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

Harris Hancock: Lectures on the theory of elliptic functions

Published $1910$.

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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

33. The addition-theorem stated

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

211. The addition-theorems tabulated

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

225. Quadratic transformations. Landen's tranformations

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

CHAPTER XVI THE ADDITION-THEOREMS

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

Addition-Theorems for the Weierstrassian Functions

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

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

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

370. The addition-theorem derived directly from the addition-theorems of the theta-functions

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