Elements of the Theory of Elliptic Functions

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Elements of the Theory of Elliptic Functions

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TRANSLATIONS OF MATHEMATICAL MONOGRAPHS VOLUME

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the Theory of Elliptic Functions N. I. IAKHIEZER

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American Mathematical Society · Providence · Rhode Island

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Translated from the Russian by H. H. McFaden Translation edited by Ben Silver 1980 Mathematics Subject Classification (l 985 Revision). Primary 3301, 33A25; Secondary l 4K25, 11 F03, l 4K20, 30C20, l 4K07, 33A30, 32H25, 33A65, 65A05. ABSTRACT. This book presents a systematic account of the theory of elliptic functions and some of its applications. The main content is intended for engineers who have to work with elliptic functions. Reading this book should not be difficult for people who know the elements of mathematical analysis and the theory of functions in the scope of the first five semesters in the physics-mathematics departments at universities and technical colleges with an advanced program in mathematics. Bibliography: 29 titles. 24 figures, 27 tables.

Library of Congress Cataloging-in-Publication Data Akhiezer, N. I. (Naum Il'ich), 1901[Elementy teorii ellipticheskikh funkfsii. English] Elements of the theory of elliptic functions/N. I. Akhiezer; translated from the Russian by H. H. McFaden; translation edited by Ben Silver. p. cm. - (Translations of mathematical monographs, ISSN 0065-9282; v. 79) Translation of: Elementy teorii ellipticheskikh funkfsii. Includes bibliograpical r~ferences. · ISBN•O-8218 ..4532-2 (alk. paper) I. F.u'ncti~;ms, 'Elliptic. I. Silver, Ben. II. Title. III. Series. QA343.A38 I 3 1990 . 89-18452 515 1 .983-dc20 CIP Copyright

© 1990 by the American Mathematical Society.

All rights reserved. Translation authorized by the All-Union Agency for Authors' Rights, Moscow The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America

Information on Copying and Reprinting can be found at the back of this volume. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. @ This publication was typeset using AMS -TEX, the American Mathematical Society's TEX macro system. 10 9 8 7 6 5 4 3 2 I

95 94 93 92 91 90

Table of Contents Foreword to the second Russian edition

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CHAPTER 1. General Theorems on Elliptic Functions §1. On periods of single-valued analytic functions §2. Proof of the Jacobi theorem §3. Theta functions §4. Liouville's theorems §5. The Weierstrass function r(u) §6. The differential equation of the function r( u)

I 1 2 4 6 9 12

CHAPTER 2. Modular Functions §7. Invariants §8. Modular forms §9. Fundamental regions of the group I §10. The modular function J(r) § 11. Inversion of elliptic integrals of the first kind

15 15 18 22 27 32

CHAPTER 3. Weierstrass Functions §12. The Weierstrass function ((u) §13. The Weierstrass function a(u) § 14. Expression of an arbitrary elliptic function by means of a(u) and by means of ((u) § 15. The addition theorems for Weierstrass functions § 16. Representation of every elliptic function in terms of the functions r(u) and r'(u) § 17. Elliptic integrals

35 35 37

CHAPTER 4. Theta Functions § 18. Representations of theta functions by infinite products § 19. The connection between sigma functions and theta functions §20. Expansion of the functions ((u) and r(u) in simple series

Ill

38 40 43 45 49 49 52 54

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TABLE OF CONTENTS

§21. Expressions for e 1 , e2 , and e 3 in terms of the zero values of the theta functions §22. Transformation of theta functions §23. The modular function J( r)

55 56 58

CHAPTER 5. Jacobi Functions §24. The elliptic integral of the first kind in the forms of Jacobi and Riemann §25. The Jacobi functions §26. Differentiation of the Jacobi functions §27. The Jacobi function Z(w) §28. The Euler theorem §29. Normal elliptic integrals of the second and third kinds in Jacobi form §30. Complete elliptic integrals of the first kind §31. Complete elliptic integrals of the second kind §32. Degeneracy of elliptic functions §33. The simple pendulum

65

75 77 80 82 84

CHAPTER 6. Transformation of Elliptic Functions §34. The problem of transformation of elliptic functions §35. Reduction of the general problem §36. The first principal first-degree transformation §37. The second principal first-degree transformation §38. Landen's transformation §39. Gauss's transformation §40. Principal transformations of nth degree

87 87 89 93 94 95 96 98

65 67 70 71 72

CHAPTER 7. Additional Facts About Elliptic Integrals §41. Elliptic curves of general form . §42. The function p(u) with real invariants §43. Reduction of elliptic integrals to Jacobi normal form in the real case §44. Complete elliptic integrals as hypergeometric functions §45. Computa~ion