note that ®(iK') = 0. It follows that
@2(0)
These values of a, ft, f, d when written in the equation (1) give us the formula
v)®(u - v)=S2(v)(d2(u)-R2
which is fundamental in the Jacobi theory (see Jacobi, Werke, I, p. 227. formula 20).
ART. 295. We introduced in Art. 208 the followin notation:
H(2 Ku) =
We also saw in Art. 215 that
j_ =Qi(Q)
Vk Hi(0) /r, ^ 0(0)
and in Art. 217 that
The addition formula above for the function 0 may be written (1) &02&o(u + v) &0(u -v}=
THE ADDITIOX-THEOKEMS. 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
(3) &
(4)
All four of the above formulas were also derived in the table (C) of Art. 211. ART. 296. If we divide equation (2) above by (1) we have
that is,
#0
or
sn[2K(u + v}~\= sn ^ ^u cn ^ ^l dn2 Kv + cn2 Ku dn 2 Ku sn 2 Kv
If we divide the equation (3) by (1) we have
cn 2 Ku cn 2 Kv - sn 2 Ku sn 2 Kv dn 2 Ku dn 2 Kv
1 - k2sn22Kusn22Kv and similarly when (4) is divided by (1) we have
dn\2K(i + 01 = dn 2 Ku dn 2 Kv ~ k2sn2 2 Ku sn 2 Kv cn 2 Ku cn 2 Kv
1 -k2sn22Kusn22Kv
If we write u and v for 2 Ku and 2 Kv, we have
sn(u + v} = snucnvdnv + cnudnusnv
1 — k2sn2usn2v Further, since ,
—-snu = cn u dn u, du
it follows that dsnv^ d sn u
sn u f- sn v
. r> •> o
1 — K~sn~u sn^v
We have thus shown that sn(u + v) is a rational function of snu, snv and the first derivatives of these functions (see Art. 158).
Remark. — If for brevity in the formula above we put sn u = s, snv = s'; cnu = c, cnv = c'; dnu = d, dnv = d', it becomes
350 THEORY OF ELLIPTIC FUNCTIONS.
We further have
cn2(u + v)=l- sn2(u + v)= ^
(1 - k2s2sf2)2 (ccf- ss'dd'}2
so that
Cn(u + V) = ±c-^-ss'dd'
Writing v = 0, and consequently s' = 0 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) ®(u + v) ®(u -v)= ®2(u) ®2(v) - we have at once
This formula differentiated logarithmically with respect to u and v respec tively becomes
®'(u + v) . ®'(u - v) _ 2®f(u) ==_ 2 k2sn u cnudnu sn2v ®(u + v) ®(u-v) 9(u) ~ 1 - k2sn2u sn2v
®'(u + v) ®'(u - v) 0 Qr (v) = 2 k2sn v en v dn v sn2u ®(u + v) ®(u - v) ~ " ®(v) ~ 1 - k2sn2usn2v
Through addition we have
Since
it fpllows that
Z(u + v) = Z(u) + Z(v) — k2sn usnv sn(u + v).
Noting that
Z(u)=E(u)-uL
A
we also have
E(u + v)= E(u) + E(v) - k2 sn u sn v sn(u +
THE ADDITION-THEOREMS. 351
ADDITION-THEOREMS FOR THE WEIERSTRASSIAN FUNCTIONS.
ART. 298. The addition-theorem for the ^function may be derived as follows: We note that the difference
is a one-valued doubly periodic function which becomes infinite of the second order at the origin and the congruent points. For all other points this difference is finite. The points u=±v + 2luaj + 2 //&/(/*, // integers) are the zeros of the function <$u — pv.
Another function which has the same zeros is
0U
Further, since
a(u + 2 w) = - e2 '("+"> au, o(u + 2 a/) = - e2 »'(«+"') <ru, it follows that
and <j)(u + 2co')
We note that the functions $(11) and pu — $>v have the same periods.
The developments of these functions in the neighborhood of the origin are
pu - &v = — - §>y + (O2)),
It is further seen that the function
au
is doubly periodic and becomes infinite in the same manner and at the same points as <@u — $rv. Other developments are
fy
o(v + u} = av + ua'v + — o"v + • • - , Z
G(— v + u) = — av + ua'v - — a"v + • • • , , M = a2v + [ov<j»y -
02(V)U2+((U*})
or
U 0V
352 THEOEY OF ELLIPTIC FUNCTIONS.
Since
o£v we may write
#i(t*)--5-
This value substituted in
— ®v — <f>i(u) = .Constant,
shows that the constant is zero. We therefore have
a formula of great elegance and importance.*
ART. 299. If the formula above be differentiated logarithmically respec tively with regard to u and v, we have
(A) - (u + v) + — (u - v) - 2 - (u) = o a o
and
(B) - (u + v)-- (u -v)-2~(v) =
o o a <@u —
Through the addition and subtraction of formulas (A) and (B) are de rived the formulas f
(C) s (M
and
•o a a Z <@u — %>v
< —
These formulas are the addition-theorems for the function — (u) = £(u).
a
Compare them with those given in Art. 297. The function %u does not have an algebraic addition-theorem. J
If we differentiate again the formula (C) with respect to u and v, we have
(E)
= _
2 (pu - gw)2
and
(F)
* See Schwarz, loc. cit., p. 13.
t Schwarz, loc. cit., p. 13.
j Daniels, Amer. Journ. Math., Vol. VI, p. 268.
THE ADDITION-THEOREMS. 353
It follows, since
- g2 &ti - g3 and p"u = 6 g?u - J g2, that the formula (E) becomes
(E') g>(w ± v) = ou - ^U
while formula (F) may be written (F') o(u ± v) = &v + fcM
- pi;)2
ire /love thus expressed $>(u ± v) rationally through $>u, pv, p'u, $>'v (see again Art. 158).
ART. 300. Through the addition of the formulas (E') and (Fr) we have
(G) &(u ± v) = 2(VU<?V
2(<pu - yv)2
The function p(u + v) is only infinite if u is equal or congruent to — v. Since #>w is finite at this point, it follows from the formula
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 follows that
and that
du ( yu —
Noting these results we may obtain another formula for p(u ± v) as follows: The function
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 u = 0 the function becomes
infinite as — • If then we add to the above expression the function
u2 v?w, we have a doubly periodic function which remains finite everywhere
354
THEORY 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 — <@v. We may consequently write
v 1 \v'u =F 6/vl2
u ± v) = - • -!-*— - pu - gw.
4 I pu - $>v J
ART. 301. If in the formula just written we put u + v f or u and — v for v, we have
It follows that
+^
- gw J
/fri + v)+ p'y (u + v)- §w
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, pu, 1, gw,
= 0.
By differentiating with regard to v the formula
/ \ 19 fp'u -F (ip'v
(^ -t v)= &u - _Ji - L_
2 6w ?w — ?v
we have
2 du dv
Remark. — If in the formula (Fx) of Art. 299 we write w in the place of v and observe that
-03=4
0,
it is seen that or
From the relation
it follows that
and consequently that
= 6
THE ADDITION-THEOREMS. 355
Further, since
- e2)(<?u - e3), and therefore also tf'u = 2[(pu - ei)(&u - e2) + (pu - ei)(pu - e3) + (pu - e2)(pu-e3)],
it follows that
$>"a) = 2(ei- e2)(ei- e3).
We consequently have
$>u - el and similarly (u ± ^ _ ^ = (e2- ejfa- e3) ,
ART. 302. The reciprocal of formula (G), Art. 300, is
± v) 2($>u&v - J g2) (pu + &v) - g3
Noting that
(&'uY= 4 $Pu - g2pu - g3 and (p'v)2 = 4 g>3v - g2?v - g3, it is seen that
gw) - Q3]2 - [4 g?3?^ - g2<?u - g3] [4 $3v - ^2^?' - fl'a]
and consequently
ff(u ± v) If we write *£ = v, we have (2i0=
4 p3^ It also follows that
- 3 4u
, A
ttj — g/it —
- g3
From the formula just written we have
356 THEOEY OF ELLIPTIC FUNCTIONS.
Integrating we have / j \ A
^(2u)=2?-(u)+1-f\ogv'u + C G o 2 du
= 2^)+i^£ + C. a 2 p'u
Developing both sides of this expression in ascending powers of u, it is seen that the constant (7 = 0. We therefore have
This formula multiplied by 2 du and integrated becomes
log a(2u)= 4 log au + log <@fu + log c,
so that
a(2u)= c(au)*p'u.
It follows that
from which it is seen that c = — 1 and consequently
o(2 u)
ART. 303. Historical. — It was known through the works of Fagnano, Landen, Jacob Bernoulli and others that the expressions for sin (a + /?), sin (a — /?,) etc., gave a means of adding or subtracting the arcs of circles, and that between the limits of two integrals that express lengths of arc of a lemniscate an algebraic relation exists, such that the arc of a lemnis- cate although a transcendent of higher order, may be doubled or halved just as the arc of a circle by means of geometric construction.
It was natural to inquire if the ellipse, hyperbola, etc., did not have similar properties. Investigating such properties Euler made the remark able discovery of the addition-theorem of elliptic integrals (see Nov. Comm. Petrop. VI, pp. 58-84, 1761; and VII, p.. 3; VIII, p. 83).
Euler shows that if
f # i r # _ r-
/ / I / — I
«/o v R(£) Jo v R(fz} J®
where R(£) is a rational integral function of the fourth degree in f, there exists among the upper limits x, y, a of the integrals an algebraic relation which is the addition-theorem of the arcs of an ellipse and is the algebraic solution (cf. again Euler, Nov. Comm. Vol. X, pp. 3-56) of the differential
ecLuation
Euler states that the above results were obtained not by any method, but potius tentando, vel divinando, and suggested that mathematicians seek a
THE ADDITION-THEOREMS. 357
direct proof. The numerous discoveries of Euler are systematized in his work Institutiones Calculi Integralis, Vol. I, Sectio Secunda, Caput VI.
The fourth volume (p. 446) contains an extension of the addition-theorem to the integrals of the second and third kinds. This work must there fore have proved of great value to Legendre in the development of his theory. In every case geometrical application of the formulas was made by Euler for the comparison of elliptic arcs.
The suggestion made by Euler that one should find a direct method of integrating the differential equation proposed by him, was carried out by Lagrange, who by direct methods integrated this equation and in a manner which elicited the great admiration of Euler (see Miscell. Taurin. IV, 1768; or Serret's (Euvres de Lagrange, t. II, p. 533).
The addition-theorem for elliptic integrals gave to the elliptic functions a meaning in higher analysis similar to that which the cyclometric and logarithmic functions had enjoyed for a long time.
ART. 304. We may consider next some of the general investigations which led Euler to the discovery of the addition-theorem and then give his solution and the one of Lagrange.
If we differentiate the equation
(I) Ax2+ 2 Rxy + Cy2 + 2 Dx + 2 Ey + F = 0, we have
(II) (Bx + Cy + E) dy + (Ax + By + D)dx = 0.
From (I) we have
x = - %L±D. ± 4
^rl ^1
Bx + E
± i V(Bx + E)2 - (Ax2 + 2Dx + F)C.
These values substituted hi (II) give
(III) !fe_ + -4±L= = 0,
where
F(x) = (Bx + E)2-(Ax2+ 2 Dx + F)C,
G(y) - (J?i/ + D)2- (C?/2+ 2 Ei/ + F)A.
If .4 = C and D = E, then G(i/) becomes F(*/). The differential equation (III) becomes thereby
VF(x) and its algebraic integral is
(F) A(x2 4- y2} + 2 Bxy + 2 D(x + y) + F = 0.
358 THEORY OF ELLIPTIC FUNCTIONS.
Suppose next that R(x) = ax2 + 2 bx + c is given and it is required to find the integral of dx
We must so determine the constants A, B, F, D that
ax2+ 2bx + c =(Bx + D)2- A(Ax2+ 2 Dx + F).
By equating like powers of x, we have three relations existing among the four quantities A, B,F, D. We may therefore determine B, F, D in terms of A.
It follows that the differential equation
VR(X) VR^)
is always integrable through an algebraic equation (F) of the second degree which is symmetric in x and y and contains an arbitrary constant A . By the comparison of this algebraic equation with a transcendental equa tion which we shall determine later, we derive the associated addition- theorem.
If further we observe that - aR(x) = (b2- ac) fl - / ax + b VI and put
ax + b = z, then Vb2-ac
r -7=r = u> saY>
Jx*. \/R(xn)
becomes, if we take the minus sign with the root,
au = I dz where s2= I - z2,
Jz0,s0Vl - Z2
or dz___
du
If s is not a one-valued function of z, there must be a second branch of the function, which in the Riemann surface is represented on a second leaf, so that if zi represents the variable z in this leaf, we have
du
ART. 305. It is evident that we may write the differential equation
dx . dy = Q
Vox2 + 2 bx + c Vay2 + 2 by + c
0,
in the form ^ dj]
Vl - £2 Vl - f)2 or
THE ADDITION-THEOREMS. 359
If r, is a function of £ which satisfies this differential equation, then is
where C is the constant of integration. Integrating by parts we have at once
c = ~
V 1 - if v 1 - £2J
or ^ / —
This is the algebraic integral of the differential equation and corresponds to the integral (I') of Art. 304, which latter equation was derived through experimenting by Euler. To determine the corresponding transcendental
integral write
(1) u= '—-==, where a =Vl - ^, and
•/ft 1 V 1 - ^
(2)
/^n * rl
v = I ' , where - =Vl — -n2.
Jo, i \ 1 - r/2
It follows that £ = sin u and y = sin v. The differential equation
VI - $2 Vl - rf
becomes du + dv = 0.
We therefore have
(**>• d; + A.t ^ =
Jo, i \/l - c2 Jo, i \ 1 - r/2
or w + v = c,
which is the transcendental integral of the above differential equation. We so determine the constant C in the algebraic integral
that for c = 0, o = +l the variable TJ takes the definite value rl0. It- follows at once that
C = rl0.
When the values £ = 0, a = + 1 are written in the upper limit of the integral (1), it is seen that
u = 0,
and since u + v = c, it follows that
fjo.Vl^ dr) c = I '
•/ V - 2
Ctl - ry
7j0 = sin c - sin (t* + v).
360 THEORY OF ELLIPTIC FUNCTIONS.
On the other hand, since
we have gin (^ _j_ v) = sjn u cos v 4. sjn v cos Wj
which is the addition-theorem for the sine-function.
ART. 306. In a similar manner Euler derived the addition-theorem for sn u as follows.
Suppose we have given the quadratic equation
(I) Ay2 + 2 By + C = F(£, T?) = 0,
where A = a0£2+ 2ai£ + a2,
J5 -J
By arranging the terms according to powers of £, the same quadratic equa tion may be written
A'$2 + 2 £'£ + C' = F(?, T?) = 0, where A' =arf+ 260>? + co,
5'= a^2+ 2 6^ + ci, C' = a2>?2 + 2 62^ + c2. Differentiating (I) we have
dr jf. . dr -, ^ TJ^ + — ^ = 0,
d£ dri
or (A7 + B') d£ + (Ar; + B)di) = 0.
It follows * at once that
= 0.
Ai)+B A'
On the other hand we have
or At] + 5 = B2- AC,
where both signs may be associated with the root; and similarly we have
A'£ + B'= We thus derive t the equation
(II)
VB2-AC VB'2-A'C
* See Euler, loc. cit., or Enneper, Elliptische Funktionen, p. 186.
t See Euler, Institutiones Calc. Int., Vol. I, Sectio Secunda, Caput VI; or Lagrange (1766-69), (Euvres (Serret, Paris, 1868), t. II, p. 533. Halphen (Fonct. Ellipt.,Vo\. II, Chap. IX) calls such an equation an Euler-equation and remarks that by the dis covery of the general integral of this equation "Euler sowed the first germ of the theory of elliptic functions " (in 1761).
THE ADDITION-THEOREMS. 361
or
2+ 2&!* 4- 62)2- (a0£2+ 2 a,* + a2)(c<*2 + 2c^ + c2)
+ , ^ =0.
V(ai^2+26i7? + c1)2-(a0^2+260>?+co)(a27/2 + 2627? + C2)
If we put ai=&0, «2= c0, b2= ci,
the expressions under the roots take the same form, while equation (I) becomes *
(I') a0cV+ 2 &o^(£+ T?) + c0 (£* + >?2) + 4 fc^ + 2 ^(£+7?) + c2= 0. If the differential equation which we wish to integrate is
(III) -^L= + -^=. - 0,
where 72(0= P0*4 + Pi*3 + P2^2+ ^3^ + P*, we may make this equation identical with (II) by writing
B2- AC = R(?),
or (&0£2+ 2 61^ + 62)2-(oo^+ 2a^ We therefore have the conditions
PI=
P2= 260^>2+ ±bi2— aQc2— 4aiCi— a2c0, P3= 46i&2— 2aic2 — 2a2ci, P4= 622- a2c2.
Thus in addition to the three conditions a\ = bo, a2 = CQ, 62 = c\ we have the above five conditions among the nine quantities aQ, bo, c0, ai, 61, ci, «2, b2, c2.
It is evident that when these conditions have been satisfied there remains an arbitrary constant in the equation (I'), which equation is the algebraic integral of (III).
ART. 307. In particular let the equation (III) have the form
(III)' ^ + drl = 0.
Noting from above that ai = bQ, a2 = c0, b2 = ci, we have
&o2— « Oc0= k2, (2 61- c0)&o- a-oci= 0, 4612-a0c2-co2-260ci=- (1 + k2), (2bi- CQ)CI- b0c2= 0,
Ci2- C0C2= 1.
* See Cayley, ?oc. ci/., p. 341.
362 THEORY OF ELLIPTIC FUNCTIONS.
We observe that (III7) remains unchanged if £ and T? are replaced by — c and —i). It follows that (I') must remain unaltered by this transforma tion. We must therefore have
b0=Q, ci=0. The relations just written are consequently
— coC2= 1, 4&i2 — &oC2- Co2+ 1 + &2 = 0, — aoCo= k2, or
1 k2
co2
Writing these values in equation (F) we have
Co CQ CQ
or
[1 + &2£V+ c02(£2 + r)2)]2= 4[/c2-(l
Arranged in powers of — , this equation is Co
2(1 + fc2£V)(£2 + 7?2)- 4(1
co4
= 7] Vl -
c0 ~ 1
or
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
or . „
(i)
where Z(£) (1 -
and required that £ be determined as a function of u. He further introduced an auxiliary variable v, such that
<* '-
* Darboux, Ann. de I'Ecole Norm., IV, p. 85 (1867).
THE ADDITION-THEOREMS. 363
We therefore have from (III')
du + dv = 0,
u + v = c, v = —u + c, where c is a constant.
It follows that j_ -
23=- Vfi(,),
du so that £ and >? are functions of u, both being integrals of the equation
We next form d 1 Z'(~) d 1
dw 2
--(1 + fc2)£ + and
We have immediately
Through division it follows that
_
2 9Z^-J # x - ~ -+Cj
?M du
du * du This expression, when integrated, becomes
du = g
~
where C is a constant. Further, since
we have at once
which is the algebraic integral of (III')-
364 THEORY 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), wre have
(N)
AXz(g) ft + A Jo,i VZ(£) Jo,i
This is also an integral of (III') but in transcendental form.
Suppose next that y becomes TJO for the values £ = 0, \/Z(£) = 1. It follows from (M) that
and from (N) that
If we write
£ = sn u, 7) = sn v,
— £ 2 = en u, \/l — rj2 = en v, dnu, VI — k2j]2 = dnv,
then from (P) we have
7)0 = snc.
But since c = u + v and also T?O= C, the' equation (M) may be written
snvcnudnu + snucnv dnv , \
1 = sn(u -f- v).
Write
D = 1 — k2sn2u sn2v
and note, since 1 = sn2u + cn2u, that
D = cn2u 4- sn2u dn2v = DI, say, and
and also that
D2= DiD2. It follows that
Z)2 — (sn u cnv dnv + snvcnudnu}2
__ ^__^^_^_____________^__^_^__— _ y
or (cf. Art. 296)
C I y\ = c^^ cnv — snu snv dnu dnv t 1 — k2sn2u sn2v
THE ADDITION-THEOREMS. 365
Similarly, if we note that
D = dn2u + k2sn2ucn2v = D3 and that
D = dn2v + k2sn2vcn2u = Z)4, we may derive from
dn?(u + r)= 1 - k2sn?(u + v) the formula
dn(u + r) = ^n u ^n r ~ ^2sn u cn u snv cnv 1 — k'2sn2u sn2v
ART. 309. A direct process for finding the algebraic integral was given by Lagrange as follows:
For brevity write X = a + bx + ex2 + dx3 + ex4,
Y = a + by + cy2 + dy3 + ey4.
The differential equation to be integrated is of the form (I) ^L + -^L = 0.
Vx VY
Considering x and y as functions of u, we have as in Art. 308
and -- du du
It follows * that
2^ = b + 2cx du2
= b + 2cy
If next we introduce two new variables defined by
p = x + y and q = x - y, we have
• = X - Y = bq + cpq + ±qd(3P2+ q2) + $ epq(p2 + q2). It is seen at once that
A-ittLriftiirt,
du2 du du
...tf+2 ,
fdu?du q*\du) du P)du
* See Cayley, loc. tit., p. 337.
366 THEORY 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
(II) f-^S I_J_ I = c + d(x + y) + e(x + ?/)2.
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, 6, c, d, e) = ( - g3, - g2, 0, 4, 0)
and consider the integral
(10 ds — + dt — = o.
V4 s3 - g2s - g3 V4 t3 - g2t - g3 The algebraic integral is seen at once to be
s — t Writing
, ds , dt
du = — =^ = . dv =
V4 s3- g2s - g3 V4 t3- g2t - g3
the transcendental integral is
(T) u + v = c,
where s = <@u, t = pv = $>(c — u)= @(u — c).
When these values are substituted in the algebraic integral, it becomes
(A) Wu-v'(c-UW_ 4fw _ 4j,(e - *)- C.
\_ @u — %>(c — u) J
From (A) it follows (Art. 300) that C = 4 g?(c), and from (T) we have
or
2 (@u — @v)2
THE ADDITION-THEOREMS 367
ART. 311. Equate to zero the determinant* 1, pu, p'u
= pfw(pv — pu) + pfv(pu — pw) 4- p'u(pw — pv) = 0.
1, pv, p'v 1, pw, p'w
Squaring we have
(p'w)2(pu — pv)2— {p'v(pu — pw)- p'u(pv - pw) }2= 0, or (pu — pv)2[4p3w — g2pw - g3] - [p'vpu - p'upv - pw(p'v — p'u)]2= 0,
an equation which is satisfied for w = u, v and also (Art. 301) for —w = u 4- v; that is for pw = pu, pv, p(u 4- v). The equation
(pu - pv)2{s - pu}{s - pv} {s - p(u + v)}= 0
has the same zeros, viz., s = pu, pv, p(u 4- v); and since the coefficients of (pw)3 and s3 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)2 and s2 give immediately
- (p'u - p'v)2= 4 (pu - pv)2\-pu - pv - p(u + v) }, or
^ ( pu — pv ART. 312. In Art. 193 we derived the formulas
u(+l)=-3K or sn(- 3/0=1,
-- — 3 K — iK' or sn ( - 3 K — iK') = - »
n/
u( oo, GC ) = — iK' or sn ( — iKf) = oc ,
u(Q, 1) = 0 or sn(0)= 0.
Ti(— 1) = — K or sn(—K) = —lj
1\ -.. .„, „ .r.,N 1 - ) = — K — iK or sn (— K — iK) = ,
kj k
u(<*j, — (x) = — 2K — iK' or sn(— 2 K — iK') = oo, u(Q,-l)=-2K or sn(-2K)=Q.
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 i) = u, we deduce the following formulas:
2 snucnudnu
D cn2u — sn2u dn2u
~D~ dn2u — k2sn2ucn2u
cn2u dn2u
D Writing sn u = s, en u = c, dn u = d, we have
1 + dn 2 u =
D ' D
2d2
D ART. 314. Dimidiation. — From the above formulas we deduce at once
or
cn2u =
1 + dn2u 2 I + dnu
dn2u + cn2u k'2 + dn2u + k2cn2u
1 + dn2u I + dn2u
Changing uto^u we have formulas* for the determination of sn(%K), sn(% iK'), sn (% K), etc.; for example
£-\A
2 Vl
• = -%/=r=^f •
+ dnK
ij? = /dniK'+cnJK' /- ikl - U /I + k 1 2 V 1 + drciK' V I -ikl \ k
[Table of Formulas, No. XVII.]
In a similar manner we have
_ Vk + ik
where we have written
k = Vl + k' Vl - k', 1 = Vk + iA and noted that
Vk - ik' + Vk + ik' = Vl - k' + Vl * See Table of Formulas, No. XVII.
THE ADDITION-THEOREMS.
369
ART. 315. To determine the value of the ^function for the quarter- periods, we note that
c^ ,
k =
We have for example
SK
(t)
€1-63
- «s)d
— 63) (e i— e2);
en
= - 2i(el-e3)ikk'(k-ik')
or
a formula which is incorrectly derived and given by Halphen, Fonct. Ellipt., t. I, p. 54. ART. 316. We also find that
to) =
sn2(v + K) + (gi— es)dn2v = e +
[v = u Vei - e3]
It foUows, if we write 5(0 = 4(« - ed(t - e2)(t - e3), that
\
U +0})=
and similarly
4 jpM - €1
1 S'(e2)
- - L^Z~:
4 g?w - e2
4 pi* - e3
370
THEORY OF ELLIPTIC FUNCTIONS.
If further we let P\(u} = $>u — e\ (X = 1, 2, 3), we may derive at once the formulas *
P2(u
4
= («!- e2)
Pi(u)
Ps(«
= («2 — e3)
Pi(u)
« aw
EXAMPLES
1. Show that
2. Show that
3. Prove that
+ v) =
snv cnu dnu — snu cnv dnv
1 12 k2cnu cnv
cn(u + v) cn(u - v) dn2u dn2v - k'2
cn(u + v) cn(u — v) _ 2 snu cnu dnv sn(u + v) sn(u — v) sn2u — sn2v
, . 1 d i 1 + k snu snv
4. Prove that sn(u + v) — sn(u — v)= log
k du 1 — k 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(0) H(i? - u) H(v + u)
v — sn*u
* See also Art. 327.
THE ADDITION-THEOREMS.
371
9. If am a = a, am b = /?, am(a + b) = cr, show that
(1) sin a sin /? A a + cos o = cos a cos /?,
(2) cos /? cos <T + Act sin /? sin a = cos a,
(3) ACT + A-2 sin a sin /? cos <r = Aa A/?.
10. Show that the algebraic integral of
where
X = a^ + 4 a^3 + 6 a2x* + 4 a.^ + a4, y = a0y4 + 4 a^3 + 6 a,?/2 4- 4 a3y + a4, may be expressed in the form of the symmetric determinant
|
X ~\~ It |
||
|
o, |
^ 2' |
xy |
|
1, |
a0> alf |
a2- 2c |
|
x+y |
alt a2+ c, |
«3 |
|
2 |
||
|
vy, |
a2- 2 c, a3, |
«4 |
(Lagrange.)
= 0,
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 ou from a certain theta-function and we then proceeded to the other sigma-functions. In Chapter XV the function ou was denned through an infinite product which followed from the definition of the ^-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
We write v = &, where 2 5 = 2 pa) + 2qajf. The quantities p and q are integers, and here one of them at least is taken odd, so that & is different from a period.
Since a> is a half period, we may write
pa> = ei (i - i, 2, 3). The formula above becomes
In Art. 290 we derived the formula
a(u + 2 6>) = =
where 2 rj = 2 py + 2 qy', and the negative or positive sign was to be taken according as oco was different from or equal to zero.
In the present case we must therefore take the negative sign; and if u — a> is written for u, it follows that
a(u + a>) = — e2JlU o(u — a>). We consequently have
o^u azw \
* Hermite (p. 753 of Serret's Calculus, 2d volume, 1900) writes: " Nothing is more important nor more worthy of interest than a careful study of a process by which, starting with notions previously acquired, one comes to the knowledge of a new func tion which becomes the origin of a new order of analytic notions."
372
THE SIGMA-FUNCTIONS. 373
If a\u is defined through the equation
aoj we have
(\ 2 au/
The quantities 77 and if are defined as in Art. 259. As there are only three incongruent half-periods, we have the three new functions
ffitt (/ = 1, 2, 3).
When u = 0 it is seen that a/u = 1. We defined in Art. 272 the function au through the relation
_ _ d2 log au d_ a'u _ aua"u — (a'u}2
d2u du au (au)2
If then we require that the sigma-functions be one-valued, analytic func tions which have the character of integral transcendental functions, it is seen that $>u, pu — e\ may be expressed through the quotient of such functions (Art. 262).
ART. 318. By means of Laurent's Theorem we may express at once the function au through a Fourier Series as follows:
If f(t) is a one-valued, finite and continuous function within and on the boundaries of a ring inclosed between two circles, it may be developed in a series * consisting of an infinite number of positive and negative terms in the form
f(t)= J ck(t — a)k (ck constant).
fc— 00
We shall next assume that the interior circle is arbi trarily small, so that the above series is convergent for the entire larger circle with the exception of the point a.
Let F(u) be a function defined for the whole or a Yis 72
part of the w-plane and suppose that this function is one-valued, finite, continuous and simply periodic having the period p, say.
We then have
2xt
p
If we write (cf . Art. 67) t = e p , or u = -—. log t, we have
* Osgood, loc. cit., p. 295.
374 THEOEY OF ELLIPTIC FUNCTIONS.
The function f(t) is one-valued, for if a definite value I is given to t, then u = ^— .log I + kp (k an integer).
But for all such values the function F(u) retains the same value, since p is its period. It follows that F(u) is one-valued. Further, if t describes a circle, so that t = re^, then is
or u = b + m<j) (b and m constants);
and consequently u describes a straight line [Art. 60].
From the relation u — — = log t it is seen that for t = 0 and also for
P t = GO we have u = oo ; and since u = oo is an essential singularity of F(u)t
it follows that t = 0 and t = oo are singularities of f(t) .
Since zero is a singular point of f(t) , we have from above the expansion
fc— 00
and therefore A=+X
ART. 319. We write
and we shall so determine the constants A, B, C that <f>(u) has the period 2 w. This function ^(w) is one-valued, finite and continuous for the finite portion of the it-plane. From the formula
we have, since
ou,
the formula
It follows that
2 TJ(U + to) + 2 Bco + 4 CUM + 4 Cw2 = (2 & + l)7rt, where fr is an integer; and consequently
2 T} + 4 Cw - 0, or C = - - 2 ;
2 w
and 2 ^ + 2 Bco + 4 Co>2= (2 k + I)TTC;
or, if k = 0, „ = 7ii_
The remaining constant A being arbitrary, may be taken equal to zero.
THE SIG MA-FUNCTIONS. 375
We then have , -i
-- - u- H -- u
= one
We further write u = 2 wv and put
0(u) =•*•(!,'). Since <j>(u + 2aj) = <j>(u), it follows that
<$2a)(v + 1)] = 0(2o>v), or
Vr(t> + l) = tM,
and consequently from the last Article
0 (P= 1),
a series which is uniformly convergent within the finite portion of the r-plane.
To determine the coefficients Ck, we note that
o(u + 2o/)=- e ,
and consequently
^- (M+2o/)
or
w' '^w'2 at'
2jj'(ti+a/)-2ijtt --- jj+st —
Since TJOJ' — wy' = —, it follows that
Writing — = T, we have to .
<56(2 o>y + 2 a/) = - e~ or
^r(v + T)= - r
Since
/:= + x
we therefore have
fc=+
V
or
376 THEOEY OF ELLIPTIC FUNCTIONS.
If the coefficients of e2*™ on either side of this equation are equated, we have - CA = CA-I e2^-1^,
which is a transcendental equation of differences. In the formula ^^ _ (7A_1g2(A-i)«t+B» .
change ^ to A + 1 and write log CA = #A- We then have
Suppose that ^(A)= A0+ A! A + A2A2 and consequently that ^(/l + 1)- %(*) = AI+ 2A2A + A2= 2^rir + ?ri.
It follows that
A2 = Ttt'r and AI= 7ri(l — r).
As A0 remains arbitrary, we choose it equal to zero. These values sub stituted in y(A) give
Let us further write Bx - j(A) = EL We then have #A+1= Bm- /(A + 1)= B
-Bi
We note that
Further, since BA = ^(/) +
or
we have
Writing eE«= C, it follows that CA = Ce"1'^-0^^^, and consequently
Further, since
•^r(v)=
it is seen that
CTI^ = a(2ow)=
?rtt 4
/2fc-l
2
THE SIGMA-FUNCTIONS. 377
_ru
Letting — e 4 C = c and substituting k + 1 for k, we have
We note that era 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.
ART. 320. If we write £ = w, M", co', we have directly from Art. 317 the formulas
OOJ OOJ
where w" = w + a;' and ^r/ = y + yf.
The argument 2 cj(v + }) corresponds to the argument u + a). We may consequently write
so that
or
, i)ta rt k = +oo
C/>^)l7£tJT7'"-r ~7T" T TT c 22
If we write
then is
and similarly k=+x
a2u = t32e2rjujl'' 2) c *--*
fc=+oo
fc=-M
378 THEORY OF ELLIPTIC FUNCTIONS.
If with Weierstrass we write
e*n* = h and ealv = z, we have
£=-00
Tit — — 7T -ZT-
Using the notation of Jacobi: h = e " = e K = q, and writing with him
= 2 g* sin nv — 2 5^ sin 3 xv + 2 g¥ sin 5 TTV — • • • ,
fc=+oo (2fe+l)2
- 2) 9 4 ^2*+1
A;=-oo = 2^* COS KV + 2 g* COS 3 7TV + 2 g^ COS 5 7TV + • • •
= 1 + 2 5 cos 2 TTV + 2 g4 cos 4 TTV + 2 q9 cos 6 ^v+ • • • ,
= 1 — 2 ^ cos 2 TTU + 2 54 cos 4 TTV — 2 <?9 cos 6 TTV + • • • , we have
ART. 321. By differentiating both sides of the formula above for era and then writing u = 0, it is seen that
du 3=^-
p «V(0)
THE SIGMA-FTXCTIOXS. 379
Further since ^i(0)= 1 = tfo(0)= ^s(O), it follows that
#2(0) 03 (0) 00(0)
In Art. 340 it is shown that
$!'(0) = 2 77/Z* JJ (1 - ft2")3. n = l
When this value is substituted in the formula above for mi, we have
n = x
In a similar manner if we write for 4#i, /?2, /?3, their values we have
cos 2 «r
<72u=^-2jj—
n= 1 G3li = e2v** JJ 1^
Take the logarithmic derivatives of oiu, o2u, o^u and equate the coeffi cients of u on either side of the resulting expressions. We then have
Since ei + e2 + e-3 = 0, it follows from Art, 286 that
3 h (i +
We note that au is an odd function, while aiu, a2u and o%u 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
/ 9/V
<@u — e\ = ( — L^ ) j <@u — e2 = [ -^^ ) i <@u — 63 = ( — ^— J \ au / \ou / \au /
are multiplied together, we have in virtue of the equation
the formula
To determine the sign to be used before the root, write u = 0 and it is seen that the negative sign must be employed. We thus have
(1)
°*u
O6U
In Art. 302 it was seen that
It follows from (1) that
cr (2 u) = 2 au GIU a2u
ART. 323. We may next note how the sigma-f unctions behave when the argument u is increased by a period. Since
OOJ OOJ
it follows that
OU)
<7W (70J
or
and similarly *
Formulas for a\(u + 2w"), etc., are found in the Table of Formulas, No. XXVI.
* See Schwarz, loc. cit., p. 22.
THE SIGMA-FUNCTIONS. 381
ART. 324. Let X, //, v represent in any order the integers 1, 2, 3; then by Art. 262 we have
(\ 2 — j = $u — ou /
ou
no two of the quantities X, a, v being supposed equal.
By eliminating <@u from the second and the third of these formulas we have
I j ( ) = W/l &v))
or
_ Cy) (j2M = 0.
It is also seen that
(e2— e3)oi2u +(e3— 6i)a22u +(e\— e2)cr32u = 0.
DIFFERENTIAL EQUATIONS WHICH ARE SATISFIED BY SIGMA-QUOTTENTS. ART. 325. If the formula
is dirTerentiated, and for <@'u its value in terms of the sigma-functions is substituted, it follows that
ou ou ou ou ou
or
du ou ou ou
If we differentiate the equation
ovii &u - e it follows that
2 a^u (L_ a^ = (g?M - ev) - (pu — eu} , ovu du avu (pu — ev}2
(o_vU\ \ou)
or
1 tL- = — (e^— ev) ~^—
du ouu ouu
382 THEORY OF ELLIPTIC FUNCTIONS.
From
it follows that
2 j avii
d / ou\ _ _ <@fu _ ou ou ou du\oxu) (<@u — ex)2 /<?A*A4
(ou)
or
d ou OfU ovu
du OxU oxU OxU Since the equation
o2u — o^u + (e^— ev)o2u — 0 may be written
o.2u , v o2u
and further since we have
OfU
Cj 1 (
au oxu/ \o
(~r — ) = 1 ~ fc~ ^A)[ — d^ (T^/ |_ \GxU/
In the same way it may be shown that I -I \2 T~ 2~ir~
\d IL~) = ^2~ C'
2"4 el ^T A "J
and
It follows that
OU 1 OM 1 <T.,W 1
(e,-^)(ey-^) ou are all particular solutions of the differential equation *
(A) (^ Y = [1 - (e,- ex)?2] [1 - (e, -
ART. 326. If we write v = 2, /£=!, >l = 3 and
-. s =
the differential equation (A) becomes
* See Schwarz, loc. cit., Art. 25; or Daniels, Am. Journ. Math., Vol. VI, p. 180 and Vol. VII, p. 89.
THE SIGMA-FUNCTIONS.
383
Further write
and
(ei-e3)c2=*2, or ^ei-
We then have
If
w= r
J0 V7(l -
- k2x2)
then is x = sin am UOT x = sn(u, k). We therefore have
(1)
Further, since
and as
we have
(2)
and similarly
(3)
ART. 327. If we write *
= 0,
- e3
— 011
— e
we have at once and
* See Enneper, Elliptische Funktionen, p. 160; or Tannery et Molk, Fonct. EUipt., t. II, Chap. IV.
384 THEORY OF ELLIPTIC FUNCTIONS.
It is also evident that
3 <@u = £iQ2(u) + £n(?(u) + £vQ2(u), 2 a))) = £),Q(U), ) where we write without regard to order
V f ft f
) (OX) (ou) (Ov lor oj. d) d) .
£fiV'(u) = - (eft- ev) ART. 328. Through the equations *
CD
an
— , v ,«,«,- 32,
<7W GU
the values of the three quantities \/$u — e^ are denned as one- valued func tions of u.
If we give to u the values w, 01", a>', it is seen that
(2)
GOJ
/ — <7iW 6'w Gd)
GO)" 0(DO(i)"
Ve3-€,=
GI(I)' e'
GO) 0(0 OO)
e\— 63 =
63— e2 =
GO)
0(1)
Through these formulas the six quantities on the left-hand side are uniquely determined. We note that
On the other hand
i (*) —
if
'€3- e2
fe2- e3
iaj
(see Art. 288).
Hence among the six quantities above there exist the relations
v 63— 62= —i ve2— 63, V 63— e\= — i \/e\— 63, *ve2— e\= —i
or
— e2= \e2— e^e^— e\= \e\— e3,e2— ei = e\— e2.
We have thus reduced the six roots without any ambiguity to the three roots \/e\ — e2, \^e\— 63, ^/e2— 63, which three roots are real and posi tive if the discriminant of 4 s3 — g2s — g3 = 0 is positive.
* Schwarz, loc. cit., Art. 21.
THE SIGMA-FU3TCTIONS. 3S5
Remark. For the sake of a greater symmetry some recent writers on this theory have written w\, oj2, MS for the quantities which at the outset with Weierstrass we denoted by w, a>" ', w'. 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
^€3 — e2 = i V^2 — €3, v €3 — ei= — i >ve\ — €3, V 'e2 — e\ = i \/e\ — e2.
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'.
ART. 329. From the equations (2) above it follows that
f^ e' „
(A) aw = ,, , war' =
Vie**"""
= 4 . 4/
Ve2- 63 vei— 63
We note here (see also Art. 345) that the quantities
4/ - - 4/ -
Ve2 — 63 Vei — e2 (where i = e^).
can take only such values whose squares
e2 - 63, ei - e3, 61-62
are uniquely determined through the equations (2) of Art. 328. Hence each of the fourth roots may take two and not four values; but as soon as the value of any one of these quantities is known, the values of the two others are uniquely determined through the formulas (A).
If in the formula
e~*uo(u) + u)
G\U = - * - ! - L%
aw
we put u = — J w, we have
|
It follows that we may write formulas (A) in the form |
^, •(¥) |
|
"\/ei— 63 Vei— ^2= — j— ?' Ve2— e3 Vei— e2 = /oA a3(") |
|
|
• „-, .......v^. |
which expressions may be used to determine the products of any two of the three fourth roots.
386 THEORY OF ELLIPTIC FUNCTIONS.
ART. 330. We may next derive a table of the four sigma-functions when the argument is increased or diminished by a quarter-period. It is assumed that the definite values derived above are given to the square and fourth roots that appear.
Take, for example, the formula
e~1iu(j(co + u) _ 6^0(0} — u) o\ii = --
Git) OCO
We have at once
o(u ± co) = ± Further, since
it follows that
-e2
The formulas given in the Table of Formulas No. XXXIV should be verified.
ART. 331. It is seen that
a(u + 2co) = _ cru
o3(u + 2 co) GSU
and consequently
<ju + 4a> ou
(73(u -\-4co) o%u
Also, since a^u + 2 w'^ = —> it follows that 4 CD and 2 a/ are periods of <73(w + 2 to')
— • A closer investigation shows that 4 co and 2 co' are a primitive pair of
periods of this function; for in the period-parallelogram with the sides 4 co and 2 w' the function <j3w becomes zero only on the points co and 2 co + co',
being zero of the first order. Hence — becomes infinite of the first order
<73lt
on these points. Since only two infinities lie within the period-parallelo gram with the sides 4 co and 2 co' , and since the smallest number of infin ities within a primitive period-parallelogram is two, it follows that 4 co, 2 co'
form a pair of primitive periods of
ART. 332. It follows at once from the formulas above that
a(u -f co) = 1 o\u
This may also be seen from the formula of Art. 326
THE SIGMA-FUNCTIOXS. 387
Since K =\/e\ — e 3 • a> we have
1
V ' e\— For u — 0, it follows that
1
ei — €3
and further that all values of u which satisfy the equation
OIL 1
<73u ei— e3 are contained in the form
a) + Ipw H- 2qw',
where p and q are integers positive or negative, including zero. We might define K more generally through the equation
K = ei— e3(cu + 4pw
where it is assumed that 4 p + 1 and 2 5 have no common divisor.. The quantity V fe\— €3 is to have the same value as given in formulas (2) of Art. 328 or the opposite value according as q is even or odd. ART. 333. It also follow from the equation
sn am
— e% • u, k)
that — 1 = — sin am (K, k),
Vei— e3 \ <?i- e3
or sn(K, A') = 1 (see Art. 218).
The coamplitude is defined by Jacobi (Werke I, p. 81) through the formula (see Art. 221)
— e3 • u, k) = am (K —Vei- e3 • u, k},
— e3 • u, k) = am \Ve\ — e3 (to + 4po; + 2 5^' — u), k].
Since 4pa> is a period for all the sigma-f unctions, it may be dropped from the argument u.
We then have
K =v/ei— e3(u) + 2qa)r),
and
coam (\/e\ — e3 • n, k) = am [V«i — e3 (aj + 2 qa)r — u), k].
388 THEOKY OF ELLIPTIC FUNCTIONS.
We may note that
r / - / , r> / \ 7! oi(— u + co + 2,qco'.k) cos am [V^!- e3(co + 2qco'- u),k\ = -^ - - - * / '
<?3( — u + co 4- 2qco , k)
_ (jl(u — oj — 2qco', k)
a3(u — co — 2qcof, k) Since
a3(u — a))
we have
r / - T~\ / - coscoam vei — e» • u, k\ = Ve1 — e2 L
and since
o(u + 2tt>r) _ ou
we have finally *
cos coamK/e!— e3 • w, A;]= (— l^V^— e2
<T2^
Making q = 0, we have the set of formulas given in the table, No. LIV. ART. 334. In Art. 79 we wrote (Cf. Schwarz, loc. cit., Art. 33)
co = pco + qco', to' = p'to + q'co', to" = to + a)',
where p, q, pf, q' were any integers such that pq' — qp'= 1 ; and it was seen that 2 co, 2 co' and 2 co, 2 co' formed equivalent pairs of primitive periods. We shall further write
TJ = pr) + qy', Ij' = p'-t) + q'rf, TJ" = rt + TJ'. If in the place of the quantities
o),a)', co"= co + to'', r), T)', 7)"= T) + rf\ we substitute
co, wf, Z>"= at + £'; Tj, ij' , rj" '= y + T)',
it follows at once (Arts. 276, 271) that the invariants g2, 93 and the func tions §m, ou remain unaltered. Also owing to the equation
(p'u)2= 4[pu - pa>][pu - v<o"][pu - pcof] = 4[$>u - ei][yu - e2][pu-e3]
the collectivity of the three quantities e\, e2, e3 remains unchanged and consequently also the collectivity of the three functions
(W\2==pu_eji (J = i,2,3),
\ou]
although the indices 1, 2, 3 may be permuted.
* See Schwarz, loc. cit., p. 30; or Daniels, Am. Journ. 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
0), CO
01, Op,
u
v = — r 2co
in the place of
CO, CO
n, V,
B\j 69,
V = —
co
where ^, /*, 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, p, v for each of the six different cases which may arise (see also Halphen, loc. tit., 1. 1., p. 262):
|
Residue, mod. 2 |
|||||||
|
I |
P |
9 |
pf |
q' |
/ |
ft |
y 3 |
|
1 |
0 |
0 |
1 |
1 |
2 |
||
|
II |
1 |
0 |
1 |
1 |
1 |
3 |
2 |
|
III |
1 |
1 |
0 |
1 |
2 |
1 |
3 |
|
IV |
1 |
1 |
1 |
0 |
2 |
3 |
1 |
|
V |
0 |
1 |
1 |
1 |
3 |
1 |
2 |
|
VI |
0 |
1 |
1 |
0 |
3 |
2 |
1 |
ADDITION-THEOREMS FOR THE SIGMA-FUNCTIONS.
ART. 335. In a similar manner as was done in the case of the theta- f unctions (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
= 0
(tfU — $>U2) (£>M3 — &Ul) + ($>U —
we make repeated application of the formula
a(u + v) a(u - v) '
390
we have (D
THEORY OF ELLIPTIC FUNCTIONS.
u3) o(u2- u3)
o(u + HI) a(u — HI) o(u + u2) o(u — u2) a(u + U3) a(u — u3)
u2) a(ui - U2) = 0,
an equation which is true for all values of the arbitrary quantities u,
U2, U3.
Through the equations
(2) u + ui= a, u — ui=b,
u + u2=af, u — u2=b', u + u3=a", u — u3=b",
u3= c, u2- u3= d, ui=c', u3— ui= d', u2=c", ui— u2= d",
we may define three systems of four quantities each
o,6,c,d; a',6',c'X; a",V',c",d", among which the following relations exist (cf. also Art. 210) :
(3)
a =i(a + b b =i(a" + b" c =ia"-b"
a -
d=i(a'-b'-c'-df) (3') a2+ b2+ c2 + d2= a'2+ b'2+ c'2+ d'2= a"2 + b"2+ c"2 + d"2, If in equation (1) instead of the quantities
U + U\j U — U\j U2+ U3, U2— U3
we write respectively
[1] a, 6, c, d- [2] a + 5, b + S>, c, d',
[3] a -f %, b + 5", c - wf, d', [4] a + 5", 6 + a>", c + 5', d - 5',
[5] a + 5 + 2 £7, 6 + 5, c + &, d — 5; [6] a + 5, 6 + to, c + a>, d -
THE SIGMA-FUNCTIONS. 391
we have the following relations given by Schwarz, loc. cit., § 38:
[A.]
[\\aaabacad + aa' ab' ac' ad' + aa" ob" ac" ad" = 0,
[2}a{iol)acad + afr'otfac'ad' + a>a" a,b" ac" ad" = 0,
[^o&ojbaj od + afr'ap'a^'ad' + a &" a fi" a j." ad" = 0,
[4] Offl ffjb a^c a$ — a^'ap'a^a^' + (e/jt-ev)a)a"aib"ac"cfd" = 0, [5] («j— «^|B*£*iC*j4+(«^ej)4FXa#^ [6] ajfLobofloid afr'ab'aic'aid' + (
From [A.] formula [2] follow without difficulty:
[B.] (1) aiwo(u + v + w)a(u — v) = a(u
(2) owa(u + v + w^a^u — v) = a(u
w)av.
Professor Schwarz, loc. cit., p. 50, gives eighteen other such formulas. Write in [A.], [2] the values
a=0, 6= 0, c = u + v, d = u — v, a' = u, b' = — u, c' = v, d' = v,
a" = v, b" = - v, c" = u, d" = - u,
and we have
[C.]
a(u + v)a(u — v)= o2u oj?v — apu o2v.
The other eight formulas given in the Table of Formulas LXII should be verified.
We note that these formulas are the analogues of the formulas (D) of Art, 211. Scheibner (Crdle, Bd. 102, p. 258) has derived the Weierstras- sian formulas from those of Jacobi. A method by which the formulas of both Jacobi and Weierstrass may be derived is given by Kronecker (Crelle, Bd. 102, p. 260); see also Briot et Bouquet, Traite des fonctions elliptiques, pp. 485 et seq.
EXPANSION OF THE SIGMA-FUNCTIONS IN POWERS OF THE ARGUMENT. ART. 336. In Art. 281 we saw that
23.3-5. 7
and in Art. 279 we saw that the coefficients of u were rational functions of
g2 and ^3.
392
THEORY OF ELLIPTIC FUNCTIONS.
We may determine these coefficients as follows : * If the equation
/d@u\2
\du)
be differentiated respectively with respect to u, g2} g$, we have
2ajerP
(a)
= [12 (»*)*- <,
2 g2
)2 _L
We also have
du
dpu
du
du
dgw du
1 1
We further note that
(2)
du
„
du
dpu
-3pu + —
6 2w
+ ±l^u "1 ?
If the equation (2) is integrated with respect to u, it becomes
-»a»+-igri
or
92'
a3 log on
(3)^2
Noting that
a3 lo
aw dw
, a log aw a3 loj
a2 lo
it is seen that the constant of integration that would appear in (3) is zero.
a2
Since — - log ou = — @u, we have from (a)
* See Weierstrass, Zwr Theorie der elliptischen Functionen, Berl. Monatsb., 1882, pp. 443-451; TFerfce, Bd. II, p. 245, and also Forsyth, Quarterly Journ., Vol. XXII, pp. 1 et seq.; Hermite, Crelle, Bd. 85, p. 248; Meyer, Crelle, Bd. 56, p. 321; Enneper, Ellipt. Funct., p. 166.
THE SIGMA-F UNCTIONS. 393
and observing the identity
i a2 /dlo<rau\2= a log <m a3 log au , /a2 log ^iA2
2du*\ 9" / dw aw3 ' \ du2 ) '
it is seen that the equation (3) may be written
2 a3 log mi , 1Q a3'log au _ 3 62 /6 log <m\2 . 3 64 log mi , 1
92 *** = ~ + ~ f ^
This equation when integrated twice with regard to u becomes
2 6 log au . -, Q 3 log <m 3 /a log<7jA2 3 a2 log <m .1 2
^ - ^ - + 18^3 - f - = o - ^ - + o - n - + o^2W ;
a^3 dg2 2\ du / 2 du2 8
or
2 al^u lg aji^ _ | ^ a^, + i 2 a^3 agr2 2 a^ a^2 8
the constant of integration being zero. It follows finally, since
a log au = J_ dau a.r au dx
that
C/2
Using this as a recursion formula Professor Schwarz (loc. rit., p. 7) has calculated the terms of au, up to the 35th power of u. If with Halphen * we write
' - o! ' *7! ' "(2n+l)!
we have
7 _ -j 9 afon- 1 .2 2 a6n- 1 (2 U -
On— 14 y3 I r — J/2 ~
To simplify the computation write and consequently
02= 2i2, 03= ~^3, o
n-l"[ (2 71 xJ --
/?3 j
2 71 - 1) (n - 1) xJ o -
a/?3 6
* Halphen, loc. cit., t. I, p. 300.
394 THEORY OF ELLIPTIC FUNCTIONS.
It follows from (1) that 62= - h2, 63= - 4ft3; and from (5) we have
=22-32
23^32 + 107
Expansions for sn u, en u and dn u were given in Art. 226. These functions may be expressed as quotients of theta-functions. We have not, however, expressed the theta-functions in powers of u. As we have already given the expansions of gw, £u, etc., in powers of u, it seems somewhat super fluous to expand a\u in powers of u. From the formula _
NV^= °&,
ou it follows that
\— - ex [_u2
ou—
20
or
Methods including recursions formulas for the further expansion of these functions are found under the references given above. In particular attention is called to the formulas that result from the partial differen tiations with regard to the invariants (given by Halphen, loc. tit., t. I, Chapter IX; Frobenius and Stickelberger, Crelle, Bd. 92, p. 311).
EXAMPLES. 1. Show that
a(2 u) = 2 °(u
oa> o<jL>a(jL)
2. Show that
o(u + v} o(u -v) d2 . d2 '
^ - rf-j - '• = — log OU ~ — log (7V.
o2uo2v du2 dv2
3. Prove that (if a>^ aj^ a)v, = cu, to" , to' without respect to order)
(1) SIQ(U + wi)
(2) ?xo(u + cjj
(3) -u + an}
(4)
THE SIGMA-FUXCTIONS. 4. Verify the formulas
* fu + v\= ^oM-nM
- (ev - efi (ev - ^)c20l/u)c20
^(u + v) 5. Show that
m = x
-fa I1
cos-
UJ
n
1
sin"
6. Show that
-nr-
2 <$u — €) O)U ou du au
(«.-
2 (<u —
u ou du
_^_
C/?i
du2
7. Show that
. fc)
, k) . E($, k) k sin 0 cos
395
where F(0, /v) and ^(^ A*) are Legendre's integrals of the first and second kinds; and that
fr) _ F(^fr) dk
, k k
fed -
' - (3 ft* -
dfc
A-) + fe A fe Sin ^ COS ^ • 0, , A:)
, fc)
, A-) A-
A- A- sin 0 cos 0 _ n
(^k] ~
Write <£ = - in these equations and note the results.
CHAPTER XVIII
THE THETA- AND SIGMA-FUNCTIONS WHEN SPECIAL VALUES ARE GIVEN TO THE ARGUMENT
ARTICLE 337. The theta-functions were expressed in Art. 209 through the following formulas :
m=l
= 2 sn
in nu JJ (1 - q2 m) JJ (1 - 2 g2 m cos 2 TTM + ?4m),
m=l
2 gicosTTuJX (1 ~ 52m) II t1 + 2 52m cos27rw + q4m),
m=l m=l
m=l
For brevity we put
m = oo
Oo= II d - a2*).
Since these quotients are absolutely convergent (Art. 17), we may write
QoQs = and consequently
m=l
It follows * that
* See the 16th Chapter of Euler, Introdutio in analysin infinit.
396
TRANSCENDENTAL CONSTANTS. 397
Making the argument equal to zero in the theta-functions, it is seen that
ART. 338. From the following formulas (see Art. 208),
m = x m = + x
1 + 2 2) (- I)mqm*cos2m-u = 2) (- \)^e'm^ie
-w = i 2) (-l)me *
(2m+l)2 .
~~
m = 0
m = x (2m+l)2 m= + x (2m+l)2
2) 9 4 cos (2 m + l)-w = 2) e 4
m = 0 m= —x
t?3(u) = 1 + 2 2 2m* cos 2 nant =
m=l
we have
m = »
#0= 1 +22) (- I)w9m= 1 -2
m=l
= 2-2) (- l)m(2m+ l)q 4 - 2-^ (1 - 3 q2 + 5 qQ- 7 q12 + - • •)
m = 0
(2m+l)2
4
= x (2m+l)*
4
m=0
(2m+l)8 >M _I_ ^2^ 4
7M = 30
+2
398 THEORY OF ELLIPTIC FUNCTIONS.
ART. 339. Since the functions $o> &i, $2, $3 depend only upon one variable q, it is natural to expect that they are connected by three rela tions, which we would suppose are of a transcendental nature. Two of these relations *, however, as we shall show in the sequel, are algebraic, viz.,
The first of these follows at once from the equation (Cf. Art. 193)
&2 + fc/2= 1.
To derive the second we use the equation of Art. 295,
#2#3#l(^ + V)&0(U - V) = &i(u)&0(
Expanded in powers of u, it becomes
( )
&M$*(v) <#/#o • u + ° ' ^2+ * ' ' c I )
0
If the coefficients of u2 on either side of this equation are equated, we have f
an expression which differentiated with regard to v becomes
If we put v = 0 in this equation, we have
or
^2 ^3
* They are both due to Jacobi, Werke I, pp. 515-17.
t See Koenigsberger, Ell. Fund., p. 380; or Burkhardt, Ell. Funkt., p. 120.
TKAXSCEXDEXTAL CONSTANTS 399
ART. 340. It may next be proved that
a r 4xi du2
Take, for example, the equation
<x(u, r) (a= 0,1,2,3).
When differentiated with regard to r, it becomes
_
dU2
By Maclaurin's Theorem
and consequently also
dr 6r 2 dr
If these values are substituted in (1), we have
o + o -
dt 2 dr
or writing u = 0,
In a similar manner it may be shown that
/'=4;r (,1 = 0,2,3),
a-
and also that
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 — /r, and consequently that
It is also seen from the results of the preceding Article that
400 THEOEY OF ELLIPTIC FUNCTIONS.
ART. 341. If the formula
_
be differentiated with regard to u, we have
en u dn u = — ^
#2
If in this expression we put u = 0, it follows that
_J or i =
It is thus seen that From the formula if also follows that
, 2KV2Kkk'
[S ~vT
We note * (see also Art. 345) that
or since we have
and also
It is seen that k and kf considered as functions of g = e1*1 are one-valued functions of T. From this point of view Kronecker found the origin of some of his most beautiful discoveries and Poincare was also thus led to the discovery of the Fuchsian Functions.
* See Jacobi, Werke I, p. 146.
TRANSCENDENTAL CONSTANTS. 401
Hermite* wrote -\ik = <!>(T) and $&=+(*), where from above and T/r(r) are one-valued functions of - which may be expressed as quotients of two infinite products. These functions are of such importance that we may consider them more closely and at the same time introduce other interesting formulas for the elliptic functions.
ART. 342. From the equation
m=l m=l
it follows that
7n = l m=l
Since
1 _ 22(2m-l) COS
2 g2™-1 cos 2~u+
t?o(2 u, 9*>-n
or
(1) and similarly
(2)
i
We also have from the product of &0(u, q) and ^i(u, 5) the formula
and since
1 - 2 it follows that
m = * (I - o2m>)2 q)&i(u,q}=q*Tl l _ \J
771 = 1
further noting that
if a - 9m)=ii a
W=l 771 = 1
we have
(3) 0o(M,g)#i(M,9)= 8*^^ v/9)-
^3
* Hermite, Resolution del' equation du cinquieme degre. (Euvres, t. II, p. 7; and also Swr /a tfuorie des equations modulaires, CEavres, t. II, p. 38; see also Webber, EUiptische Functionen, pp. 147 and 327.
402 THEORY OF ELLIPTIC FUNCTIONS.
If for u we write u + J in this equation, it becomes (see Art. 208) (4) &B(u, q)»2(u, q) = 9* &2(
If for q we write qe** = -q = qe-*, the quantity g*2o becomes q*e~*^
Qs Q2
and the equations (3) and (4) become
'
- (5) &3(u,q)&i(u,q)=qle 8 i(ut e2 Vq),
Q2
(6)
The six formulas above are given by Jacobi (Seconde memoire sur la rotation d'un corps. Werke, II, p. 431). In the formula
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
in (4n
n = —oo
and similarly
Since
sn
Vk ^o A;
dn2Ku=
it follows from the formulas above that (7)
V2 V7^0(2 w, q2) ^k ] ( _ 1)^2 n' CoS 4
TRANSCENDENTAL CONSTANTS. 403
where the summations on the right are over all integers from n = — oc to n = -f QC . The summations are taken over the same integers in the following formulas:
(8)
(10)
(ID c^wsv^^^
V tf&^Vq) •
(12)
in (4 n If we put u = 0 in (8) and (9), we have
Jacobi (Werke II, pp. 233-235) has given several different forms for these two quotients of infinite series.
If we write u = 0 in (10) and (12) and determine the resulting indeter minate forms, we have *
2 \/2
- l)n(4n
~ l)n(4n + I
ART. 343. By equating the expressions for the theta-f unctions 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(w) we have after division by q*
(1) sin-w(l-52)(l-252cos2^ + 54)(l-54) = sin-w — g2 sin 3 TTM + g6 sin 5 TTU — g12 sin
* See Hermite, (Euvres, t. II, p. 275.
404 THEOEY OF ELLIPTIC FUNCTIONS.
If in this equation we put u = J and divide by i V§, we have *
or writing q6 = t, it follows that
m=oo m=+Go 3m2+m
(2) jj(i - **) = 2) (- i)»« *'• .
w = l m= — oo
Upon this formula depends the trisection of the elliptic functions.
If further we divide equation (1) by sin nu and then put u = 0, we have
[(1 - g2)(l -g4)(l -g6) • • • ]3 = 1 -3q2 + 5qQ- 7g12 + 9g20- Writing g2= Z in this equation, it follows that
If we compare the equations (2) and (3), it is seen (cf. Jacobi, Werke, I, p. 237) that
(1 - q- £2+g5+g7_ ql2+ . . .)3= ! _ 3g + 5g3
Further in equation (1) put Vq in the place of q. We then have
- g2)(l - g3) . . . sin7rw(l - 2qcos27tu + q2)
(I - 2q2 cos 2 7m + g4) (1 - 2 #3 cos 2 KU + g6) . . . = sin nu — q sin 3 TTW + g3 sin 5 TTW — g6 sin 7 TT^ + • • • .
Write in this equation u = J and observe that
QsQoQi2Q22 = ;it follows that
If we compare the two expressions for &Q(U), we have
Q0(l -2q cos 2xu + q2) (1-2 g3cos 2 TTW + q6) . . .
= 1 — 2q cos 2 7m + 2 q4 cos 4 7m — 2 g9 cos 6 ?m + In this equation write u = 0 and observe that
It follows that
---q*). . . 1 „ ,
~ 29 -
* See Euler, Introductio in analysin infinit., § ^23.
TRANSCENDENTAL CONSTANTS. From the formulas
405
2Ku
\ 1 - 2q2m~1 cos2u + g4m'
1 4- 2g2mcos 2u + g4m L^ 1 — 2g2m-1 cos 2 u + q4m~
it follows that (1) log sn
(2) log en - - = log (2 q* y — cos i*j + 2 2* J~ 4. (_ \m cos 2 mw'
(3) log dn ^^ = log Vk' + 4
From (1) and (2) we have 2 Kw'
t2m-l
^-;cos (4m — 2)
sn
log — r-1 sin fc
i ^ rv. i ^ (7* p. ^-\ i
= log = log -*= + 2 > —
e»
2 Ku
log
cos z^ _
We also have from (1), (2) and (3) the formulas (4) Alogsn— = ?-
t^l^iKM
m = x
sn
sin
1 + q i
(5) — T-
cosu
sin 2 mu,
(6) -^-logrfn^
2K
2 KM 2 KM
en
O
= 8
, an
;sin(4?tt — 2)M.
406
THEORY OF ELLIPTIC FUNCTIONS.
If in (4) and (5) we put - — u for u, we have ft
sn u
cn
2Ku
S1D
2Ku
71 71
To these we add the equations of Art. 231
TT 2 Ku sin sn —
= -4-4
--— sin (2m 1 — q2 m ~ 1
= l
(10)
en
m=l
and the equations of Art. 228
--ain (2m _
= 1 ^
(13)
cos2mu.
In Equa. (12) write u = 0 and in Equa. (9) put u = -; it follows that
Y2m-l
Similarly writing u = -in (11) and u = 0 in (12), we have
-i
lm-1
If in Equa. (13) we put - — u for u we have
= 1+4
m = l
TRANSCENDENTAL CONSTANTS. 407
and substituting u = 0 in (10) and u = 0 in the equation just written, it is seen that
+ n'2m ,„ = * V.
If further we differentiate (8) with regard to u and then put u = - , we have
and if Equa. (7) be differentiated with regard to u, it becomes for u = 0
m=«
= #4(0) =1 + 8
- 1 Subtracting (18) from (17) we have
Jacobi (Werke, I, pp. 159, et seq.) has given forty-seven such formulas as those above.
ART. 344. In Art. 89 mention was made of the fact that many of the properties of the 0-functions had been recognized by Poisson. For example, in the 12th volume of the Journal de VEcole Poly technique, p. 420 (1823), he established by means of definite integrals the formula
tf-
2 e~9}C/x + 2 6~163r/
T^V
To verify this formula by means of the elliptic functions, let x = — •
-I jr
tity x becoming- = — . Hence if in the formula
.'' L\.
Instead of k we take the complementary modulus fc' = \/l — k2, the quan-
'M
v/^=1 +
V ^
we change k to &', we have
_5 _ if =l +2e x + 2e *
and consequently the formula of Poisson.*
* In this connection see a remark by Abel, Crelle, Bd. 4, p. 93.
408 THEORY OF ELLIPTIC FUNCTIONS.
ART. 345. In Arts. 260 and 320 we derived the relations au = ?e2^2#iO), [u
where
Noting that
we have, if we put
. *'»=
e\— 63
— 63
If follows immediately that
/2o> = * n
7T
or
/2co VT=
— 63 —
We also have
TRANSCENDENTAL CONSTANTS.
2(1 4- 2
409
= #2(0)
2 >
/
It is further seen that
or
and similarl
2(61 +
92= -(TT-
t?28(0) + t?38(0)],
»-5?f-
= 4 e
and
=(e3-
= _.
16 Wi2
ART. 346. The formulas of the preceding Article may be written
(1)
(2)
(3)
(4)
T) -
or.
(5)
410 THEORY OF ELLIPTIC FUNCTIONS.
Noting that the coefficient of u3 in ou is zero, and that the coefficient of u2 in oxu is — J e^ it follows by a comparison of the coefficients on the right-hand side of equations (5) and (6) that
(8) .
2
From (7) and (8) we have at once the relation of Art. 339,
'" " #2"(0) fl3"(0)
" " '
ART. 347. The formulas of Art. 329, in virtue of the relations just derived, may be written
The six formulas of Art. 328 thus offer a means of deriving the values of the functions o\, <72, <73, having as arguments the quantities oj, o/', CD'.
The results as set forth in the Table of Formulas, XLIV, should be veri fied. We have for example
e 2
Such formulas may also be had as follows:
Since z = e***, where u = 2 a>v, when -u takes the values
the values of z are i, iq*, q*',
and since W'— in" to = — ,
7)0) T)U , Tl n ,
we have •&-- - ^- + j (1 + r),
so
that when u takes the values
0)
1)0) Tj(l) T)ll)
2 2 2
becomes e, e iq±, e
TRAXSCEXDEXTAL CONSTANTS. 411
If for example for u in the formula for on (in Art. 291) we write u = u", we have
2i l-g2
'
The formulas expressing aiu, a2u, a$u through infinite products are given in Art. 321.
EXAMPLES 1. Show that
(Jacobi, Werke, II, p. 431.)
2. Through a comparison of the coefficients in Formula (6) of Art. 346 show that
(a- 1,2,3; #4 3. Show that
tei — e*)2 3
~ 27
4. Verify all the formulas given in the Table of Formulas, XLI and XLIL
5. Show that
4)3 27 <3 1 ^8(0) + #8(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 C dt
r dz and f—
J (z2- f?)V(l - z2)(l - /c2z2) J (t-
In the neighborhood of the point z = /?, if ^ is noZ a root of S2 = Z (Z) = (l - z2)(l - /c2z2)= 0, the expansion of
v (1 - z2)(l - /c2z2) is by Taylor's Theorem
A
- z2)(l -/b2z2) where
1
It is evident that Legendre 's normal integral becomes logarithmically infinite for z = /? in both leaves of the Riemann surface as the two quan tities
-LAlog(Z-0) and --LAlog(z-/?); &p * p
and for Z = — /? in both leaves as
-J_Alog(Z+/3) and ^Alog(z + /9).
If /? is a root of (1 - z2)(l - k2z2) = 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(J-6) and =logtf-6).
v463- gf2^> - 93 412
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 a\ and a2> say> since the sum of the residues of the integrand must be zero.
We may write the integrand in the form
(z- «i)(z - «2)v2(z)
We shall choose the points [«i, v/Z(a1)], [a2, x/Z(a2) ] in the upper leaf of the Riemann surface and we must determine the constants A0, A\, A2 so that the integral does not become infinite at the two corresponding points [a i, — \/Z(ai)], [a2, — VZ(a2)] in the lower leaf. Accordingly we must have
(A0+ Aiai- A2NZ(tti) = 0,
= 0.
In the neighborhood of the point z = a\ we have by Taylor's Theorem
(z-a2)VZ(z) and consequently
It follows that
Res/(z, s)= ^°
[which owing to equations (1)] = In a similar manner we have
(ai— o:2) Z(di)
2 1
Res /(z, s) O ^2 A
(«2- ai) v/Z(a2) Eliminating J.x from the equations (1), we have
and eliminating A0 from the same equations, we have
414 THEORY OF ELLIPTIC FUNCTIONS.
It follows that
A2 a2\/Z(al] -a1VZ(a2} + (VZfe) - v'Z(ai) )z+ («2-« i)V/Z(z) - «i (z -
= A2 \ VZ(ai)WZ(z) + VZ(tt2)WZ(z)1 «2- ai L (z - «i) VZ(z) (z - «2)\/Z(z) J
When 7(z, s) has this form, the integral / /(z, s)dz is of the third kind, being
logarithmically infinite at the points («i, VZ(«i)), (a2, VZ(a2)).
This integral may be considered the fundamental integral of the third kind and written
Il(z, VZ(z); «i, \/Z(«i); «2> VZ(^)) or more simply II(z; «i; «2).
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 «i7 «2 if we add integrals of the first kind to H(z; «i; a2).
ART. 350. Take three points ai, Z(ai); «27 VZ(«2); «3j Z(a3) on the Riemann surface of Chapter VI and form the integrals
H(z; ai; a2), n(z; «2; «3), H(z; «3; o^).
Further, let A2, A2(1) and A3(1) be the constants that correspond to A2 above.
We may so choose the constants A, /*, v that the expression
(1) ^II(z; «i; a2)+/*n(z; «2; «3) + v II(z; o:3; ai)
does not become infinite at any point of the Riemann surface and is con sequently an integral of the first kind or a constant.
We note that in the neighborhood of the point «i the expression becomes infinite as
a\— ot-2 ai~ a
and consequently remains finite at a i if
Similarly the expression remains finite at a2 and a3 if
+ = 0 and
a2— a\ a2— «3 a3— a2
ELLIPTIC INTEGRALS OF THE THIRD KIND. 415
The third equation is a consequence of the first two. If the ratios of /, p, v have been determined from these equations the integral (1) is an integral of the first kind * or a constant.
ART. 351. We have seen in Chapters VII and XIII that the integral of the first kind has in common with the integral of the second kind the property of being a one-valued function of position on the Riemann surface Tf. This is not true of the integral of the third kind; for consider in the Riemann surface the fundamental integral above. In the neighborhood of the point z = «i we saw that the integrand could be written in the form
P(z- ai).
It follows that the integral over a small circle including the point a\ as center is
2A.
while the integral over a small circle including the point z = «2 is
2i 7T1.
If then two paths of integration (1) and (2) starting from the point po include both or neither of the points «i and «2, we come to the point p with the same value along either path.
Hence to construct a Riemann surface upon which the fundamental integral of the third kind will be one-valued we draw small circles around a\ and a2 and join these circles by a canal so as to form a connected curve. To make the surface simply connected we join this canal with the canal «, say in Tf (of Art. 142), by another canal AB. The new surface we denote
by T".
Denote the difference in values of the integral H(z; a\\ a^} taken on the left and right banks of the canals in T" by II(/) — II((0). It is seen that for the canal AB any path of integration must encircle both or
neither of the points «i and «2 to get from the left to the right bank. It follows that Fig> 74> along the canal AB we have H(X) - U(p) = 0.
* See Clebsch und Gordan, Theorie der Abel'schen Functionen, p. 118; or Klein- Fricke, Theorie der elliptischen Modulfunctionen, Bd. I, p. 518.
416 THEORY OF ELLIPTIC FUNCTIONS.
^ To go from the point D to the point C in the figure we must encircle either a\ or a2. In 'either case we have
This difference may be made - 2 m if in the fundamental integral we give to the arbitrary constant A2 a value such that
A* 1
0.2 —- Oil 2
ART. 352. Let us consider next the elementary integral of the third kind in the Weierstrassian notation
; a, VS&); 00) = C" J
+
2 (t - a) VS(t) where S(t)= 4 Z3 - g2t - gs.
Writing /? = VS(a) we note that in the neighborhood of the point (a, /?) we have
so that in the neighborhood of t = a
U(t- a; oo) =log(J- a)- ± ^ (t - a) + • . -
2ft da
and that the residue corresponding to the point t = a is +2 m. In the neighborhood of the point at infinity we have
1 _ 1 a a2
~~ T "i~ To "i ~TT~
- « t
S(t) and consequently in the neighborhood of infinity
ELLIPTIC INTEGRALS OF THE THIRD KIND. 417
Further, if we put t = reie
and v =y - = pe**,
we have p = V / -> <£ = — i 6,
so that a double circle about the point at infinity in the ^-plane corresponds in the opposite direction to a single circle taken around the origin in the r-plane. Hence (see Art. 120) the residue corresponding to the point at infinity is — 2 TTI.
ART. 353. It is also seen that if in the T'-surface we draw canals from the points a\ and infinity to the canal «, say, we form another simply connected surface T" in which the integral II(£; a; x) is one-valued. On the first of these canals we have
and on the second H(A) - TL(p) = — 2 xi.
If the point a coincides with one of the branch-points ei, say, then in the neighborhood of t = e\ the integral 11(2; e\\ oc) becomes infinite as log \^t — e\] while in the neighborhood of t = oc this integral becomes infinite as log \/t.
Further, if we put
II(*; «2; ai)= H(t; «2; oc)- Ufa ttl; oo)
CVS(t) + VS(a2) dt C
\/S(t)+VS(al) dt
it follows from Art. 349 that II(£; «2; «i) becomes logarithmically infinite at the arbitrary points a2> «i but has a definite value f or t = oc . If here the point a\ is in the lower leaf directly under a2, so that a2 = a\, = — Vo(«i), then the above integral
<*i)\'S(t) which is the integral cited at the beginning of this Chapter.
ART. 354. To study the moduli of periodicity of the integrals of the first, second and third kinds, Riemann * took two functions u and v and considered the integral
J UdzdZ'
When u and v are integrals of the first and second kinds the integrand u —
dz
is one-valued in the Riemann surface Tr ; when one of these functions is an
* Riemann, Theorie der Abel'schen Functionen; see also Appell et Goursat, Fonctions algebriques, Chap. III.
418
THEORY OF ELLIPTIC FUNCTIONS.
integral of the third kind, the integrand is one-valued in the surface T" . Riemann's mode of procedure is essentially the following : The integra tion is taken first over the entire boundary of the simply connected sur face in which the integrand is one- valued, and secondly over a curve which gives the same value of the integral as the first curve; for example, the circle or double circle around the point at infinity. Since the latter curve contains in general no discontinuities of the integrand, the associated integral is zero. Consider the two integrals
where u and £u are integrals of the first and second kinds respectively and where the integration is over the complete boundary of the surface Tf taken in the positive direction.
Let the moduli of periodicity of II(£; a; oo) on the canal a be II (X) — !!(/>)= 2 Uj and on the canal b let H(p) - II (A) = 2 u'. Further, note that the integrands of both I\ and 1 2 are continuous at the point t = GO.
ART. 355. The Riemann surface T" projected on the w-plane is (see Art. 197) Represented in the figure. It is evident that
\
Fig. 75.
or
U°udU+ fudtt ,
uz *J around a.
CU\udTI-(u
9S UQ
UQ
= - 2 oj'
2iriu(a)
2co
+ 2 niu(a)
on 6
2 niu(a).
This integral around the circle at infinity is zero. It follows that
and similarly from I2,
ELLIPTIC INTEGRALS OF THE THIRD KIND. 419
Noting that TJO/— cur)' = ^,
it is seen * that o = yu(a) — co^(a),
The quantities u and u't the moduli of periodicity of the integral II (Z; a;oo), have values that are the negative of those given, if the canals a and b are crossed in the opposite direction, or what is the same thing, if the direction of integration around these two canals is taken in the opposite direction.
EXAMPLES
1. Derive the results analogous to those given above for the integral II(z; «1;a2), the surface T" being that given in Art. 351 (see Forsyth, Theory of Functions, §238).
2. Let u = II(z; at; «2) and v = H(z; «3; «4) and discuss the moduli of periodic ity in the associated Riemann surface (see Koenigsberger, Ellip. Funct., p. 278).
ART. 356. We wrote (see Art, 196) t = pu, VS(f) = - p'u; it follows, if a = $>UQ, VS(a) = — P'UQ, that
•V8®VS(t)+ VSM dt _ I ruy'u 4- 9'\i 2(t - a) V~S(?) 2J V" ~ fi?M-
The quantity UQ must not be congruent to the origin. In Art. 299 we saw that
1 p'u + p'lip = a'(u - Up) _ o^u O'UQ_
2 $>U — $>UQ o(ll — UQ) OU
Through integration it follows that
0(u0 - ») ^
1 (-"<?' u + V'tlo du =
2J $11 — tfllQ OUOUQ
The constant C is to be so determined that in the development (see Art. 300)
l r-g^±^dtt = _ iogtt + c-
2 J pu - <?uo C is zero.
We then have (see Schwarz, loc. cit., § 56)
U(t; a; oe ) = log a(u°~ u) + u
* See Schwarz, loc. cit., § 59.
420 THEORY OF ELLIPTIC FUNCTIONS.
It follows at once, if m is an integer, that
',a; oo)- II(a;$;oo) = u - - UQ — + (2m + l)m,
(JUQ OU
a result which corresponds to the interchange of argument and parameter in the Legendre-Jacobi theory of Art. 258.
ART. 357. Legendre Traitc des fonctions elliptiques, t. I, p. 18, repre sented the elliptic integral of the third kind in the form
II (n, fc, <« = C- - ^ [see Art. 167],
J (1 •+ nsm2(>) A<>
where the parameter n may be positive or negative, real or imaginary. This integral may be written
TT/ 7 \ C du U(n,k,u) = I -— —-- J 1 + n snzu
It follows that
TT/ 7 \ C —n sn2u -,
tt(n,k,u)-u = I—- — du,
J 1 + n snzu
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 = — k2sn2a,
where a being susceptible of both real and imaginary values, leaves n arbitrary.
J Q -t
Multiplying the right-hand side by — > the form of the elliptic
STL a
integral of the third kind adopted by Jacobi is
uk2sna cna dna sn'2u -,
ART. 358. In Art. 294 the following equation was derived:
92(0) ecu + a) ecu -a) = l _ k2sn2u Sn2at
If we differentiate logarithmically with regard to a, we have
2 k2sna cna dna sn2u _ ®'(u — a) _ Q'(u + a) , pQ^a) 1 - k2sn2u sn2a ~ ®(u - a) ®(u + a) 0(a)
from which it follows at once that
TT, v 1, ®(u - a) . 0'(a)
ELLIPTIC INTEGRALS OF THE THIRD KIND. 421
Interchanging u and a we further have
n(a,u)-llo8e<*-M>+ag^, 2 50(a + -w) 0(u)
from which it is seen that
U(u,a)- H(a, u}= uE(a)- aE(u).
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 II (u, a) is denned, that (1) H(M, a) = - H(- u, a) and
(2) n(o,o)=o.
Further, since snK = I, en K = 0, dnK = k', we have
(3) IL(u,K)=Q.
For a = iK' we have sn a = ao = en a = dn a, so that
(4) no/, ;£') = «;
and since
sn(K ±iK'} = -, cn(K±iK')= ^F~t dn(K ± iK')= 0, A; 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 II (u, a) through the theta-functions, as do also the formulas
(7) IKK + iK', a) = (K + iK') Z(a) +
ma
2K
(8) 11(2 iK1, a) = 2 iK'Z(a) + |^,
(9) U(u + 2K, a) = II(M, o) + 2 KZ(a),
(10) U(u, a + 2K)= n(u,a) = U(u,a + 2iK'),
(11) n(M + 2iK',o)= H(M?O)+ 2U(K + ^',0)- 2H(K,a)
- U(u,a)+2iK'Z(a)+ ^-
A
From the equations (9) and (11) it is seen that the moduli of periodicity of II (u, a) are respectively
2 K Z (a) and 2 iK'Z (a) + ^-
422 THEOKY OF ELLIPTIC FUNCTIONS.
ART. 360. From the definition of U(u, a) given in Art. 357 we have
d II (u, a) _ k2sna cnadna sn2u du 1 - k2sn2a sn2u
= Z(o) + i Z(u - a) - J Z(u + 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
V" k2snucnudnusn2a „ . ,
The addition of these two equations gives
Z(u) + Z(o) — Z(u + a) = k2snu sna sn(u + a),
which is the addition-theorem of the Z-f unction (see Art. 297). ART. 361. From the formula
Vk' we have by writing in in the place of u
V ^« v- — — / /
Vk' or, (see Arts. 204 and 220)
0(0, k')
If we take the logarithmic derivative of this equation, we have tZ(tu + K) = ^-r + Z(« + K', k').
If these expressions are written in the formula •
U(iu, ia + K)= iuZ(ia
0^a + ^u + K) we have
H(iu, ia + K} = TL(u, a + K', kf).
ELLIPTIC INTEGRALS . OF THE THIRD KIND. 423
If a is changed into ia, it follows that
H(iu, a + K)=-U(u,ia + K', k').
These results may be derived directly by a consideration of the integral which defines H(w, a) [see Jacobi, Werke I, p. 220]. ART. 362. In Art. 227 we saw that
It follows directl from the formula
that
Ku 2Ka\ 2Ku
n ~
, q cos 2 (it + a) _^ q2 cos4(u + a) _, 1-92 2(1 -g«)
_ gcos 2(u — a) _ q2 cos4(i£ — a) _ 1 - q2 2(1 - 54)
2Ku
Ku \ - I _ 9 |"g sin 2 a sin 2 u q2 sin 4 a sin 4 u ~ 2Ka I !-52 2(1 -5^)
q3 sin 6 a sin 6 u . "]
3d -96) "J
THE OMEGA-FUNCTION. ART. 363. Jacobi (Werke, I, p. 300) put
/ E(u)du = log li(w).
i/O
If we integrate the formula of Art. 297
E(u + a) + E(u - a) = 2 E(u) -
1 — k2sn2a sn2u
from u = 0 to u = u, we have at once
log - + iog - = 2 log o(M) + log (1 - k2sn2a sn*u),
O(aj O(a)
or O(M + a) fl(?< — a) 1 790 o
— - - - — { — J — - — - = 1 — k2sn2a sn2u.
424 THEORY OF ELLIPTIC FUNCTIONS.
Further, if u and a are interchanged in the above formula, it becomes
E(u + a) - E(u - a) = 2 E(a) - 2 k2sna cna dna sn2u ^
1 — k2sn2a sn2u which integrated from u = 0 to u = u is
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
n
log 0(m) = log cn(u, k')-~ + log Q(u, k'),
ft
or _«2
e 2 cn(u, k') Q(w, A;'). ART. 364. From the formula E(u + 2 mK) = E(u)+ 2 mE we have
f|^
n(u + 2mK)= 2mE 0(2 mK}
If we put M = — 2 mK in this formula, and note that
0(- u) = Q(u), 0(0)= 1, we have O(2 mK) = e2m'EK,
and also to(u + 2
Or -
0(w + 2mK)= e 2V(u).
Eu*
This formula shows that the function e 2K &(u) remains unchanged when the argument is increased by the real period 2 K. Further, if in the formula
e 2 cn(u, k')tt(u, k'), we write u + 2 nK' in the place of u, we have
(u+2nK'}2
Q(iu + 2niK') = (- l)ne cn(u,k')Q(u + 2nK',k'),
or
e 2 n(iu + 2niK') = (-l)ne 2 cn(u,k')e *K' Q(utW).
ELLIPTIC INTEGRALS OF THE THIRD KIND. 425
It follows that
~r( ^
= (- l)*e 2K' O(tu). If in this expression we put —iu for u or u for iu, we have
6 Qw
from which formula it is seen that the expression
e 2K>
remains unchanged when u is changed * into u + 4 ART. 365. We derived in Art, 263 the formula
ei— e3 s from which we have at once through logarithmic integration
- e3 • u) Writing these values in the formula
,
2 O(« + a)
it is seen that
• u, ^. a) = I log e^^u
2 n[vei- e3(u + a)J
- e3 • a)
2 <T3(w + a) a3a
[See Schwarz, loc. tit., p. 52.]
ART. 366. The following relations may be derived from the addition theorems of the theta-functions given in Art. 211, formulas [C]:
e«(0)H(n + a)H(u - a) _ 2 _ 2
» + a)H. (u - a) =
* See Jacobi, Werke,, I, p. 309.
426 THEOEY 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
r»
Jo
I r
8nacnadnad 1 , H(q - u)
o sn2u - sn2a 2 to H(a + u) 9 (a)
uk2snacnadnacn2u d =l^ ®i(u - a) 0'(a)
k2cn2ucn2a + k'2 2 g ®i(u + a) ' U 0(a)
acnadna dn2u , = 1 j Hi(t6 — a) , 0'(cQ dn2udn2a — k'2 2 HI(W + a) 0(a)
These integrals * may all be expressed through the integral II (u, a) and an elliptic integral of the first kind ; for example
f
Jo
sn a en a dn a i _ TT/ , '/TM— ucnadna
sn a
ADDITION-THEOREMS 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
TT/ \ TT/ \ TT/ \ IT ®(u — a) 0 (v — a
ft(u,a)+ n(tva)-H(« + v,a) = -log -) -- (r>/ . rv .
2 @(w + a)@(v + a)@(w + 'y — a)
For brevity we shall put
+ v + a) = p( }
+ v - d)
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(^ + v) 0(jM - v) = 02(/£) ©2( we have at once
* See nofe by Hermite in Serret's Calcul, t. II, p. 840.
t Legendre, Fond. 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
o/U + V \1L , 0 2^ + ^ 2/U + V i_ \1
t2 — a 1-Fsn2-— sn2 — - + a)
*•(", *, a) =] x^ 7 V 2 ({J ? ^ /I
1 - A;2sn2 /'^^-Jsn2 f ^^ + a J 111 - k2m2 ^^sn2(^--a
From the formula
, x / x sn2u — sn2v
sn(/jL + v) sn(/i - v) = f-— —
1 — k^sn^juisn^v
we further have
— V
Taking the products of these two equations each multiplied by —k2 and adding a common term on either side, we have *
2 \ 2
multiplied by { 1 — k2sn asnusnvsn(u + v — a]
2 \ 2
•^-^•Sf^f^-a)
Writing —a for a in this equation, we have a second equation, which divided by the first gives
- a
1 + k2sn a sn u sn r sn(u -t- r -L q) 1 — k2sn a sn u sn v sn(u + r — 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
p, ^_ 1 — k2sn a snusnv sn(u + v — a) 1 + k2sn a snusnv sn(u + v + a)
ART. 368. It follows also from the expressions given in the preceding Article that
_ a)02(, _ a)_ 02(0)
1 — k2sn2(u — a) sn2(v — a) u + a)0*(* + a) = 0*(0) .
1 — k2sn2a sn2 (u + v — a)
M + „ + „) = e*(0) e(« + »)e(« + P + 2«) . 1 — k2sn2a sn2(u + v + a) From these equations we have
F(u v a)= H * ~ k2sn2(u + a) sn2(?; + a) } { 1 - k2sn2a sn2(u + v - a) }"P U 1 - k2sn2(u - a) sn2(v - a) } { 1 - k2sn2a sn2(u + v + a) } J
ART. 369. Since II (u, a) — II (a, u) = wZ(a) — aZ(u), we have H(w, a) + IL(u, b) - n(^7 a + 6)
= H(a, w) + H(6, w) -H(a + 6, w) + w{Z(a) + Z(6) - Z(a +6) } = J log F(a, 6, u) + w fc2sn a sn b sn(a + b),
which is a theorem for the addition of the parameters. ART. 370. In the formula (see Table (B) of Art. 211)
y) = &(x + y y + «)^i(«)^i
write 3. . 2^? = 2& and ^ = _ 2Ka and + 2Ka respectively.
71 71 7T 7T
Divide the first result by the second and we have
l _ H(o)H(M)H(p)H(M + v-a)
®(u - a] ®(v - a) ®(u + v + a) _ @(o) 0(u) Q(t;) Q(i^ 4- y - a) ; 0(w + a) 0(v + a) @(w + v - a) ~ - H(o)H(M '
F( x _ 1 — fe2sna snu snv sn (u + v — a) 1 + k2sna snu snv sn (u + v + a)
Remark. — By writing as we have done
n =-k2sin26,
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 desf auctions 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 = - k2sin20, (2) n = 1 + k'2sm26,
where 6 is real in both cases. Owing to the fact that
tan-1 it = -i 2
1 — t
the inverse tangent appears in the second case instead of the logarithm.* ART. 371. If we put
we have from Art. 355
U(tl; t0;
OU\OllQ
t0; x)=
t0; *)= If u% = ^1+ u2, it follows that
; oc)- log/ (i^3, 112,1*!, u0),
where
Olif\—
2
P'UO) Wo )
The last formula is verified by using the equation (see Art. 335, [B.]) owa(u + v + w) 0i(u — v) = o(u + w) a(v + w) o#i aw — oi(u + w) a),(v + w) auaVj
and the formulas given in the Table of Formulas, No. LXII, combined with the formula
7,
-^-
02U
It follows that
; t0', oo)
^2-^0 [See Schwarz, loc. tit., p. 90.]
* As an application of Abel's Theorem, Professor Forsyth (Phil Trans., 1883, p. 344) has given a very elegant method for the addition of the elliptic integrals of the third kind. See also a paper by Rowe (Phil. Trans., 1881, p. 713).
430 THEORY OF ELLIPTIC FUNCTIONS.
EXAMPLES 1. Show that
IL(u + %K,%K} = i(l - k')(u + \ K) - i log dn u + $ logX/Jfc7,
U(u + i itf', i itf') = i i(l + k) (u + \ iK'} - i log «n M + i log -
V/c
iX') - 4 log en w+ i log
A;
. Show that
H(M + K, a) = H(M, a) + KZ(d) + i log
a)
II(it, a+ A) = II(w, a) — k2 sna sin coam a • u+ % log — - - *
dn(u+ a)
3. Verify the formulas
du_ _ dlo
~ da 2~*®(u-a)
£^) + IlogH(! da 2 H(u — d)
_M<Oog©M__Llog<
— k2sn2(ia)sn2u] da 2i ®(u — id)
rdna cot am a du _ d log H(a) _ 1 , ®(u + a) 1 - k2sn*a sn2u ~ '
CK sna cna dna du = , ^ _ . d log 0 (a) 1 , H(it + a) Ju sn2u — sn2a
rfc2sn (10) en (id) dn (id) sn2u du _ d log @ (la) 1_ , G(M + i'a)
4. Show that
H(u, d) + H(v, a) - TL(u + v, a)
2 £l(u + a) O(v + a) O(w -f v — a) and that
- a) O(M + v+ a)
(w — a) O(v — a) O(w J2 / \
i-^ L = 1 — k* sna snu snv sn(u+ v — a),
CHAPTER XX
METHODS OF REPRESENTING ANALYTICALLY DOUBLY PERIODIC FUNCTIONS OF ANY ORDER WHICH HAVE EVERYWHERE IN THE FINITE PORTION OF THE PLANE THE CHARACTER OF INTEGRAL OR (FRACTIONAL) RATIONAL FUNCTIONS
ARTICLE 372. We have seen that the simplest doubly periodic functions, which in the finite portion of the plane have everywhere the character of integral or (fractional) rational functions, are the functions pu, snu, etc. We shall show in the present Chapter that all other doubly periodic func tions which have the properties just mentioned may be expressed in terms of these simpler functions.
We shall study in particular five kinds of representations:
(1) Representation as a su?n of terms each of which is a complete derivative.
(2) Representation as a rational function of, say, $>u and p'u [Liouville's Theorem].
(3) Representation in the form of a quotient of two products of theta- functions or sig ma-functions.
(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- mite 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(w) and was defined through the equation
m = + » 2ri'mu . 6
X(M) =
We saw that this function satisfied the functional equations (l) X(t* + a)-X(iO,
--(2u+6) (2) X(M + &)=? °
We also saw that this function vanished on the point -~ — = c and on all congruent points, but nowhere else.
431
432 THEOEY OF ELLIPTIC FUNCTIONS.
By writing X(w + c)= Xi(w) we formed in Xi(w) a function that van ished for u = 0 and congruent points. It is seen that
We next wrote (Art. 96)
and we saw (Art. 98) that every one-valued doubly periodic function F(u) with periods a and 6 and which had everywhere in the finite portion of the plane the character of an integral or (fractional) rational function could be expressed in the form
F(u)= C +
+ Z0"(u- uk) ---- ± -Zo*"1' (u -
where k extends over the n infinities uk of F(u) that are situated within a period-parallelogram, the order of these infinities being Xk respectively;
C is an arbitrary constant, while bk,v is the coefficient of - —in the
(u- uky
expansion of F(u) in the neighborhood of the infinities u = uk (k = I, 2, . . . n). If r is the order of the function F(u) (see Art. 92), then r = AI + X2 + -^3
+ • • • + Jn-
The function ZQ(U) is infinite of the first order for u = 0. We may next write ;u2+juM
where X, p are constants. It follows that
i(w) u iu
We therefore have
ZQ(U)+ 2XU + fJL = Zi(w),
Zo/(ti)+2A-Z1'(ti)J
Z0/r(w)= Zi"(tt), etc. The formula above becomes
k=n
F(u)= C +5) K,i lzi(w - u*> ~ 2 ^(w ~ u$ ~ fc=iL
- 6- Z'i - i* - 2 A4 Zu - Mfc
DOUBLY PERIODIC FUNCTIONS OF ANY ORDER. 433
k=n k=n
The constants ^ bk,i(2 Xuk — //) and ^fbk,22 X
k=l k = l
may be embodied in the constant C, making, say, C\. We also note that
It follows * that
t- •Ci.4
1!
• ±
i) •
ART. 374. To introduce the Jacobi Theory write a = 2K and b = 2iKf.
It follows at once that
and
If we make A = 0, // = -^-, we have from above 2 /£
and also
On the other hand we had
H(u).
We may therefore write in the formulas above ATL(u) instead of
where A is an arbitrary constant, and Zi(u)= ^ •
H(t»)
It follows that we may express every doubly periodic function F(u) with
the characteristics required above through the function u^ »
M ( w)
* See Hermite, Ann. de Toulouse, t. II (1888), pp. 1-12.
434 THEORY OF ELLIPTIC FUNCTIONS.
ART. 375. To introduce the theory of Weierstrass write a = 2 w and b = 2 at',
so that Xi (u + 2 aj) = Xx (w)
and «
We shall so choose the constants A, /* that instead of the function we may employ the function on. We have the relations
a(u + 2 o>) = — a(w + 2<o')= —
We further have
It follows that
Comparing this result with
ff(u + 2 it is seen that we must write
^Xco = 2y and 4 Xto2 + 2 /*&> = 2^w + m,
where id has been added to change the sign. We have at once
A--2L and P--&,
2aj 2aj
and consequently also
This function satisfies the first of the functional equations which au satisfies.
We have further
2,^u+2,^+,rf< -^u-2
^(u + 2w')=-e ' w " e w
or, since yw' — TI'UJ = — ,
we have -^(u + 2w') = —
It is thus proved that ^(u) satisfies also the second functional equation satisfied by au. We may therefore put
ty(u)= Ban, where B is an arbitrary constant, and
au
DOUBLY PERIODIC FUNCTIONS OF ANY ORDER. 435 ART. 376. It is evident from above that we may write F(u) in the form *
We here have F(w) 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 uk.
ART. 377. We saw in Art, 299 that
2 pit - puk
where we assumed that uk is not congruent to a period; otherwise £uk and $>Uk 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,
*
v* Bkw:(u - wo - v* Bk<»;u - v
+* *i **
2 *-* pu - 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 ^u^Bkw = 0. It then follows at once that ^ BkM{(u-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 Bk(1^u to 2)*5fc(1)C(w ~ ft*) that we may u — uk). But here also it is seen that
>u = o, since Bk™ = 0.
have V
Thus without exception it is seen that Bkw£(u — Uk) is rationally express ible through pu and p'u. k
* * See Kiepert, Crelle's Journ., Bd. 76, pp. 21 et seq.
436 THEORY OF ELLIPTIC FUNCTIONS.
Further, since the derivatives of £(u — Uk) are all rationally expressible through <@u and @'u, it follows that
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 u = 0 and congruent points, then this function F(u), say, is an integral function of pu and <@'u. To prove this note that since u = 0 is the only infinity within the first period-parallelogram we have k = 1 and u\ = 0. Further, since 2) 5*(1) = 0, it follows that #!(!)= o. We thus have
By definition we had *
£:«=
and consequently
'u + p'upu),
/r+2^ + ^V), "+ 3 >+ 3
It follows that F(w) is an integral function of g?(w) and g?'(w).
ART. 378. Let F(u) be a doubly periodic function of the second sort
so that
F(u + a)= vF(w),
F(u + 6) = i/F(tt). The logarithmic derivative of F(w),
is a doubly periodic function of the first sort. The function (£>(ii), as seen in Art. 4, becomes infinite on the zeros and on the infinities of F(u). Let ^i°? ^2°, • • • j um° be the zeros of F(u)', and at m° let F(u) be zero of the
* See Kiepert, Dissertation (De curvis quarum arcus, etc., Berlin, 1870).
DOUBLY PERIODIC FUNCTIONS OF ANY ORDER. 437
/\- order (i = 1, 2, . . . , m). Let u\, u2, . . . , un be the infinities of F(u); and at Uj let F(w) be infinite of the /// order (/ = 1, 2, . . . , ri). We may therefore write
F(u) = (u - Ul°)«Fi(u) (i = 1, 2, . . . , TO),
where Ft-(w) is neither zero nor infinite for w = wt°. It follows that
and consequently Rea^(u) = /U;
and similarly Re^(u) = -W.
U=U;
It is thus seen that (f>(u) has only infinities of the first order. It was seen in the previous Article that if the development of ^(M) in the neighbor hood of its infinities is given, we may express (j>(u) through the ^-functions. It follows also here that the quantities Bk(v+l) are all zero, and conse quently
- U2°) + ' • ' +^(U - Un°) u - U2)~ ' • • -j*mr(u - Un).
Also, since
it is seen that
F'(u)
j '
Through integration it follows that F(u) = ec>u+c> +(u ~ MI°)A
Every doubly periodic function of the second sort and consequently also every doubly periodic function of the first sort may be expressed in this manner. This representation corresponds to the decomposition of a rational function into its linear factors (see Arts. 12 and 26). Instead of the function ty(u) we may write either H(M) or au.
Further, since the sum of the residues of a doubly periodic function of the first sort (Art. 99) is zero, we have
2Res</>(«)= Sx-Sju = 0, or 2/i = SJM = r,
where r is the order of the function F(u). It follows also that a doubly periodic function of the second sort F(u) has as many zeros of the first order as it has infinities of the first order, a zero or infinity of the yth order counting as v zeros or infinities of the first order.
438 THEORY OF ELLIPTIC FUNCTIONS.
ART. 379. We may write
(A) F(u) = ecuc' ~ ~ 2 - - - a(u-
a(u — u\)o(u — U2) • • - o(u — ur)
where some of the quantities UIQ, u2°, . . . , ur° may be equal and some of the quantities HI, u2, . . . , ur may be equal. This representation of a doubly periodic function is very convenient when all the zeros and infinities are known.
We have assumed that the points m° and Uj all lie within the same period-parallelogram. This assumption, however, is not necessary; for if 2 co be added to or subtracted from the argument of one of the ^-functions which enters in the expression above, then only the factor before the frac tion is changed.
For example.
a(u- up-2u))=- e-2*(t*-tt,-o,) 0(u _ Up)f
or o(u — Up) = - e^(u-Up-aj) 0^u _ ^+ 2 CD)].
It follows that every elliptic function of the rth degree may be expressed in the above form in an infinite number of ways.
ART. 380. It we write u + 2 a> in the place of u, then a(u — ur) be comes— ^(ti-ii'+w) o(u — uf), where U' = UIQ, u2°, . . . ,ur°;ui,u2, • . . , ur. Hence, since F(u + 2 w)=F(u) [if we suppose that F(u) is a doubly periodic function of the first sort], it follows that
(B) F(u) = i(*+?+< 6 ^ a(u~ Ul°)a(u ~ U2°} ' ' ' °(U ~ Ur°} -
-2^2% e i=1 a(u — Ui)o(u — u2) • - • o(u — ur)
The two expressions (A) and (B) must be equal. We must consequently have
or e
/t=r i=r \
and similarly e i=l i==1 — 1.
In virtue of these relations we also have
(1) 2 co, + 2 T? (j^Ui - j?uA = 2 Mm,
/i = r i = r \
(2) 2 co)f + 2 ^ ( 2)^i - 2) w.'° ) = 2 M'o,
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
But since yaj' — o»/ = ^ xi, it is seen that
c = 2 M'i) - 2 A/y. If c is eliminated from (1) and (2), we have
,MI - 2ju.-° r 2 M<»' ~ 2 M'w.
t = i 1 = 1
For the sake of greater simplicity we may write — m! for M and m for Af '. We then have
c = 2 mTj -}- 2 m T) f
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 ur° by ur° + 2 ma> + 2 m'aj'. If this is done, then for the new system of zeros and infinities we have m = 0 = m' and consequently
n and c = 0.
»=i 1=1 We then have
F(u) = C °(u ~ u^°(u ~ ^2°) ' • • o(u — ur°) ^ o(u — ui}a(u — u2) • • • o(u — ur)
It is thus seen that F(u) depends upon the quantities 2 w, 2 a>' ', C; u\j U2, . . . , ur\ and upon r — 1 of the quantities uf (we note in partic ular that of the r quantities uf there are only r — I 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(^) 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
%[F(u) + F(— u)] = ^o(u), say, is an even function, and that %[F(u)— F(— u}] = ^i(u) 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)= l[F(u)+F(- u)] + l[F(u)- F(- u)], or F(u)= ir0(u)+^i(u).
Corollary II. — We may next prove that every even elliptic function of order say 2 r may be rationally expressed through gw. Such a function may be represented in the form
We may also write
a(u - Ui°)a(u + uf)
o(u - Ui°}a(u + UJQ) __ a2uo2Ui°
0(U — Ui)(7(u + Ui) 0(U — Uj)o(u + Uj)
02U02Ui
(J2Uj°
We therefore have
4=1 1=1
a formula by which it is shown that -^oO) is rationally expressed through We may therefore write
where RQ denotes a rational function of its argument. Further, if
is an odd elliptic function, then, since <@'u is also an odd elliptic function,
is an even elliptic function = RI($>U), say,
so that ty\(u)= 8>'u R i (&u) ,
where RI denotes a rational function of its argument.
DOUBLY PERIODIC FUNCTIONS OF ANY ORDER.
441
ART. 381. As an interesting application of the above representation of an elliptic function we note the following: In determinant al form we write the formula
G2UG2V 1, %>V
We may also express through sigma-quotients such expressions as 1, pu, p'u 1, pv, $>'v = A(w), say.
The infinities of pu and p'u are congruent to the origin, $>u being infinite of the second and p'u of the third order for u = 0. The determinant is a doubly periodic function of the third order in u with zeros HI°= v, u2° = w and u3°= - v — w. Further, Ui° + u2° + u3°= 0 = 2 (infinities), the infin ities being the triple pole zero.
It follows then that the determinant must be of the form *
r a(u + v + w)a(u — v)a(u — w)a(v — ir) _ <
v-' „ ~ .. — —1 Ut
Multiply both sides of this expression by u3 and then make u = 0, and we have
p g(y + w)0(v — w) _ a2va2w
1,
so that C = - 2. It follows that
o(u 4- v + w)a(u — v)a(u — w)a(v — w) _ 1
1 , tpv, o*
1 &W £>'
Appell and Lacour (loc. tit., p. 63, Ex. 2) give an incorrect value to the constant C.
Further, since p'uj = 0 = $/o/, if we write in the expression above v=aj and w = a/, it becomes
a(u
-f (t)')a(u — a))a(u — ajf}o(co — CD')
or
and consequently
r i >\
fj(oj + (*) )
— a> —
o(u
U — CD')
0*1*0*0)
= 4 (<pu — e2)(<pu - e-i)(<?u - 63).
* See Daniels, Am. Journ. Math., Vol. VI, p. 266.
442 THEOEY 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) = d + 5)£*<i> C(u - uk) + 2)5,(2) p(u - uk)
k k
In Art. 272 we saw that
and consequently
If we take the summation over this expression with regard to k and note that the summations with regard to w and with regard to k may be inter changed, we have
We further note that
V 1
(u - U
(w - ^) = - 2! V
**
,
+
,\ 1 1?
< ; - rz -- J?
f •(» - t*fc- ^)2 w;2 )
(u - uk- w)9 - - l- - -, etc.
It follows at once that
- Uk
^ ( (u - uk- wy
+|^__^_ + ||_^L
•*)
DOUBLY PERIODIC FUNCTIONS OF ANY ORDER. 443 If for brevity we put
(u - nk)2 (u -
the above formula may be written
u) = Ct + /(u)
u- ' A;
ART. 383. We may next consider the fifth kind of representation of the doubly periodic function F(u). We saw in Art. 287 that
2ft*-*-*
ion
where 22 = ^ = e w . We have at once
0 + Z~l = t 4- 1 Z - Z~l t - I
If we write
(M-M*)- ^ «t^"
6 " = — , where tk= e a '
tk it follows that
+ Next let
and observe that /i (0 = 0 f or t = 0 and f or t =00. We may then write the formula for F(u) in the form
A- = 1, 2, . . . , n \ 1,2, ... ,4- I/ We have the following expansion (Art. 286) :
444 THEORY OF ELLIPTIC FUNCTIONS.
It is further seen that
1 = --L_ aild tk - tik
t
~~\2 (t - tk)*
\fk~ V
Next let
^(2)
It is evident that /2(0) = 0 = /2 (<*>)•
The terms in F(u) which correspond to v = 1 are
The terms in F(u) which correspond to v = 2 are
If we differentiate the formula above for pu we have a suitable expres sion for p'u in the form of an infinite summation, which may be written
where /3(0 is a rational function in t having the property that
/3(0)=o=/3M.
We continue this process and finally write
the f unction /(O being a rational function in t such that
We therefore have
F(u)= Ci-
Since < has the period 2 w, it is evident that F(w) has the period 2w; also noting that < becomes h2t when w is increased by 2 a/, it is seen that 2 a/ is also a period of F(u) provided the above series is convergent.
DOUBLY PERIODIC FUNCTIONS OF ANY ORDEK. 445
ART. 384. We may establish the convergence of the series in the pre vious Article as follows: Since f(t) = 0 for t = 0, we observe that t = 0 is a root oif(t) = 0, so that we may write
• • -f 6^ It is always possible to choose t so small that
|W| + | MM + • • • + IV" I < i-
It follows that* the denominator in the fraction above is greater than J, while the numerator is finite. We may therefore write
f(t)<At,
where A is a finite quantity. It is further seen that
f(h*t) < Ah*t, f(h*t) < Ah*t,
It follows that the series f(t) + f(h2t) + f(h4t) + • - - is convergent; and in the same way it may be shown that f(h ~2 t} + f(h ~4 0 4- • • • is conver gent. We have therefore established the convergence of the series express ing F(u).
ART. 385. We may also express F(u) in the form of an infinite product whose factors are rational functions of t.
In Art. 380 we derived the formula
a(u —
where MI° 4- u 2°+ • • • + ur°= HI+ u2+ In Art. 291 we saw that
If for brevity we write
it follows that
t -, i _ ^2n(^ i _ /,2n_L
2- 2 « <7 " '
with corresponding formulas for a(u — ilk).
446 THEORY OF ELLIPTIC FUNCTIONS.
We next write
n
and note that /i (0)= 1 =/I(QO). We have at once
^-
F(u)=Ce "
That the product on the right-hand side is absolutely convergent may be proved by writing
where /0(0 = 0 =/o(°°); it then follows by Art. 17 that the above product is absolutely convergent if
m = +oo
is absolutely convergent. The convergence of this series is easily estab lished by using a geometric progression whose ratio is h2.
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 <@u and p'u, say
(f>(u) = RI($>U, %>'u), where R i denotes a rational function of its arguments. It follows that
~\ 7~> *\ ~D
,,, \ o/ii / , o/ii //.
Writing for <@"u its value p"u = 6 <^u — \ g2, it is seen that (/>'(u) may be rationally expressed through pu, <@'u. We therefore write
where R2 is a rational function of its arguments.
Any rational function of pu and g/w may be written in the form
T> / / \ ^i (v R ! (&u, ®'u) = 1|S
where G\ and G2 are integral functions.
DOUBLY PERIODIC FUNCTIONS OF ANY ORDER. 447
Further, since
(p'u)2
it is evident that we ma write
where S, T and W are integral functions of <@u\ or finally
where U and F'are rational functions of <@u. We have accordingly
(1) <f>(u) and similarly
(2) 6'(u)
where U\ and Vi are rational functions of pu. We note that V and V\ cannot be simultaneously zero; for U(pu) and U\(pu) are both even functions of u, while if <p(u) is even <j>'(u) must be odd and vice versa. From (1) and (2) it follows that
(3) ?'u = and (4)
In general both of these equations (and always one of them) have definite forms, since V and V\ cannot both be simultaneously zero. If then the values <t>(u) and <$u 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) 0{«
where g denotes an integral function of its arguments. If further we square the equation (3) and give to p'u2 its value in terms of #ra, we have
(II) gi{s>u,<t>(u)} = 0,
where g\ is an integral function.
On the other hand if V, say, is zero, we have from (1) the equation
(I') g { <pu, $(u) } = 0, and from (4)
(II') £i{^,<£'00} = 0,
where g and g\ are integral functions. We thus always have two algebraic equations among the three functions pu, <j>(u), <j>'(u).
Under the assumption that the pair of primitive periods 2 o>, 2 wf of pu are at the same time a primitive pair of periods of </>(u) it may be shown that the two equations (I) and (II) or (I7) and (II') have only one common root in pu.
448 THEOKY OF ELLIPTIC FUNCTIONS.
The following indirect proof is due, I believe, to Weierstrass: Suppose that a pair of values belonging to <fr(u) and <j>'(u) has been chosen and suppose that the equations (I) and (II) have two common roots, say
<@u = Si and <@u = 82.
Suppose that u\ is the value of u which satisfies the equation
$>ui= si.
Then also, since <@u is an even function, the value —u\ satisfies the same equation.
From the equation (3) above we have
The two values that are had through the extraction of the root are +@'u and —<@'u and there is only a choice of u between +u\ and —u\. We shall suppose that +u\ gives
By a comparison of (a) and (b) it is seen that
<l>(u)= <f>(ui). Next suppose that u2 is the value of u which satisfies the equation
then also — u2 satisfies the same equation.
In the same way as the equations (a) and (b) were formed, we have
and
„'„
V(s2)
It follows that </>(u)= <j>(u2), and consequently corresponding to <j>(u) to which a definite value was given at the outset, we have shown that
In the same way from the value of <f>'(u) which was chosen at the outset
we have
(ii)
DOUBLY PERIODIC FUNCTIONS OF ANY ORDER.
449
In Art. 37a it was seen that if the relations (i) and (ii) are true, then u\ — u2 is a period of <t>(u). It follows that, if 2co and 2w' are a pair of primitive periods of this function,
u\ — u2 = 2 mw + 2 m'a)',
where m and mf are integers. We have thus shown that the two equations (I) and (II) have only one common root. The method to be followed is the same if we take the equations (F) and (!!')•
It may be shown * that if two algebraic equations have only one root in common, then this root may be expressed rationally in terms of the coeffi cients of the two equations, so that therefore here
where R$ 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
C0+
0(U - W*)-
M - uk) = #(u - U Since 2) ^(1) = 0, we may write V Bk™ log <J(M - uk) = V
~~ —
We also saw in Art. 299 that
- 5) BkWr(u - Uk) =-'u k
It follows that
= 2, 3, . . . , lk - 1].
log o(Uk ~ M) + Constant.
an cZZtpfic function of u.
where ^>i(w) is a doubly periodic function with periods 2cu, 2w'. Further, since (Art, 356)
log
ou auk * See Baltzer, Theorie der Determinanten, p. 109.
450 we have
THEORY OF ELLIPTIC FUNCTIONS.
j*F(
u)du - u
The moduli 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 kind;
2. An elliptic integral of the second kind;
3. A finite number of elliptic integrals of the third kind;
4. A rational function of pu and p'u.
EXAMPLES
1. Show that any integral function of pu and p'u may be written in the form
<J(M)W
where uf, u2°} . . . , un° are the zeros of the function.
2. Show that any rational function of pu and p'u may be written
- a^'u + - • - + an&fr-Vu
F(u) = A a°
60 +
+
3. Write
•>(n-i).
(n'li c->(n~^u
Vuu • • • i 5^ «i
u}
-u) ... a(un —u)a(u
Show that
where C is independent of u.
Multiply both sides of this expression by un+l and determine C.
4. Express F(u) through the function Z^w) of Art. 374, and derive the expres sion corresponding to the one of Art. 387 for the integral / F(u)du in terms of Z(w) and the theta-f unctions.
CHAPTER XXI
THE DETERMINATION OF ALL ANALYTIC FUNCTIONS WHICH HAVE ALGEBRAIC ADDITION-THEOREMS
ARTICLE 388. The problem of this Chapter has already been solved for the case of the one-valued functions. Weierstrass * has also solved it for the many-valued functions by making use of the principles which we shall attempt to give in the sequel. Using a method due to him (see references in Chapter II) we must first show that a function <j>(u) which has an algebraic addition-theorem may be extended by analytic continu ation over an arbitrarily large portion of the plane without ceasing to have the character of an algebraic function; that is, in the neighborhood of any given point the function may be developed in a convergent series accord ing to powers of a certain quantity which may stand under a root-sign, and in which series the number of negative exponents is finite. We assume that the function may be defined in the neighborhood of a certain region about the origin and we choose a point UQ such that one branch of the function <j)(u) has the character of an integral function at the point UQ.
We may therefore write
(1)" <J)(U)=
Next put « = «„+«',
v = UQ + v', UQ being a constant.
Since <p(u) has by hypothesis an algebraic adoption-theorem, we have an equation of the form
where G denotes an integral function of its arguments. We therefore have
O, <£(«o + V), <t>(2uQ+ u'+ v')}= 0. Further, if we write
U = UQ,
V = UQ + Uf + V',
it is seen that
G{<£(MO), <t>(u0+ u'+ v'), (t>(2u0+ 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 freefy in the preparation of this Chapter.
451
452 THEORY OF ELLIPTIC FUNCTIONS.
If (j>(2 UQ + u' + v') be eliminated from these two equations, there results an algebraic equation of the form
u
' + v')}= 0.
We may consider <J>(UQ) as a new constant. Writing
(J)(U0+ U')= </>l(
we see that
If in equation (1) we write WQ+ M instead of u, we have
from which it follows that by a change of the origin the function <j>(u) may be changed into the function (f>i(uf) in such a way that the function (f>\(uf) has the character of an integral function at the point u'= 0 in the branch of the function under consideration.
Hence without limiting the generality of the given function <p(u)t we may assume that the point u = 0 in the branch in question of the function <j)(u) is a point at which </>(u) 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 </>(u) in the neighborhood of u = 0.
If then | u | < p, the function <j>(u) has the character of an integral func tion in the branch considered.
If | u < i p, v I < J fi, then is \ u + v \ < p, and we have
for the region considered.
If in this equation v is put =u it follows that
which is an algebraic equation between ^>(u) and (f>(2u) with constant coefficients. We may write this equation
(2) G1{^(w
If in this equation the value of u is limited so that u < J 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 (/>(u) its expression as a power series in terms of u is written in equation (2) which is then solved with respect to (f>(2u). We know that one root of this equation represents the branch of <p(2 u} under consideration if | u \ < J p. 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 u for u in the equation (2) we have
Eliminate <j>(2 u) from this equation and equation (2) and we have an algebraic equation of the form
If the variable u be limited to values such that \u\ < & then by repeating
the above process it is seen that the function may be continued to the region of a circle with radius 4 p.
By repetition of this process we come finally to an algebraic equation
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 (j>(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
is developed in powers of (j>(u + v). It takes, say, the form (3) ^«+f)+Pi.i[*(«),^
where the P's are rational functions of <j>(u), (j>(v). In this equation write
u + &i for u
and v — k L for v,
where ki is a variable quantity which may be limited to small values.
454 THEORY OF ELLIPTIC FUNCTIONS.
By this substitution u + v remains unchanged, and the above equation becomes
(4) <j>m*(u + v) + Pi,i[0(tt + ki),<l)(v - k1)}pm^-l(u + v)+ • - - = 0.
The equations (3) and (4) are algebraic and have at least one root in com mon, viz., <f>(u + v) 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
2,i[<£(w), </>(u + &i), $(v), (j>(v - ki)]^~l(u + v)
where ra2 < m\.
We write in the above equation
u + k2 instead of u
and v — k2 instead of v.
That equation then becomes
(j)m*(u + v) + P2,i[(j)(u + k2), <t>(u + k!+ fc2), <j>(v - k2)} $(v - kl- k2}]<]>m*-l(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 k%. 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
0«,(u
9 (u
+ • • • =o,
where m^ < m2.
This process may be continued. Each following rrik is less than the pre ceding. Finally we must either have ra^ = 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
( v - k2), . . . , <t>(v - kr}, <j>(v - k1-k2),
. . +P, [same arguments] = 0,
-kr) J
the P's being rational functions of their arguments. We may assume that the degree of this equation cannot be decreased by the above process. It
MANY-VALUED ELLIPTIC FUNCTIONS.
455
follows that all the coefficients of the equation remain unaltered when u is increased by a certain quantity k, and v diminished by the same quan tity k. Some of the coefficients of the above equation may be constants, but they cannot all be constant, for in that case (/>(u + v) would be a con stant.
Suppose that Pv is one of the variable coefficients, which is therefore a function of both u and v.
We may write Pv=f(u,v),
and will show that Pv is a function of u + v.
We know that Pv = f(u, v} has the property that
f(u + k,v-k) = f(u,v). We may choose k so small that
or
du dv
It follows that /is a function of u + v.
We shall put f(u + v) = ^v(u + v) and shall show that ^ is a one- valued function, while </)(u} may be an arbitrarily many-valued function.
Draw a circle about u = 0 with a radius R, where R may be taken as large as we wish. If wre then succeed in showing that tyv (u + v) is one- valued within this circle with radius R, the theorem may be considered proved, since R may be taken arbitrarily large. We knowT that in the neighborhood of u = 0, the function (£>(u) 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 <j>(u) under consideration has everywhere the character of an integral function. This may be done as fol lows: We suppose that all the branch-points of <f>(u), or of the analytic continuation of the branch of <l>(u) 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 drawrn 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 lines 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 (j>(u) and <f)(v) are one-valued along the middle lines of the strips which form the cross. We shall now take the k's defined above so small that | ki \ + | k2 | + • • • + | kr | is less than half the width of the more narrow of the two strips. Then if u moves along the middle line of one of the strips, while v moves along the middle line of the other, all the arguments which have been used in the formation of Pv are situated within the cross. If u and v are added geometrically, it is seen that Pv = f(u, v) = "fyv(u + v) is a one-valued function for all values of u + v within the square that circumscribes the circle with radius R. It follows, since R is arbitrarily large, that fa is a one-valued analytic function of its arguments.
ART. 390. If we write v = 0, then fa(u + v) becomes
4,t \_ P \<t>(u)><t>(u + *i)» • ' • ,*(" + *! + ' ' ' +& "(-*i), . . . ,<£(-&!- - - - -kr)
From this it maybe shown as follows that <p(u) and fa(u) are connected by an algebraic equation:
The function fa(u) is expressed rationally through <t>(u),<j>(u + ki), . . . , <p(u + ki+ k2+ - • • + kr). By means of the addition-theorem <j>(u + k i) may be expressed algebraically through <f>(u) and <£(&i), and similarly </>(u + k2), etc.
We thus have an algebraic equation of the form
(5) H(<f>(u), +v(u)} = 0.
From the four algebraic equations
, 0(v), cf>(u 4- v)] = 0, , +v(u)] = 0,
H[<f>(u + v), tyv(u + v)] = 0,
we may eliminate <f>(u), <j>(v), </>(u + v) and have the algebraic equation
v = 0.
MANY-VALUED ELLIPTIC FUNCTIONS 457
Further, if we differentiate equation (5) we have an algebraic equation
(6) H^u), +v(u), ^f(u), 1r'(u)] = 0. We also have the,eliminant equation
(7) E[<t>(u},<f>'(u)] = 0. (i)
If from the equations (5), (6) and (7) we eliminate <f>(u) and <j>'(u) we have the eliminant equation
= 0. (ii)
It follows then that ^V(M) has an algebraic addition-theorem.
Since the algebraic equation (5) exists connecting <f>(u) and "fyv(u), it follows that <j>(u) is an algebraic function of ^v(u). We have thus solved the problem of determining the function <f>(u) in its greatest generality. The function <p(u) is the root of an algebraic equation, whose coefficients are rationally expressed through a one-valued analytic function tyv(u), 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
«ri
II, rational functions of e w , or III, rational functions of $>u and
TABLE OF FORMULAS
(The formulas of Jacobi and of Weierstrass in juxtaposition)
I.
/b2sin2<£
= amw p. 241,
z = snu, Vl - z2= cos0 = cnu, Vl - &2z2 = dn u. . p. 241.
Vl — k2 sin2(f> = A0, u = F(k, z)= F(k, (f>). . . . p. 285.
am 0 = 0, snO = 0, en 0 = 1, dnO = l. . p. 245
am(- u)=- amu, sn(-u)=-snu, cn(-u)=cnu, dn(-u}=dnu. sn2u + cn2u = 1, k2sn2u + dn2u = 1. . . p. 247.
II.
=^ or = dnu. . p. 243.
du du
= (sn/w)2== ^ ~ ™2u)(l ~ k2sn2u). ... p. 247.
sn'u = cnudnu, ........ p. 247.
cn'u = — snudnu,
dn'u = — k2sn u en u.
(sriu)2= (1 - sn2u)(l - k2sn2u), ..... p. 247. (criu)2= (1 - cn2w)(l - k2+ k2cn2u), (driu)2= (1 - dn2u)(dn2u - 1 + /b2).
(See also No. LVI). 458
TABLE OF FORMULAS. 459
III.
dt
p. 215.
F* - V4*3- g2t - g3 t = gm, p. 298.
p. 325. — e3).
4t*-g2t-g3= 4(t- ei)(t-e2)(t-e3). . . pp. 191, 200.
Cl+ 02+ €3= 0,
e22+ «32) = - i^2,
= 0. . p. 408.
460 THEORY OF ELLIPTIC FUNCTIONS.
IV.
K =
-/V=
jn vn —
dz
. p. 212.
F(k, nx + p)=2nK + F (k, p). a,mK = ^, am2K = T: = 2amK, am(p ± 2 nK) = am p ± nn. p. 241.
V.
K = 0, dnK = k' p. 245.
k'2= I p. 213.
VI.
K"£
_dz = C2
-k'2z2) Q Vl -k'2s
. p. 213.
r^
J0 VZ
iVZ
- z2)(l -k2z2).
iK', ... p. 289.
VII.
sn(K + iK') = , cn(K + iKr) = - — , dri(K + iK') = 0. p. 246. k k
TABLE OF FORMULAS.
461
CO
f
J*.
where
VIII.
ei, 0,
dt
— 9 2$ — 93 Jei 2 V (t —6i)(t — €2)(t
S - 4t3- g2t - g3.
aj"= w + at'
IX.
X.
iK'=Vei-
108(1 -
= -^±- p.215.
p. 215.
pp. 93, 384.
p. 215.
pa>'=<*3, . . . p. 216. pV=0. . pp.315, 355.
. . p. 201, . . p. 201.
p. 201,
462 THEORY OF ELLIPTIC FUNCTIONS.
XI. sn(— u) = — snu, ...... p. 245.
cn(— u) = cnu, dn(— u) = dnu.
sn(u + K)=--, .... p. 245.
dnu
dnu
k'
dnu
sn(u + 2 K) = — snu, cn(u + 2K) = — cnu, dn(u + 2K)= dnu.
sn(u + iK')= — — , ...... p. 246.
ksnu
k snu
snu
sn(u 4- 2 iK') = sn u, cn(u + 2 iK') = — cnu, dn(u + 2 iK') = — dn w.
sn(w + K + iK') =
kcnu
A*
- -^— • k cnu
dn(u + K
+ 2 K + 2 t'K') = - sn t*, cn(u + 2 K + 2 iK') = en tt,
TABLE OF FORMULAS.
XII.
pu -
± 0>0= 63+
- 60(63-
-63
XIII.
- e3 (See also formulas LIV.)
463
p. 317.
. . pp. 355, 369.
. . . pp. 216, 298. p. 305.
p. 307
p. 307.
464
THEORY OF ELLIPTIC FUNCTIONS.
cn(iu,
isn(u, k') cn(u, k')
1
XIV.
cn(u, k')
i cn(iu, k)
cn(iu,
cn(iu, k)
m(iu + K,k)= — — — , . , dn(u, kf)
,. TS 7N ik'sn(u, kf) cn(iu + K,k) = — — Li
dn(u, k)
if , IT- 7 \ k'cn(u,k') dn(iu + K,k)= / ' J ; dn(u, k')
p. 247.
p. 261,
i • -Tsr T\ — icnfu, k'} sn(iu + iK', k) = — - / ' ', K sn(u, k)
dn(iu
u,
1
sn(u, k')
XV.
p. 246.
|
Function |
Periods |
|
|
sn u |
4 K and |
2iK' |
|
cn u |
4K and |
2K + 2iK' |
|
dnu |
2K and |
4iK' |
p. 245.
|
Function |
Zeros |
Infinities |
|
sn u |
2 m K + 2 niK' |
2mK+(2n + l)iK' |
|
cnu |
(2m + l)K + 2 niK' |
a |
|
dnu |
(2m + 1)K + (2n + l)iKf |
<( |
(m, n integers including zero.)
TABLE OF FORMULAS.
465
2iK'
0
sn en dn
XVI.
u + (0, 1,2, 3)K + (0,1,2,3)^'
K
2K
3K
m(u + 2mK + 2m'iK') = (- l)msnu, cn(u + 2mK + 2m'iK') = (- l)m+m'cnu, dn(u + 2 raK + 2 rn'iK') = (- l)m'dn u, (m, m' integers including zero.)
p. 245
|
1 |
dn u |
- 1 |
— dnu |
|
k sn u i dn u |
k cnu ik' |
ksnu — idnu |
kcnu -ik' |
|
ksnu ikcnu |
kcnu — ikk'sn u |
k sn u ik en u |
kcnu — ikk'sn u |
|
ksnu |
k en u |
k sn u |
k cnu |
|
en u |
— cnu |
||
|
dnu k'sn u |
dn u — k'sn u |
||
|
dnu -k' |
en u |
dn u - k' |
|
|
dnu |
dn u |
||
|
1 |
dnu |
- 1 |
— dnu |
|
k snu — i dn u |
kcnu - ik' |
ksnu idnu |
kcnu ik' |
|
k snu — ik en u |
k en u ikk'sn u |
k snu — ik en u |
k cnu ikk'sn u |
|
k sn u |
kcnu |
k snu |
kcnu |
|
cnu |
— cnu |
||
|
dnu — k'sn u |
dn u k'snu |
||
|
dn u k' |
dn u k' |
||
|
dn u |
dn u |
466
THEORY OF ELLIPTIC FUNCTIONS.
XVII.
[See p. 368.]
|
•M |
0 |
1 |
i |
I |
|
|
i |
Vl + k' -VF |
0. |
Vl +k' Vk? |
||
|
Cn dn |
- 1 |
Vl +k' -Vk' |
-kf |
Vl + Iff -Vk' |
_ I |
|
— i |
Vk - ik' |
1 |
Vk + ik' |
i |
|
|
Vk |
Vk |
Vk |
Vk |
Vk |
|
|
-Vl + k |
-Vik' |
-iVl -k |
V - ik' |
Vl +k |
|
|
Vk |
Vk |
Vk |
Vk |
Vk |
|
|
dn |
-Vl + k |
Vk'(k'+ik) |
-Vl -k |
-Vkf(k'-ik) |
-Vl + k |
|
977 |
4- 7* |
I |
1 |
I |
T |
|
i'T |
vT-T -iVkf |
k -ik' |
Vl -k' -iVkf |
I VT |
|
|
dn |
-ikl |
vT^TP -iVkf |
k 0 |
Vl -k' iVv |
T 11 -ikl |
|
i |
Vk + ikf |
1 |
Vk - W |
— i |
|
|
Vk |
Vk |
Vk |
Vk |
Vk |
|
|
Vl + k |
V — ik' |
-iVl -k |
-Vik' |
-Vl + k |
|
|
Vk |
Vk |
Vk |
Vk |
Vk |
|
|
dn |
Vl + k |
Vk'(k'-ik) |
Vl -k |
Vk'(k' + ik) |
Vl + k |
|
977 |
n |
I |
i |
I |
n |
|
en |
1 |
Vl + k' Vk> |
o |
Vl +k' -Vk' |
.. 1 |
|
dn |
1 |
Vl +kf. Vk' |
k' |
Vl +k' Vk' |
1 |
u =
K
2K
* In the table 7 = lim
=0 ksnu
TABLE OF FORMULAS.
467
XVIII.
&(^\=el+Ve1-e2Vel-e3, . . ,. .... p
. 369.
= - 2(6i- 63)v/6i-62- 2(ei- 62) Vei- e3,
- 63,
(2") = ~
- 63 Vei- 63
+ w= 2i(6i- 63) 62- 63 - 2i(e2- 63) ei- e3
- = 62- i e2 - 63 \6i - 62
= ~ 2(6l~ 62) s- 63+2i(e2- 63) VCl- 62.
(Halphen, Fonci. Ellip., Vol. I, p. 54.)
468 THEORY OF ELLIPTIC FUNCTIONS.
Sin TtU = 71
TO
u - m m
P. 20e
., TO = +oo
2 = v i
27ra ^^oo (u —
m)2
XX.
-,£
q = e K
p. 220.
= 1 + 2 5 cos 2 w + 2 ^4 cos 4 u + 2 g9 cos 6 u + - • • ,
3 (2TO+1)2 (2
5 4 e
m = -oo
H^K - w), . . p. 221. = ! ~ 2 ? cos 2 w + 2 54 cos 4 w - 2 g9 cos 6 w + • - • ,
H is an odd function; 0, ©j, HI are even functions.
TABLE OF FORMULAS.
469
U 1 tt»
XXI.
w = 2 pa) 4- 2 JJL'C
^y-o,±i,±2,
W 7^ 0
du
u
u — w w w
a(—u)=—au,
, . . p. 319. . pp. 318, 324.
~ W W
p. 315.
. . . . p. 323. u). . . p. 298.
XXII.
'— . . p. 324.
.... p. 323.
.... p. 323.
• .... p. 323.
XXIII.
+ 2 aj) = ru + 2 T], C(u + 2w')= £u + 2 >/. pp. 303, 338. C^>, V - C"'» ^ = ^ + V- - • • P- 301.
7?a/-w)/=^, if R(-} is positive p. 339.
2 V^/
470 THEORY OF ELLIPTIC FUNCTIONS.
XXIV.
i(u + K)=®(u), 0i(w + iK') = mi(u), pp. 222, 223.
K)= 0i (w), H(tt + K)= HI(M),
0(w + 2 mK) = 0(w), 0(w + 2 wiK')
(- l)wH(w), H(w + 2?niK')
rr/ .«_.'
. A TWTTt.
TABLE OF FORMULAS.
471
XXV.
o(u, w, a/
(0)
ooj
,. pp. 378, 304.
Hi(0)
304,
377.
(HftT
(7W
.
0(0)
0(U
al(u
= e
XXVI ....... pp. 340, 380.
' = - 2
2 wr
2a>')
2 0/
25)= (-
2 5 = -
|"2 5 = 2 pa> + 2 ra>', p, r any integers including"}
472
THEOKY OF ELLIPTIC FUNCTIONS.
XXVII p. 224.
Function
Zeros
(2m + l)K + 2niK'
(2m + l)K + (2n +
2mK + (2 n +
2 mK + 2 niK'
(m, w integers including zero.) XXVIII.
snu = —-=
dnu
S(u)
XXIX.
XXX.
|
Function |
Zeros |
|
|
t^oW |
m + nr + - 2 |
|
|
^l(«) |
m + WT |
|
|
*,(*) |
m + i + nr |
|
|
^3(tt) |
m + J + nr + |
T 2 |
(m, n integers including zero.) * = ^ = - (P- 23°)-
p. 244.
p. 229.
TABLE OF FORMULAS.
473
XXXI.
|
Function |
Zeros |
||
|
G\U |
(2m |
+ l)w + 2 no/ |
|
|
G2U |
(2m |
+ l)o> +(2n + |
IK |
|
°zu |
2 mw + (2 n + |
IX |
|
|
GU |
2maj + 2na)' |
(m, n integers including zero.)
OU
GU
u = _
p. 380.
XXXII p. 384.
v/e-^=£^ = <^,/. v-ir^;=2ff=.«^W';
G<JJ OOHJOJ
GO) ooj aw
€*> " (JW .
ooj aw
aco
aw aw aw
aa>
acu
— e\ = —
- e3, Ve2— ei= —
— e2,
where R
iw
474 THEORY OF ELLIPTIC FUNCTIONS.
XXXIII p. 230.
= q-le-2rnut
nr) = (-
m
nr) = (-
_ /->— n2/, — 2n7fiu
TABLE OF FOKMULAS.
475
XXXIV.
p. 386.
= ±-
— 62 ei —
= ± e2
Vi
4/ 4
[Schwarz, ^oc. c^., p. 26.]
476 THEORY OF ELLIPTIC FUNCTIONS.
XXXV. . . . pp. 220, 229, 378, 397.
m=oo
w = l
m = oo (2m + l)2
2 q cos
m=0
m=l
XXXVI p. 230.
(1 - 2g2w-1cos27m + g4w~2),
sin TTW JJ (1 - 2 g2m cos 2xu + g4m), 2 Qo^ cos TLU JJ (1 + 2 q2m cos 2 TTW + g4m),
m = l m = oo
XXXVII p. 396.
m = oo m-oo
O- n u - 92m>' Qi=n(i + 22'n>'
m = l wi"1
TO = oo m = oo
Q2= n a + ?2m-i)> ^= n (i - 52m-1)-
w = l w»-l
QiOaQs-1, 16gQi8=Q28-Q38. • .pp. 396, 409.
TABLE OF FORMULAS 477
XXXIX.
If v = — , z = e™ r --
2 mti (1 - h2mz~2)2 (l -h2^z2)2Y
p. 336.
p. 337,
nVrS- p-341'
m = l i
P.342.
- yj 1 + q2mz~2 "L-j. 1 + ^2mg2 ( 342,
11 1 + q2m 11 1 +52m ' PP* 1 379.
2m-l~-2
_ -
TT __ - _ TJ
11 I+y— 1 11 l
au = 3
d+22"
2yf l-92m-lg-2»yf j__ JLl !_g2m-l JL=1 1
wz=» ^ ^ 2m-l O
e2*^ TT ? d"-?— "
_ 02m-l "
478
THEORY OF ELLIPTIC FUNCTIONS.
TO=0
#o(0) = 1 + 2
m=0
+2
= 0
XL
(2m + l)2
pp. 397, 400.
- 2q
XLI ......... See p. 397.
(l), #i'd) = - #i'(0).
XLII ..... See pp. 397, 411.
#O'(T)= 2^g-1#o(0), #0'(w #i/W=-2-1#i'(0), #i/(m
#3' (r) = - 2 wg - ! ^3 (0) , #3' (m + WT) = - #i'(0) = 2 *Qo82*, #o(0)
~w2^3 (0) ,
, • pp. 397, 399.
TABLE OF FORMULAS.
479
OO) =
XLIII pp. 385, 410.
- e3
e2— 63
OO)' =
XLIV See p. 410.
- Co
w'
QS f
6Z
480
THEORY OF ELLIPTIC FUNCTIONS.
XLV.
p. 399.
, h
,QS , • pp- 398> 4i
XLVI.
p. 400.
„ • P- 244.
= 4-00
, PJX400, 403.
>2m2+w
TABLE OF FORMULAS.
481
XL VII.
G = (61- 62)2fe- e3)2(63-
•i- »t f\ 04. O >i r\r\
= Tc ~T9~^o 5 •> • P- 409. 16 to12
i { 408, 'PP- 397.
7T2
12
p. 408.
2 to
p. 409.
03=
4-
— 63
482
THEORY OF ELLIPTIC FUNCTIONS.
CO
XL VIII ......... p. 409.
u = 2cov.
XLIX i'(0)
2H +l_^l C w 2 co &i(
pp. 409, 304, 378.
L
p. 409.
TABLE OF FOKMULAS.
483
LI.
Vei— 63+ Vei — e2
=/i.
^•^•MB
• • '),
p. 408.
(l+2g4 + 231«+. • •), p. 409.
(1 +g2w)
=1
2w2
m = l
77l = X
2 pu
p. 336. p. 379.
^ 1_38g1.2+58g2.3_
m = °° r,2m m = °° «2m-l w-00
— 1
V— f*"
m = l
p. 379.
484
THEORY OF ELLIPTIC FUNCTIONS.
LIL
[Formulas (D), p. 237].
LIIL
022U — °32U + (02 — 63) (72U = 0,
o32u - ai2u + (e3- ei)o2u = 0,
oi2u — o22u +(ei— e2)o2u = 0.
(e2 — e3)oi2u + (e3 — ei)o22u + (e\ —
p. 381.
= 0.
LIV. . .... ." . pp. 305, 383, 387.
|
(• / 7N |
|
|
ou 1 (\/ l,\ |
o\u / cn (v e \ — e3 • u, k) |
|
o3u Vei-es |
ou sn(Vei—e3'U,k) |
|
G\W rrt(\./f> f> 11 If} |
&2U \/ p r ct/i^V t1] — ^3 • 'U, k) |
|
"*u |
ou sn (\^e i — e3 • u, k) |
|
O2U i (\/f> f> 11 If} |
&3W' \/f f> |
|
aau |
ou sn(\/ei—e3'U,k) |
|
o\u • CQom (\/g g - u k] |
ou 1 |
|
02U |
/ multiplied by o\u Ve\—e3 |
|
o(u} l |
tg am (v e i — e3 • u, k), 02U 1 |
|
, multiplied by |
o\u sin coam^ej — e3 • u, k) |
|
cos coam (\/e i — e3 • u,k), |
|
|
^3 \^/) ^^^ ^ 1 — ^*^ 1^.* 1'^ll-w |
03U 1 |
|
— • multiplied DV |
o\u cos am(v/ei — e3 • u, k) |
|
A coair^Vei —e3»u,k)} |
[Schwarz, Zoc. ci^., p. 30.]
TABLE OF FORMULAS. 485
LV. Homogeneity ........ p. 343.
Xco, Xcof) = Xo(u, CD, cof),
fa), fa)') = i £(u, (o, oj'),
, fa)') = — @(u, co, a)'),
^(u, co, co'),
(*"'
486 THEORY OF ELLIPTIC FUNCTIONS.
LVI .......... p. 252.
sn"u = - (1 + k2)sn u + 2 k2snsu, cn"u = (2k2~ l)cnu - 2k2cn3u, dri'u = (2 - k2)dn u-2 dn3u-,
(1 + 14k2+ k4)snu - 20k2(l + k2)sn*u + 24k*sn5u, (1 - 16/b2+ 16A;4)c/iw + 20A;2(1 - 2k2)cn3u + 24 A;4cnX (16 _ 16 A;2 + k*)dnu + 20 (k2- 2)dn*u + 24 dn5u. (See also Formulas II.)
LVII ........ p. 252, et seq.
,
5!
sn'(0)= 1, - sn'"(0)= 1 + k2, sn^(0}= 1 + 14A:2+ k4, sn<7>(0)= 1 + 135 k2 + 135 /b4+ A;6, sri(9)(0)= i + 1228 A;2 + 5478 k4 + 1228 kQ+ k8,
cn"(0)= - l,cnW(0)= 1 + 4fc2, - cn^(0)- 1 + 44 fc2 + 16 fc* cnW(0)= 1 + 408 &2+ 912 A;4 + 64 /b6, cn<10>(0)= 1 + 3688 k2 + 30768 /b4+ 15808 kQ+ 256 A;8,
= k2(k2 + 4), - dn<6)(0)= k2(k* + 44k2 + 16), dn<8HO) = k2(k&+ 408 A;4 + 912 k2 + 64), dn<10>(0)= A;2(A;8+ 3688 kQ+ 30768 &4 + 15808 A;2 + 256),
[Gudermann, CreUe, Bd. XIX, p. 80.]
kK 2 Ku Vg sinw , Vo3 sin 3u . Vg5 sin 5w ,
— sn = — + — 5 — + — * = h • • • , T
2;r n 1—5 1 — q3 1 — q5
kK 2 Ku _ \/~q cost£ \/q3 cos 3u Vq5 cos 5u
— en — - ~— H - — g -T : ~ ^ i * * * >
K i 2 Kw = 1 , q cos 2^ g2 cos 4u , q3 cos 6^
2^ ~^~ "4^ 1 +g2 1 +g4 1 + g6 ' '
TABLE OF FORMULAS. 487
LVIII.
Of O2 . Q P\ f O2 £\ O - P\
O I ^ O O I ^ O ^ O
1 /oxx v e ^2 3 5 ^3 o I 9^2 i_ H^2^3
9!^ W~^t~22^?? ~ 22 • 7 ^ 1 23.3-5^ 24-3-5-7*
[See Art. 377.]
LIX.
+ • • • 4- CnM2f|-2+ • • • • - 326-8.
25 • 3 • 53 • 13 24 • 72 • 13
• - p. 327.
[n>3]
LX.
™ = u-^u5-2dl^u7-'-- - p-328-
O (J -L o oi^o
— • QtiUrO. ... p. 6\l6.
1 - ieAM2 - -L (6 e?- g2)u*- - - - . . p. 394.
-I 4o
U = 1, 2, 3.)
488 THEORY OF ELLIPTIC FUNCTIONS.
LXI ........ pp. 236, 246.
u)&*a + i(v)-&a+i(u)#12(v),
(a- 1,2,3; #4= 00).
p. 237.
TABLE OF FOEMULA& 489
LXII.
o(u + u\)a(u — Ui)0(ii2 + ^3)^(^2 — ^3), + 0(u + 112) a (u — U2)0(uz + u\)a(u^ — HI), + 0(u + us)0(u — u3) (7 (HI + u2)0(ui — w2) = 0. . . p. 390.
a(u + v)0(u - v} = o2ua)2v — o?uo2v, ..... p. 391 (ev — etl)(j(u + v)a(u — v)= o^ua.^v — a£ua,?v,
o(u + v)0i(u — v)= oruapv - (en - e^(ex- ev)a2ua2v, (ev— ej 01(11 -f v)oi(u - v) = (ex— e^ov2uov2v —(e^- e^ofuofv, V)GX(U — v) = o^ua^v — (ei - e^a2uov2Vj
v) a (u — v) = o\u ou (7^ ovv — o^u
0(U + V) 0i(u — v) = (JxU(7U OpV avV + 0ftU 0VU Otf) 0V,
0ft(u + v)0i(u — v) = oiu 0ftu 0)V 0^ — (eu — e\) au 0vu 0v 0vv. . p, v = 1,2,3.] [Schwarz, loc. tit., p. 51.]
490 THEOEY OF ELLIPTIC FUNCTIONS.
LXIIL . . . (See pp. 273, 349, 364.)
sn(u ± v) = (snucnv dnv ± snvcnu dn u) -r- D, where D = 1 — k2sn2u sn2v.
cn(u ± v) = (en ucnv T sn u sn v dn u dn v) -r- D, dw(w ± v) = (dnudnv T k'2snu snv cnucnv) + D, sn(u + v) + S7i(i£ — v) = 2 sn ucnv dnv -r- D,
+ v) — sn(u — v) = 2 sn v en u dn u -f- D,
+ v) sn(w — v) = (sn2u — sn2v) -r- D,
+ v) + cn(w — v)= 2 cnucnv -5- D,
— v) — cri(w + v) = 2 snudnusnv dnv -r- D, + v) + d?i(^ — v) = 2 dnudnv -r- D,
— v) — dw(w + v)= 2k2 snucnu snv cnv -=- D, k2sn(u + v) sn(w — v) = (dn2v + k2sn2u cn2v) -f- Z), sn(u + v) sn(u — v) = (cn2v + s?i2it dn2v) -f- D,
+ dn(u + v) dn(u — v) = (dn2u + dn2v) -f- Z),
— k2sn(u + v) sn(u — v) = (dn2u + k2sn2v cn2u) -*- Z>,
— sn(u + v) sn(w. — v) = (cn2u + sn2v dn2u) -f- D,
— cn(w + v) cn(u — v)= sn2u dn2v + sn*v dn2u -r- Z),
— dw(w 4- v) rf?z(w — v) = k2(sn2u cn2v + sn2/y cn2w) -H D.
TABLE OF FORMULAS.
491
LXIV.
02U02V
r(u + v)+ r(u - v)- 2ru
r(u + v)-£(u-v)-2 £v
2 %>u — %>v
= ,
(6 y2u - ^
4 ?3?^ —
p. 352. p. 352.
p.352.
). 353.
pp. 366, 367.
p. 355.
492 THEORY OF ELLIPTIC FUNCTIONS.
LXIII (Continued).
{ 1 ± sn(u + v) } {l ± sn(u — v) } = (cnv ± snu dnv)2 + D, {l± sn(u + v) } JIT sn(u — v) } = (cnu ± snv dnu)2 ^-D, { 1 ± k sn(u + v) } { 1 ± k sn(u — v) } = (dn v ± ksnucn v)2-+- D, { 1 ± k sn(u + v) } { 1 T A; sri(w — v) } = (dnu ± k snv cnu)2-^ D, \ I ± cn(u + v) } { 1 ± cn(u — v)\ = (cnu ± en v)2 + D, \ 1 ± cn(u + v) } { 1 -F cn(u — v) } = (snu dnv =F STIV dnu)2+- D, jl ± rfn(^ + v) } {l ± dn(u - v) } =(dnu ± dnv)2+ D, 1 ± dn(u + v) 1 ~F dn(w — v) = k2(snu cnv T swv cnu)2-r- D.
sn(u + v) cn(^ — f ) = (snu cnu dnv + snv cnv dnu} ~ D, sn(u — v) cn(u + v) = (snu cnu dnv — snv cnv dnu) + D, sn(u + v)dn(u — v) = (snudnu cnv + snv dnv cnu)^- D, sn(u — v)dn(u + v) = (snudnu cnv — snv dnv cnu)-r- D, cn(u + v)dn(u — v) = (cnu dnu cnv dnv — k'2 snu snv) + D, cn(u — v)dn(u + v) = (cnu dnu cnv dnv + k'2 snu snv) -r- D.
sin { am(w + v) + am(w — v) J = 2 snu cnu dnv -5- D, sin |am(& + v)— am(w — v) } = 2 snv cnv dnu-- D, cos { am(w + v) + am (it — v) } = (cn2u — sn2u dn2v) -=- D,
cos { am(w 4- v) -am (w — v) } = (cn2v - sn2v dn2u) -=- D.
(Jacobi, Werke, I, pp. 83-85.)
TABLE OF FORMULAS.
LXV.
. - p. 353.
d2 2%>u - — log
d2
- gw),
-f f) - <?(tt - t;) = -
log (gni - gw),
. 354<
0 p. 354.
t ^ . . 4
= 2 ^((7^3- 3
w, p. 356.
(7(2 U) = 2
..... ..... p. 380.
(Schwarz, /oc. c^., p. 14.)
494 THEOKY OF ELLIPTIC FUNCTIONS.
LXVI.
rr$ / : — dn2udu = I V 1 — k2 sin2 (f> d<j) = E((f>,k), .... p. 285. «/o
= fVl -k2 sin2 0d0, £7'= fVl - fc/2 sin2 0 d0. Jo Jo
KE' + K'E-KK'= -, ..... p. 291. 2
J = K-E, J'=E'; J'K-K'J = f-
2i
®(u)=®(0)eS"Z(u}du ....... p. 292.
j-}du = E(u)-u K/ K
- |V- r . K/ Jo
Z'(0) = 1 - f /v
= Z'(0) - Z'(tt), /b2cn2^ = k2- Z'(0) + Z'(w),
Z(0)=0, - f , . . p. 294.
, . . p. 292.
tan am(u, k')dn(u, k') + ^ + Z(w, A;'). p. 293. 2 KK
TABLE OF FORMULAS. 495
LXVII ..... , . . pp. 302-303.
E -- ^— K I , T/ = - iV^^73 \ Ef + —^—Kf I > ei - «3 J ( el - e3 )
Vei- e* Vei - e3
Formulas for £u are found under Nos. XXII and XLIV.
u = ^
e\- e3-u) = ^ / 1 ^^+eiM], . B B p. 307.
'!• p. 308.
p. 295.
v) = Z(M) + Z(v) - fc2sn z^ snvsn(u + v), ... p. 350.
496
THEORY OF ELLIPTIC FUNCTIONS.
LXVIII.
n[z,
i p
2J
f \Z(ai); «2, v^Z(a2)= H(z; au
z —
\/Z(z)
414
Z(Z)= (1 -
2 ,vA,
(1 + nsm2</>)A<£
n( fl) =
+ a) 0 (a)
p. 420. p. 420.
II(M, a)- II(a, u)= uE(a)- aE(u), ..... p. 421
n(w,o)=-H(- w,o), H(0,a)= 0=n(w,K), w, iK') = oo, H(K7 a) - KE(a) - aE = KZ(a),
U(u + 2K,a)= H(u,a)+ 2K
K
U(iu,ia + K)= U(u,a + K',k'),
Q/ /2 K 2jCi/ y TT
e/2Ka\ Addition-theorems are found on p. 426.
p. 421. p. 422.
p. 423.
TABLE OF FORMULAS. 497
LXIX.
, \AS~CO; a, VS(a); oc) = U(t; a; oo)
dt .„,, 3
' S®=**-92t-to, . p. 419.
t = &u, VS(t) = -ff'u, a =
I'u + o'\
; a; oc) - II(a; «; oc) = u - UQ + (2n + l)m. . p. 420.
CT^o (7W
Addition-theorem on p. 429.
LXX. J E(u}du = logfl(w), ........ p. 423.
_M*
Q(iu)=e 2cn(u,k')n(u. k'), . . . . p. 424. (Vei- e3-u)= e^e^o3u, ........ p. 425.
n(M,a)=Mjg(a)+llogQ(M~g), . . p. 424. 2 O(i^ + a)
^73 . u, V^T^ . a) = 1 ^ gs(» - a) + u ,
2 (73 (u 4- a) <73a
498 THEORY OF ELLIPTIC FUNCTIONS.
LXXI.
If f(u) is a rational function of u, we may write
(l) /(«)- A .f-V+V' ---- ±
where vt- = - ........ p. 9.
u —
If <£(w) is a rational function of sin u and cos w, we may write
(1) <f>(u)= P(eiu) + 3>(u), . "... . . p. 22.
where
*./ \ 75 o ,u — di B2{ d u — di
<S>(u)=B+BliCOi
vf %\
i u-^1 f0 2 J
B3i d* u-_a< Bnti
"C< ~~ :
(2) ^(u)= C^ " " . ' • • " .". - - P- 25.
sm (it — 61) sin (u — 62) • • • sin (u — bn)
If F(w) is a doubly periodic function, we may write (1) F(u)= D
± Zo^-^Cw - ^), . . . pp. 120 and 433.
where the transcendental function Z0(w) becomes infinite of the first order for u = 0, the residue being unity.
(2) FfrHC- •••• "/" ~Mf), . p. 439.
<T(M — UI)<T(U — u2) . . . <T(M — Ut>)
where WIQ+ w2° + • • • + ur°= ui+ u2+ - • - -\- ur.
7 7n o
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