# Book:Harris Hancock/Lectures on the theory of elliptic functions

## Harris Hancock: Lectures on the theory of elliptic functions

Published $1910$.

### Contents

CHAPTER 1 PRELIMINARY NOTIONS
1. One-valued function. Regular function. Zeros
2. Singular points. Pole or infinity
3. Essential singular points
4. Remark concerning the zeros and poles
5. The point at infinity
6. Convergence of series
7. A one-valued function that is regular at all points of the plane is a constant
8. The zeros and the poles of a one-valued function are necessarily isolated
Rational functions
9.
10.
Principal Analytical Forms of Rational Functions
11. First form: Where the poles and the corresponding principal parts are brought into evidence
12. Second form: Where the zeros and the infinities are brought into evidence
Trigonometric functions
13. Integral transcendental functions
14. Results established by Cauchy
15.
16.
Infinite products
17.
18.
19. The infinite products expressed through infinite series
20.
21.
22. The cot-function
23. Development in series
The General Trigonometric Functions
24. The general trigonometric function expressed as a rational function of the cot-function
25. Decomposition into partial fractions
26. Expressed as a quotient of linear factors
27. Domain of convergence. Analytic continuation
28. Example of a function which has no definite derivative
29. The function is one-valued in the plane where the canals have been drawn
30. The process may be reversed
31. Algebraic addition-theorems. Definition of an elliptic function
Examples
CHAPTER II FUNCTIONS WHICH HAVE ALGEBRAIC ADDITION-THEOREMS
32. Examples of functions having algebraic addition-theorems
34. Méray's eliminant equation
35. The existence of this equation is universal for functions considered
36. A formula of fundamental importance for the addition-theorems
37. The higher derivatives expressed as rational functions of the function and its first derivative
37a.
38.
39. A form of the general integral of Méray's equation
The Discussion Restricted to One-valued Functions
40. All functions which have the property that $\phi(u+v)$ may be rationally expressed through $\phi(u)$, $\phi'(u)$, $\phi(v)$, $\phi'(v)$ are one-valued
41.
42.
43.
44.
45.
46. Example showing that a function $\phi(u)$ may be such that $\phi(u+v)$ is rationally expressible through $\phi(u)$, $\phi'(u)$, $\phi(v)$, $\phi'(v)$ without having an algebraic addition-theorem
Continuation of the Domain in which the Analytic Function $\phi(u)$ 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
48.
49.
50.
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 $\phi(u)$
CHAPTER III THE EXISTENCE OF PERIODIC FUNCTIONS IN GENERAL
52.
53.
54.
55. Functions defined by their behavior at infinity
The Period-Strips
56. The exponential function takes an arbitrary value once within its period-strip
57. The sine-function take an arbitrary value twice within its period-strip
58. It is sufficient to study a simply periodic function within initial period strip
59. General form of a simply periodic function
60. Fourier Series
61.
62.
63.
64. The nature of the integrals of this equation
65. A further condition that an integral of the equation be simply periodic
66. A final condition
Examples
CHAPTER IV DOUBLE PERIODIC FUNCTIONS. THEIR EXISTENCE. THE PERIODS
67.
68.
69. The distance between two period-points is finite
70. The quotient of two periods cannot be real
71. Jacobi's proof
72.
73.
74. Existence of two primitive periods
75. The study of a doubly periodic function may be restricted to a period-parallelogram
76. Congruent points
77. All periods may be expressed through a pair of primitive periods
78. A theorem due to Jacobi
79. Pairs of primitive periods are not unique
80. Equivalent pairs of primitive periods. Transformations of the first degree
81. Preference given to certain pairs of primitive periods
82. Numerical values
CHAPTER V CONSTRUCTION OF DOUBLY PERIODIC FUNCTIONS
83. An integral transcendental function which is doubly periodic is a constant
84. Hermite's doubly periodic functions of the third sort
85. Formation of the intermediary functions
86. Condition of convergence
87.
88.
89. Historical
90. Intermediary functions of the $k$th order
91. The zeros
92. Their number within a period-parallelogram
93. The zero of the Chi-function
The General Doubly Periodic Function Expressed through a Simple Transcendent
94. A doubly periodic function expressed as the quotient of two integral transcendental functions
95. Expressed through the Chi-function
96.
97.
98.
99.
100.
101. Liouville's Theorem regarding the infinities
102. Two different methods for the treatment of doubly periodic functions
The Eliminant Equation
103. The existence of the eliminant equation which is associated with every one-valued doubly periodic function
104. A doubly periodic function takes any value as often as it becomes infinite of the first order within a period-parallelogram
105. Algebraic equation connecting two doubly periodic functions of different orders. Algebraic equation connecting a doubly periodic function and its derivative
106. The form of the eliminant equation
107. The form of the resulting integral. The inverse sine-function. Statement of the "problem of inversion"
CHAPTER VI THE RIEMANN SURFACE
108. Two-valued functions. Branch-points
109. The circle of convergence cannot contain a branch-point
110.
111.
112.
113. The case where a circuit is around a branch-point
114. The case where a circuit is around two branch-points
115. The case where the point at infinity is a branch-point
116. Canals. The Riemann Surface $s^2=R(z)$
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
118. Its form is $w=p+qs$, where $p$ and $q$ are rational functions of $z$
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 $w$
120. The order of the zero, if at a branch-point
Integration
121. The path of integration may lie in both leaves
122. The boundaries of a portion of surface
123. The residues
124. The sum of the residues taken of the complete boundaries of a portion of surface
125. The values of the residues at branch-points
126. Application of Cauchy's Theorem
127. The one-valued function of position takes every value in the Riemann Surface an equal number of times
128. Simply connected surfaces
129.
130.
Realms of Rationality
131. Definitions. Elements. The elliptic realm
CHAPTER VII THE PROBLEM OF INVERSION
132. The problem stated
133.
134.
135.
136. The elliptic integral of the first kind remains finite at a branch-point and also for the point at infinity
137. The Riemann Surface in which the canals have been drawn
138.
139.
140. The intermediary functions on the Riemann Surface
141. The quotient of two such functions is a rational function
142. The moduli of periodicity expressed through integrals
143. The Riemann Surface having three finite branch-points
144.
145.
146.
147. The zeros of the intermediary functions
148. The Theta-functions again introduced
149. The sum of two integrals whose upper limits are points one over the other on the Riemann Surface
150.
151.
152. Résumé
153. Remarks of Lejeunne Dirichlet
154. The eliminant equation reduced by another method
155. A Theorem of Liouville
156.
157.
158. Classification of one-valued functions that have algebraic addition-theorems
159. The elliptic realm of rationality includes all one-vauled functions which have algebraic addition-theorems
CHAPTER VIII ELLIPTIC INTEGRALS IN GENERAL
160.
161.
162.
163.
164.
165.
Legendre's Normal Forms
166.
167.
168. The name "elliptic integral"
169. The forms employed by Weierstrass
170. Other methods of deriving the forms employed by Weierstrass
171. Discussion of the six anharmonic ratios which are connected with the modulus
172. Other methods of deriving the forms employed by Weierstrass
173.
174.
175. The discriminant
176.
177.
178.
179. The Hessian covariant
180.
181.
182.
183.
184. Weierstrass's notation
185. A substitution which changes Weierstrass's normal form into that of Legendre
186. A certain absolute invariant
187. Riemann's normal form
188. Further discussion of the elliptic realm of rationality
Examples
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
190.
191.
192.
193. The quantities $K$ and $K'$
194.
195.
196. The relations between the moduli of periodicity for the normal forms of Legendre and of Weierstrass
197.
198.
Examples
CHAPTER X THE JACOBI THETA-FUNCTIONS
199.
200.
201.
202.
203. The zeros
204. The Theta-functions when the moduli are interchanged
Expression of the Theta-Functions in the Form of Infinite Products
205.
206.
207. Determination of the constant
The Small Theta-Functions
208. Expressed through infinite series
209. Expressed through infinite products
210. Jacobi's fundamental theorem for the addition of theta-functions
212. Reason given for not expressing the theta-functions through binomial products
Examples
213.
214.
215.
216.
217. The zeros of the elliptic functions
218. The argument increased by quarter and half periods. The periods of these functions
219. The derivatives
220. Jacobi's imaginary transformation
221.
222.
223. Linear transformations
224. Imaginary argument
226. Development in powers of $u$
227. First method
228. Formulas employed by Hermite
229.
230.
231.
232. Explanation of the term
233. Definitions
234. Representation of such functions in terms of a fundamental function
235. Formation of the fundamental function
236. The exceptional case
237. Different procedure
238.
239.
240.
Examples
240.
241. Formation of an integral that is algebraically infinite at only one point
242. The addition of an integral of the first kind to an integral of the second kind
243. Formation of an expression consisting of two integrals of the second kind which is nowhere infinite
244. Notation of Legendre and of Jacobi
245. A form employed by Hermite. The problem of inversion does not lead to unique results
246. The integral is a one-valued function of its argument $u$
247. The analytic expression of the integral. Its relation with the theta-function
248. The moduli of periodicity
249. Legendre's celebrated formula
250. Jacobi's zeta-function
251. The properties of the theta-function derived from those of the zeta-function; an insight into the Weierstrassian functions
252. The zeta-function expressed in series
253. Thomae's notation
254. The second logarithmic derivatives are rational functions of the upper limit
Examples
CHAPTER XIV INTRODUCTION TO WEIERSTRASS'S THEORY
255. The former investigations relative to the Riemann Surface are applicable here
256. The transformation of Weierstrass's normal integral into that of Legendre gives at once the nature and the periods of Weierstrass's function
257. Derivation of the sigma-function from the theta-function
258. Definition of Weierstrass's zeta-function. The moduli of periodicity
259. These moduli expressed through those of Jacobi; relations among the moduli of periodicity
260. Other sigma-functions introduced
261.
262.
263. Jacobi's zeta-function expressed through Weierstrass's zeta-function
Examples
CHAPTER XV THE WEIERSTRASSIAN FUNCTIONS $\wp u, \zeta u, \sigma u$
264. The $\mathrm{Pe}$-function
265. The existence of a function having the properties required of this function
266. Conditions of convergence
267. The infinite series through which the $\mathrm{Pe}$-function is expressed, is absolutely convergent
268. The derivative of the $\mathrm{Pe}$-function
269. The periods
270. Another proof that this function is doubly periodic
271. This function remains unchanged when a translation is made to an equivalent pair of primitive periods
The Sigma-Function
272. The expression through which the sigma-function is defined, is absolutely convergent; expressed as an infinite product
273. Historical. Mention is made in particular of the work of Eisenstein
274. The infinite product is absolutely convergent
275.
276.
The $\zeta u$-Function
277. Convergence of the series through which this function is defined
278. The eliminant equation through which the $\mathrm{Pe}$-function is defined
279. The coefficients of the three functions defined above are integral functions of the invariants
280. Recursion formula for the coefficients of the $\mathrm{Pe}$-function. The three functions expressed as infinite series in powers of $u$
281. The $\mathrm{Pe}$-function expressed as the quotient of two integral transcendental functions
282. Another expression of this function
283. The $\mathrm{Pe}$-function when one of its periods is infinite
284.
285.
286.
287.
288.
289.
290.
291. The sigma-function expressed as an infinite product of trigonometric functions; the zeta- and $\mathrm{Pe}$-functions expressed as infinite summations of such functions. The invariants
292. Homogeneity
293. Degeneracy
Examples
294.
295.
296. The elliptic functions being quotient of theta-functions have algebraic addition-theorems which may be derived from those of the intermediary functions
297. Addition-theorem for the integrals of the second kind
298. A theorem of fundamental importance in Weierstrass's theory
299. Addition-theorems for the sigma-functions and the addition theorem of the $\mathrm{Pe}$-function derived therefrom by differentiation
300.
301.
302. The sigma-function when the argument is doubled
303. Historical. Euler and Lagrange
304.
305.
306.
307.
308. The method of Darboux
309. Lagrange's direct method of finding the algebraic integral
310. The algebraic integral in Weierstrass's theory follows directly from Lagrange's method
311. Another deviation of the addition-theorem for the $\mathrm{Pe}$-function
312. Another method of representing the elliptic functions when quarter and half periods are added to the argument
313. Duplication
314. Dimidiation
315.
316.
Examples
CHAPTER XVII THE SIGMA-FUNCTIONS
317. It is required to determine directly the sigma-function when its characteristic properties are assigned
318. Introduction of a Fourier Series
319. The sigma-function completely determined
320. Introduction of the other sigma-functions; their relation with the theta-functions
321. The sigma-functions expressed through infinite products. The moduli of periodicity expressed through infinite series
322. The sigma-function when the argument is doubled
323. The sigma-functions when the argument is increased by a period
324. Relation among the sigma-functions
Differential Equations which are satisfied by Sigma-Quotients
325. The differential equation is the same as that given by Legendre
326. The Jacobi-functions expressed through products of sigma-functions
327. Other relations existing among quotients of sigma-functions
328. The square root of the difference of branch-points expressed through quotients of sigma-functions
329. These difference uniquely determined
330. The sigma-functions when the argument is increased by a quarter-period
331. The quotient of sigma-functions when the argument is increased by a period
332.
333.
334. The sigma-functions for equivalent pairs of primitive periods
335. The addition-theorems derived and tabulated in the same manner as has already been done for the theta-functions
Expansion of the Sigma-Functions in Powers of the Argument
336. Derivation of the differential equation which serves as a recursion-formula for the expansion of the sigma-function
Examples
CHAPTER XVIII - THE THETA- AND SIGMA-FUNCTIONS WHEN SPECIAL VALUES ARE GIVEN TO THE ARGUMENT
337.
338.
339.
340.
341. The moduli and the moduli of periodicity expressed through theta-functions
342. Other interesting formulas for the elliptic functions; expressions for the fourth roots of the moduli
343. Formulas which arise by equating different expressions through which the theta-functions are represented; the squares of theta-functions with zero arguments
344. A formula due to Poisson
345. The equations connecting the theta- and sigma-functions; relations among the Jacobi and the Weierstrassian constants
346. The Weierstrassian moduli of periodicity expressed through theta-functions
347. The sigma-functions with quarter periods as arguments
Examples
CHAPTER XIX ELLIPTIC INTEGRALS OF THE THIRD KIND
348. An integral which becomes logarithmically infinite at four points of the Riemann Surface
349. Formation of an integral which has only two logarithmic infinities. The fundamental integral of the third kind
350. Three fundamental integrals so combined as to make an integral of the first kind
351. Construction of the Riemann Surface upon which the fundamental integral is one-valued
352. The elementary integral in Weierstrass's normal form
353. The values of the integrals when the canals are crossed
354.
355.
356. The elementary integral of Weierstrass expressed through sigma-functions
357. Legendre's normal integral. The integral of Jacobi
358. Jacobi's integral expressed through theta-functions
359. Definite values given to the argument
360. Another derivation of the addition-theorem for the zeta-function
361. Integrals with imaginary arguments
362. The integral expressed through infinite series
The Omega-Function
363. Definition of the Omega-function. The integral of the third kind expressed through this function
364. The Omega-function with imaginary argument
365. The Jacobi integral expressed through sigma-functions
366. Other forms of integrals of the third kind
Addition-Theorems for the Integrals of the Third Kind
367. The addition-theorem expressed as the logarithm of theta-functions
368. Other forms of this theorem
369. A theorem for the addition of the parameters
371. The addition-theorem for Weierstrass's integral
Examples
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 OF (FRACTIONAL) RATIONAL FUNCTIONS
372. Statement of five kinds of representations of such functions
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
374. The first representation in the Jacobi theory
375. The same in Weierstrass's theory
376. The adaptability of this representation for integration
377. Liouville's theorem in the Weierstrassian notation
378.
379.
380. A linear relation among the zeros and the infinities
381. An application of the above representation
382.
383.
384.
385. The function expressed as an infinite product
386. Weierstrass's proof of Briot and Bouquet's theorem as stated in Art. 156
387. The expression of the general elliptic integral
Examples
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
389. The variable coefficients that appear in the expression of the addition-theorem are one-valued functions
390. These coefficients have algebraic addition-theorems. The function in question is the root of an algebraic equation, whose coefficients are rationally expressed through a one-valued analytic function, which function has an algebraic addition-theorem
Table of Formulas