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

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

Published $1910$.

### Online copies

### 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

- 11.
- 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