# Elliptic Functions

## 1st Edition

### A Primer

**Authors:**Eric Harold Neville

**eBook ISBN:**9781483151915

**Imprint:**Pergamon

**Published Date:**1st January 1971

**Page Count:**212

## Description

Elliptic Functions: A Primer defines and describes what is an elliptic function, attempts to have a more elementary approach to them, and drastically reduce the complications of its classic formulae; from which the book proceeds to a more detailed study of the subject while being reasonably complete in itself.

The book squarely faces the situation and acknowledges the history of the subject through the use of twelve allied functions instead of the three Jacobian functions and includes its applications for double periodicity, lattices, multiples and sub-multiple periods, as well as many others in trigonometry.
Aimed especially towards but not limited to young mathematicians and undergraduates alike, the text intends to have its readers acquainted on elliptic functions, pass on to a study in Jacobian elliptic functions, and bring a theory of the complex plane back to popularity.

## Table of Contents

Editor's Preface

List of Tables

Chapter 1. Double periodicity

Equivalent bases

Chapter 2. Lattices

Chapter 3. Multiples and sub-multiples of periods

Chapter 4. Fundamental parallelogram

Liouville's theorem—a doubly periodic function without accessible singularities is a constant

Chapter 5. Definition of an elliptic function

A rational function of an elliptic function is an elliptic function

Chapter 6. An elliptic function (unless constant) has poles and zeros

Identification of an elliptic function

(i) by poles and principal parts

(ii) by poles and zeros

Chapter 7. Residue sum of an elliptic function is zero

Chapter 8. Derivative of an elliptic function

Order of an elliptic function

No functions of the first order

Chapter 9. Additive pseudoperiodicity

Integration of an elliptic function with zero residues

Signature

Evaluation of Aβ — Bα for a function additively pseudoperiodic in α, β with moduli A, B

Chapter 10. Pole-sum of an elliptic function

Chapter 11. The mid-lattice points

Odd and even elliptic functions

Chapter 12. Construction of the function ζκz

Chapter 13. Construction and periodicity of the Weierstrassian function Pz

Chapter 14. Zeros of P'z

The constants ef, eg, eh

Construction of the primitive functions fj z, gj z, hj z

Chapter 15. Periodicity of the primitive functions

Primitive functions are odd functions with simple poles

Structure patterns and residue patterns

Double series for fj z

Chapter 16. Construction and pseudoperiodicity of ζz

The constants ηf, ηg, ηh

Laurent series for ζz, Oz, ζ2k-1z, ζ2kz

Chapter 17. Construction of σz

Chapter 18. Construction, in terms of ζz and Pz of an elliptic function with assigned poles and principal parts

Expression for...

Constant value of...

Chapter 19. Construction, in terms of óæ, of an elliptic function with assigned poles and zeros

Expression for...

Expression for the primitive function pjz

Chapter 20. Expression of an elliptic function in the form ...

Chapter 21. Expression for.'2z in terms of.z

Evaluation of...

Chapter 22. Expression of an elliptic function in the form S ...

Chapter 23. Elliptic functions on the same lattice are connected algebraically

Chapter 24. The six critical constants pq

f2g + g2g + h2f =0

fgfh = gfhf

gr = vfg

Chapter 25. Quarter-period addition to the argument of a primitive function

The twelve elementary functions

pq z qp z = qp'wq; pqz qrz = pqwr, prz

Periods and poles of pq z

Relations between the squares of the elementary functions

Chapter 26. The functions pz and pqz as solutions of differential equations

Chapter 27. Copolar functions and simultaneous differential equations

Chapter 28. Addition theorems for pz and .z and .z

...+ fj'z/fjz

Chapter 29. Addition theorems for fjz, jfz and hgz

Chapter 30. Symmetrical algebraic relations between fjx, fjy, fjz, x + z = 0

Chapter 31. Integration of rational functions of .z and .'z

Integration of functions rational in the primitive functions

Chapter 32. The functions .z and pqz as inverted integrals

Chapter 33. Statements of the inversion theorem

Chapter 34. The Weierstrassian half-periods as definite integrals

Chapter 35. Standardisation of an elliptic integral

The normalising factor and the Jacobian lattice

Chapter 36. Definition of the Jacobian functions

Chapter 37. Periodicity of pqu

Solution of pqu = ±pqa

Chapter 38. Parameters and moduli

The constant pqKr

Chapter 39. Leading coefficients

Linear relations between squares of copolar Jacobian functions

Quarter-period addition

Chapter 40. Derivatives and differential equations

Chapter 41. The Jacobian functions as inverted integrals

Chapter 42. The Jacobian quarter-periods as definite integrals

The functions X(c), X'(c)

The ranges of the twelve Jacobian functions for 0 = c < 1 ; 0 = u = X

Chapter 43. Addition theorems for the Jacobian functions

Chapter 44. Jacobi's imaginary and real transformations

Chapter 45. Duplication

The bipolar function bpqu

ps 2u + qs 2u = brsu

2 ps 2u = bqsu + brsu – bpsu

ps2u = ps2Kr (1 + qp 2u)/(1 - rp 2u)

Chapter 46. The Landen transformations

Chapter 47. The reduction of a rational function of Jacobian functions

Chapter 48. Integration of the Jacobian function pqu

Integration of functions of the form pqu ø(pq2u)

Chapter 49. The integrating function Pqu

Linear relations between integrating functions

Pseudoperiodicity of the integrating functions

The half-moduli Nn, Cc

Legendre's identity

Interchange of Sew and Snw under Jacobi's imaginary transformation

The constants K, K', E, E'

Addition theorems for integrating functions

Chapter 50. Integration of a polynomial in the squares of Jacobian functions

Chapter 51. The function IIs (u, a)

Relation of IIs (u, a) and óu

Chapter 52. Differentiation of Jacobian functions

Integrating functions with respect to the parameter c

Chapter 53. Degeneration of Jacobian systems (c = 0) to circular functions

Degeneration of Jacobian systems (c = 1) to hyperbolic functions

First approximations to functions with a small parameter

Chapter 54. The c-derivatives of Kc, Kn, DsKc, DsKn

The quarter-period differential equation and its solution

X'...

X' = ...

2...

f...

Legendre's identity

Chapter 55. Differentiation of Weierstrassian functions with respect to h2, h3

Chapter 56. Weierstrassian and elementary functions with an axial basis

Distribution of real values of...

Variation of .z and pq z on the perimeter of the basic rectangle JFHG

Signs of the critical constants

Chapter 57. Jacobian functions with an axial basis

Variations of pq<< on the perimeter of the basic rectangle SCDN

The parameters and the moduli are real numbers between 0 and 1

Chapter 58. Evaluation of the real integral...

Chapter 59. Reduction of the integrals...

Chapter 60. Simultaneous uniformisation of two quadratic functions y, z...

Appendix A

Appendix B

Appendix C

Exercises

Answers to Exercises

## Details

- No. of pages:
- 212

- Language:
- English

- Copyright:
- © Pergamon 1971

- Published:
- 1st January 1971

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483151915