Applied Mathematical Sciences Volume 80

Editors F. John J. E. Marsden L. Sirovich

Advisors M. Ghil J. K. Hale J. Keller K. Kirchgăssner B. Matkowsky J. T. Stuart A. Weinstein

Applied Mathematical Sciences

1. John: Partial Differential Equations. 4th ed. 2. Sirovieh: Techniques of Asymptotic Analysis.

3. Hale: Theory of Functional Differential Equations. 2nd ed. 4. Pereus: Combinatorial Methods.

5. von MiseslFriedriehs: Fluid Dynamics.

6. FreibergerlGrenander: A Shon Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory.

9. Friedriehs: Spectral Theory ofOperators in Hilbert Space. II. Wolovieh: Linear Mullivariable Systems. 12. Berkovilz: Optimal Control Theory. 13. BlumanlCole: Similarity Methods for Differential Equations. 14. IfJshizawa: Stability Theory and the Existence of Periodic Solution and Almost Periodic Solutions. 15. Braun: Differential Equations and Their Applications. 3rd ed. 16. Lefsehelz: Applications of Aigebraic Topology.

17. CollalzlWellerling: Optimization Problems. 18. Grenander: Pattern Synthesis: Lectures in Pattern Theory. Voi 1. 20. Driver: Ordinary and Delay Differential Equations.

21. CourantlFriedriehs: SupersonicFlow and Shock Waves.

22. RouehelHabelslLaloy: Stability Theory by Liapunov's Direct Method.

23. Lamperri: Stochastic Processes: A Survey ofthe Mathematical Theory. 24. Grenander: Pattern Analysis: Lectures in Pattern Theory. VoI. II.

25. Davies: Integral Transforms and Their Applications. 2nd ed. 26. KushnerlClark: Stochastic Approximation Methods for Constrained and Unconstrained Systems 27. de Boor: A Practical Guide to Splines. 28. Keilson: Markov Chain Models-Rarity and Exponentiality. 29. de Veubeke: A Course in Elasticity. 30. Sniatycki: Geometric Quantization and Quantum Mechanics. 31. Reid: Sturmian Theory for Ordinary Differential Equations. 32. MeislMarkowilz: Numerical Solution of Partial Differential Equations. 33. Grenander: Regular Structures: Lectures in Pattern Theory. VoI. III. 34. KevorkianlCole: Perturootion methods in Applied Mathematics. 35. Carr: Applications of Centre Manifold Theory.

36. BenglssonIGhillKiil/en: Dynamic Meteorology: Data Assimilation Methods. 37. SaperslOne: Semidynamical Systems in Infinite Dimensional Spaces. 38. LiehlenberglLieberman: Regular and Stochastic Motion. 39. PiecinilSlampaeehialVidossieh: Ordinary Differential Equations in RO.

40. NaylorlSell: Linear Operator Theory in Engineering and Science. 41. Sparrow: The Lorenz Equations: Bifurcations. Chaos. and Strange Attractors. 42. GuekenheimerlHolmes: NonlinearOsciilations. Dynamical Systems and Bifurcations of Vector Fields. 43. OckendonlTayler: Inviscid fluid Flows. 44. Pa:,': Semigroups of Linear Operators and Applications ta Partial Differential Equations. 45. GlaslwjJ7Guslllfsoll: Linear Optimization and Approxlmation: An Introduction to the Theoretical Analysis

and Numerical Treatment of Semi-Infinite Programs. 46. Wi/mx: Scattering Theory for Diffraction Gratings.

47. Hale el al.: An Introduction to Infinite Dimensional Dynamical Systems-Geometric Theory.

48. Murray: Asymptotic Analysis.

49. Lac,,·:ltenskaya: The Boundary-Value Problems of Mathematical Physics. 50. Wi/mx: Sound Propagation in Stratified Fluids.

51. Golubil."(yISchaeffer: Bifurcation and Groups in Bifurcation Theory. VoI. 1. 52. ChipO/: Variational Inequalities and Flow in Porous Media.

53. Majda: Compressible fluid Flow and Systems of Conservat ion Laws in Several Space Variables.

54. Waso\\': Linear Turning Point Theory.

(nJ/llil/lledliJ/lolI'il/li il/dex)

Derek F. Lawden

Elliptic Functions and Applications

With 23 Illustrations

Springer Science+Business Media, LLC

Derek F. Lawden University of Aston in Birmingham Birmingham B4 7ET United Kingdom

Editors F.John Courant Institute of

Mathematical Sciences New York University New York, NY 10012 USA

J. E. Marsden Department of

Mathematics University of

California Berkeley, CA 94720 USA

Mathematics Subject Classifications (1980): 33A25

Library of Congress Cataloging-in-Publication Data Lawden, Derek F.

Elliptic functions and applications/Derek F. Lawden. p. cm.-(Applied mathematical sciences; v. 80)

Bibliography: p. Inc1udes index. ISBN 978-1-4419-3090-3 ISBN 978-1-4757-3980-0 (eBook) DOI 10.1007/978-1-4757-3980-0 1. Functions, Elliptic. 1. Title. II. Series: Applied

mathematical sciences (Springer-Verlag New York Inc.); v. 80. QA1.A647 voI. 80 [QA343] 510 s-dc20 [515'.983] 89-6171

Printed on acid-free paper.

© 1989 by Springer Scienee+Business Media New York Origina11y published by Springer-Verlag New York, Ine. in 1989

L. Sirovich Division of Applied

Mathematics Brown University Providence, RI 02912 USA

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher, Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Phototypesetting by Thomson Press (India) Limited, New Delhi.

9 8 7 6 543 2 1

Preface

The subject matter of this book formed the substance of a mathematical se am which was worked by many of the great mathematicians of the last century. The mining metaphor is here very appropriate, for the analytical tools perfected by Cauchy permitted the mathematical argument to penetra te to unprecedented depths over a restricted region of its domain and enabled mathematicians like Abel, Jacobi, and Weierstrass to uncover a treasurehouse of results whose variety, aesthetic appeal, and capacity for arousing our astonishment have not since been equaled by research in any other area. But the circumstance that this theory can be applied to solve problems arising in many departments of science and engineering graces the topic with an additional aura and provides a powerful argument for including it in university courses for students who are expected to use mathematics as a tool for technological investigations in later life. Unfortunately, since the status of university staff is almost wholly determined by their effectiveness as research workers rather than as teachers, the content of undergraduate courses tends to reflect those academic research topics which are currently popular and bears little relationship to the future needs of students who are themselves not destined to become university teachers. Thus, having been comprehensively explored in the last century and being undoubtedly difficult . to master, the theory of elliptic functions has dropped out of most university courses and very few mathematics graduates leaving universities today know more than that many problems of applied mathematics lead to elliptic integrals, and that these therefore "cannot be solved." However, there are signs that the arid view of mathematics, which became dominant in the immediate postwar years, according to which the discipline's prime purpose is not the solving of problems but the cataloging ofaxiom systems, is now giving way to a more balanced appraisal and school pupils are once again

vi Preface

being encouraged to develop their mathematical ingenuity in using mathematics, rather than to exhibit a precocious (and, one suspects, very partial) understanding of its foundations. In this changed c1imate, one is hopeful that elliptic functions will again appear in the repertoire of university lecturers, particularly if government pressure on the universities to place greater emphasis on the efTectiveness of their teaching and the interests of the majority of their students continues to be maintained.

In this event, a difficulty which will arise is the lack of textbooks for recommendation to students for additional reading and as sources of exercises. During the last twenty years of his service as a university teacher, the author has of ten received enquiries from users of mathematics regarding books which he could recommend for self-study in this field. Upon checking the appropriate section of his and other university libraries, he has discovered the choice of suitable books to be severely restricted and, very significant1y, those available to be in heavy demand. Although a number of elementary introductions to the theory were published in the last century and a very few early in this, these are now out of print and the only works easily available assume an extensive background knowledge of the theory of analytic functions, in the application of which the ordinary professional applied mathematician can no longer be expected to be well practiced. For most undergraduate students also, a textbook which treats elliptic functions as a footnote to the theory of analytic functions is not very satisfactory, since it entails delaying the introduction of the topic until very late in the course and so limits the number of illustrations which may be made of the utility of the functions for solving practic al problems. It is prefera bIe, therefore, to permit a student to acquire a working knowledge of these functions on the basis of elementary analysis involving little more than the convergence of infinite series, in the same way that the properties of the circular and hyperbolic functions of a complex varia bIe are elucidated during the first year of a university course by reason of their manifold applications. It is true that the extremely powerful methods provided by analytic function theory permit very elegant proofs of some of the elliptic function relationships to be constructed, but to the beginner these arguments appear as c1ever tricks which only serve to deepen his mystification regarding the true provenance of the results they establish. An introductory book which can be recommended for self-study by the undergraduate student or ordinary working mathematician, therefore, will first define the functions by the more elementary processes of analysis, in a similar manner to the other well-known transcendental functions, and then proceed to derive their properties in a straightforward way, before perhaps turning, at a later stage, to a deeper analysis depending upon the general properties of analytic functions. The most elegant and economical account is rarely the most efTective for didactic purposes. This, then, is the plan I have followed in attempting to fill the gap in the range of texts available to the university student or to practicing applied mathematicians and engineers who need to remedy a serious

Preface Vll

deficiency in their tool kit. The authors of a standard set of tables (Smithsonian Elliptic Functions Tables by G. W. and R. M. Spenceley) comment in their introduction: "In the popular field of undergraduate mathematics, a new elementary text on elliptic functions with a wide variety of problems and applications, supplemented by an adequate set of tables, would be a boon to teachers and students." 1 hope that this book satisfies this long-standing need which they, and others, have identified.

Chapter 1 introduces the theta functions by their Fourier expansions and establishes their principal properties and identities. These functions have relatively few applications and, analytically, their behavior is more complex than that of the elliptic functions. However, their definition involves no more than an appeal to the idea of convergence of a complex series and this will be familiar to all serious users of mathematical techniques. Further, the rate of convergence of a theta series is very rapid in all practical circumstances, so that the reader is immediately provided with a practica bIe method for computing these functions and, later, the elliptic functions which are simply related to them. All the tables at the end of this book have been computed on this basis using a small desk computer and the re ader should experience no difficulty in extending this collection to suit his requirements, either by the addition of new tables or by the recomputation of the given tables to higher accuracy or for smaller steps of the arguments. Proving the theta function identities also offers the student useful practice in the manipulation of infinite series, especially the multiplication of absolutely convergent series and the rearrangement of terms in the product. Thus, we have followed the approach to elliptic functions pioneered by Jacobi in his lectures and which leads quite naturally to the definition of the Jacobi functions in Chapter 2. In many treatments, the Weierstrass 9-function is taken to be fundamental and the Jacobi functions are derived from this. But it is the Jacobi functions which are the more closely related to the circular and hyperbolic functions with which the reader will be familiar and it is these functions which are predominantly useful in applications to mechanics, electrodynamics, geometry, etc. Thus, in a text intended for reading by mathematical practi­tioners, it is essential that the Jacobi functions are introduced as soon as possible. Having defined these functions as ratios of theta functions, their double periodicity and the identities they satisfy follow immediately from corresponding results for the theta functions and are worked through in the remainder of Chapter 2. Chapter 3 is devoted to the inverse Jacobi functions; it presents the three standard forms of elliptic integrals described by Legendre (for real values of the integration variable only) and obtains the properties of the complete integrals as functions of the modulus k; Jacobi's epsilon and zeta functions are also brought in at this stage. The text next turns to consider applications of the theory as it has been developed thus far. Geometrical applications are studied in Chapter 4, including those to the ellipse, the ellipsoid, Carte si an ovals, and spherical trigonometry. Chapter 5 deals with physical applications and covers plane motion of a pendulum, orbits under

viii Preface

various laws of attraction to a center (including Einstein's law), the rotation of a rigid body about a smooth pivot, current flow in a rectangular plate, and parallel plate capacitors.

Almost all the argument covered in the tirst tive chapters can be understood by a reader whose mathematical equipment extends no further than the techniques covered in the tirst year of an undergraduate mathematics course or during ancillary mathematics lectures arranged for engineers and physicists. It will be found challenging, but the attractiveness of the diverse and unexpected relationships revealed and the interest of the problems which become amenable to solution by application of the theory should prove adequate compensation for the efTort expended. The remaining four chapters are provided for those who now feeI the urge to explore the subject further and, in particular, to apply in this very fertile tield a knowledge of the general theory of analytic functions.

Chapter 6 commences the study of elliptic functions having any pair of assigned periods by tirst constructing the sigma functions from the theta functions and then proceeding to detine Weierstrass's zeta function and &'-function. The inverse &'-function leads to a consideration of an elliptic integral in Weierstrass's standard form. Application of the Weierstrass functions to the solution of a number of geometrical and physical problems is made in Chapter 7; among the items discussed are general three­dimensional motion of a pendulum, the spinning top, and a projectile moving under constant gravity and air resistance proportional to the cube ofits speed.

In Chapter 8, the theory of analytic functions is applied without restraint to investigate the general properties of doubly periodic functions. Further expansions of the Weierstrass and Jacobi functions are constructed and the representation of any elliptic function as a rational function of the sigma function or the gII-function is determined. The chapter closes with an investiga­tion of the multivalued nature of the elliptic integrals, both complete and incomplete, when these are treated as contour integrals.

The tinal chapter is an introduction to the very extensive theory of modular transformations. This associates together in a comprehensive theory a number of particular transformations of the Jacobian modulus already encountered in earlier chapters.

At the end of the book, we have collected together a wide variety of relevant tables. The entries are generally to tive signiticant tigures, but are only intended to provide an indication of the numerical behavior of the quantities tabulated, without permitting linear interpolation for intermediate values of the argument. For accurate numerical evaluation, the reader will need to refer to one of the more extensive sets of tables listed in the biblio­graphy or to recompute the tables (a useful and straightforward exercise).

A collection of over 200 exercises has been distributed between the chapters and it is hoped these will provide stimulating checks for the reader on his understanding of the main text and, in addition, a useful collection of subsidiary results.

Preface IX

In regard to my notation, I have generally conformed to that which will be most frequently encountered when other readily available texts are referred to. Thus, the four theta functions are denoted by 8i (z 1.) (i = 1,2,3,4) and represent the same functions considered by Whittaker and Watson in their Course of Modern Analysis. However, I have preferred to denote the basic primitive periods of Weierstrass's &l-function by 2(01 and 2(03 (Whittaker and Watson use 2(01 and 2(02), with (02 = - (01 - (03' since in the important special case when (01 is real and (03 is purely imaginary, this leads to the inequalities el > e2 > e3, where e~ = &'((O~). This notation was adopted in many of the earlier texts (e.g., Applications of Elliptic Functions by Greenhill). In relating the Weierstrass function to the theta functions, it is then necessary to take .=(03/(01. The sigma functions are denoted by u(u), u1(u), U2(U), U3(U) and, except for an additional exponential factor, are replicas of the theta functions 81(z), 82(z), 83(z), 84 (z) respectively (with z = 1tu/2(OIl. The notation for the Jacobi functions is the one introduced by Gudermann and Glaisher; this is an international standard and, since alternative notations suggested by Neville and others have not been generally accepted, it was felt no purpose would be served in burdening the reader with an account ofthese.

I am now approaching the termination of a life, one of whose major enjoyments has been the study of mathematics. The three jewels whose efTulgence has most dazzled me are Maxwell's theory of electromagnetism, Einstein's theory of relativity, and the theory of elliptic functions. I have now published textbooks on each of these topics, but the one from whose preparation and writing I have derived the greatest pleasure is the present work on elliptic functions. How enviable are Jacobi and Weierstrass to have been the creators of such a work of art! As a lesser mortal lays down his pen, he salutes them and hopes that his execution of their composition does not ofTend any who have ears to hear the music of the spheres.

D. F. LAWDEN

Contents

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

CHAPTER 1

Theta Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.l. Theta Functions as Solutions of the Heat Conduction Equation 1 1.2. Definitions of the Four Theta Functions. . . . . . 3 1.3. Periodicity Properties. . . . . . . . . . . . . . . . . 5 1.4. Identities Involving Products of Theta Functions 7 1.5. The Identity 0'1 (O) = O2(0)03(0)04 (0). . 12 1.6. Theta Functions as Infinite Products. 12 1.7. Jacobi's Transformation . . . . . . . . 15 1.8. Landen's Transformation. . . . . . . . 17 1.9. Derivatives of Ratios of the Theta Functions . 18

Exercises . . . . . . . . . . . . . . . . . . . . . . 20

CHAPTER 2

Jacobi's Elliptic Functions ............... . 2.1. Definitions of Jacobi's Elliptic Functions ... . 2.2. Double Periodicity of the Elliptic Functions . 2.3. Primitive Periods and Period Parallelograms . 2.4. Addition Theorems ............... . 2.5. Derivatives of Jacobi's Elliptic Functions .. . 2.6. Elliptic Functions with Imaginary Argument. 2.7. Integrals of the Elliptic Functions. 2.8. Poles of sn u, cn u, and dn u

Exercises .............. .

24 24 26 30 33 36 38 40 41 42

xii Contents

CHAPTER 3

Elliptic Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50 3.1. Elliptic Integral of the First Kind. . . . . . . . . . . . . . . . . . . . . .. 50 3.2. Further Elliptic Integrals Reducible to Canonical Form . . . . . . . . . 52 3.3. General Elliptic Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . .. 55 3.4. Elliptic Integrals of the Second Kind. . . . . . . . . . . . . . . . . . . .. 60 3.5. Properties of the Function E(u) . . . . . . . . . . . . . . . . . . . . . . .. 63 3.6. Jacobi's Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65 3.7. Elliptic Integral of the Third Kind . . . . . . . . . . . . . . . . . . . . .. 67 3.8. Complete Elliptic Integrals. . . . . . . . . . . . . . . . . . . . . . . . . .. 73 3.9. Further Transformations of the Modulus . . . . . . . . . . . . . . . . .. 77 3.10. k-Derivatives of Elliptic Functions . . . . . . . . . . . . . . . . . . . . .. 81

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84

CHAPTER4

Geometrical Applications . . . . . . . . . . . . . . . . . . . . . . . . .. 95 4.1. Geometry of the Ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95 4.2. Area of an Ellipsoid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 100 4.3. The Lemniscate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 102 4.4. Formulae of Spherical Trigonometry. . . . . . . . . . . . . . . . . . . .. 103 4.5. Seiffert's Spiral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 105 4.6. Orthogonal Systems of Cartesian Ovals . . . . . . . . . . . . . . . . . .. 107

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 109

CHAPTER 5

Physical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 114 5.1. The Simple Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 114 5.2. Duffing's Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 117 5.3. Orbits under a Jl/r4 Law of Attraction. . . . . . . . . . . . . . . . . . .. 119 5.4. Orbits under a Jl/r 5 Law of Attraction. . . . . . . . . . . . . . . . . . .. 123 5.5. Relativistic Planetary Orbits. . . . . . . . . . . . . . . . . . . . . . . . .. 126 5.6. Whirling Chain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 129 5.7. Body Rotating Freely about a Fixed Point . . . . . . . . . . . . . . . .. 131 5.8. Current Flow in a Rectangular Conducting Plate. . . . . . . . . . . .. 140 5.9. Parallei Plate Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . .. 142

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 145

CHAPTER6

Weierstrass's Elliptic Function . . . . . . . . . . . . . . . . . . . . . .. 149 6.1. Jacobi's Functions with Specified Periods . . . . . . . . . . . . . . . . .. 149 6.2. The Sigma Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 150 6.3. Alternative Definitions for Jacobi's Elliptic Functions. . . . . . . . . .. 152 6.4. Identities Relating Sigma Functions . . . . . . . . . . . . . . . . . . . .. 152 6.5. Sigma Functions as Infinite Products . . . . . . . . . . . . . . . . . . .. 154 6.6. Weierstrass's Elliptic Function. . . . . . . . . . . . . . . . . . . . . . . .. 155 6.7. Differential Equation Satisfied by ,gJl(u). • • . • . • • • • . • . . . • • . .• 157 6.8. Addition Theorem for ,gJl(u) . . • . . • . • . . . . . . • . . . . . . • • . •• 161 6.9. Relationship between Jacobi's and Weierstrass's Functions. . . . . . .. 162

Contents xiii

6.10. Jacobi's Transformation of the Weierstrass Function . . . . . . . . . .. 164 6.11. Weierstrass Function with Real and Purely Imaginary Periods.. . . .. 166 6.12. Weierstrass's Evaluation of Elliptic Integrals of the First Kind . . . .. 167 6.13. Weierstrass's Method for Integrals of the Second Kind . . . . . . . . .. 172 6.14. Weierstrass's Integral of the Third Kind. . . . . . . . . . . . . . . . . .. 174 6.15. Further Conditions for .?JI(u) to Be Real. . . . . . . . . . . . . . . . . .. 175 6.16. Extraction of .?JI(ulg2 ,g3) from Tables . . . . . . . . . . . . . . . . . . .. 177 6.17. Elliptic Integrals (Negative Discriminant) . . . . . . . . . . . . . . . . .. 179

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 181

CHAPTER 7

Applications of the Weierstrass Functions. . . . . . . . . . . . . . . .. 186 7.1. Orthogonal Families of Cartesian Ovals. . . . . . . . . . . . . . . . . .. 186 7.2. Solution of Euler's Equations for Body Rotation. . . . . . . . . . . . .. 187 7.3. Spherical Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 191 7.4. Motion of a Spinning Top or Gyroscope . . . . . . . . . . . . . . . . .. 196 7.5. Projectile Subject to Resistance Proportional to Cube of Speed . . . .. 199

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 205

CHAPTER 8

Complex Variable Analysis . . . . . . . . . . . . . . . . . . . . . . . .. 207 8.1. Existence of Primitive Periods. . . . . . . . . . . . . . . . . . . . . . . .. 207 8.2. General Properties of Elliptic Functions. . . . . . . . . . . . . . . . . .. 209 8.3. Partial Fraction Expansion of .?JI(u). . . . . . . . . . . . . . . . . . . . .. 212 8.4. Expansions for '(u) and u(u). . . . . . . . . . . . . . . . . . . . . . 217 8.5. Invariants Expressed in Terms of the Periods. . . . . . . . . . . . . . .. 217 8.6. Fourier Expansions for Certain Ratios of Theta Functions. . . . . . .. 218 8.7. Fourier Expansions of the Jacobian Functions . . . . . . . . . . . . . .. 221 8.8. Other Trigonometric Expansions of Jacobi's Functions. . . . . . . . .. 223 8.9. Representation of a General Elliptic Function with Theta and Sigma

Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 226 8.10. Representation of a General Elliptic Function in Terms of .?JI(u). . . .. 227 8.11. Expansion of an Elliptic Function in Terms of '(u) . . . . . . . . . . .. 230 8.12. Quarter Periods K(k) and K'(k) as Analytic Functions of k. . . . . . .. 232 8.13. The Complete Integrals E(k), E'(k) as Analytic Functions of k. . . . .. 236 8.14. Elliptic Integrals in the Complex Plane . . . . . . . . . . . . . . . . . .. 238

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 242

CHAPTER9

Modular Transformations. ......................... 245 9.1. The Modular Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 245 9.2. Generators of the Modular Group . . . . . . . . . . . . . . . . . . . . .. 247 9.3. The Transformation t' = 1 + t. . . . . . . . . . . . . . . . . . . . . . . .. 249 9.4. General Modular Transformation of the Elliptic Functions . . . . . .. 250 9.5. Transformations of Higher Order. . . . . . . . . . . . . . . . . . . . . .. 252 9.6. Third-Order Transformation of Theta Functions. . . . . . . . . . . . .. 253 9.7. Third-Order Transformation of Jacobi's Functions . . . . . . . . . . .. 254 9.8. Transformation of Weierstrass's Function. . . . . . . . . . . . . . . . .. 257

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 258

XIV Contents

APPENDIXA

Fourier Series for a Periodic Analytic Function . 261 APPENDIXB

CaIculation of a Definite Integral. . . . . . . . . . 263 APPENDIXC

BASIC Program for Reduction of EIIiptic Integral to Standard Form . . . . . . . . . . . 265 APPENDIXD

Computation of Tables . 267 Table A. Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 270 Table B. Nome and Complete Integrals of the First and Second Kinds as

Functions of the Squared Modulus . . . . . . . . . . . . . . . . . .. 278 Table C. Jacobi's Functions snx, cnx, and dnx for a Range of Values of

the Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 280 Table D. Legendre's Incomplete Integrals of First and Second Kinds. . . .. 285 Table E. Jacobi's Zeta and Epsilon Functions . . . . . . . . . . . . . . . . .. 293 Table F. Sigma Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 298 Table G. Weierstrass's Zeta Function W1 '(u, w1, iKWd ............ , 302 Table H. Weierstrass's Function wI9(u,W1,iKWd . . . . . . . . . . . . . . .. 303 Table 1. Weierstrass's Function wI9(u + w3, w1, iKW1). . . . . . . . . . . .. 307 TableJ. Weierstrass's Function wI9(u,W1,!w1(1 +iv)) . . . . . . . . . . .. 311 Table K. Stationary Values and Invariants of 9(u,wt>iKW1) and

9(u,W1,tw1(1 +iv)) . . . . . . . . . . . . . . . . . . . . . . . . . . .. 315 Table L. Weierstrass's Function wI9(ulg2 ,g3)' . . . . . . . . . . . . . . . .. 316 Table M. Stationary Values and Invariants of 9(ulg2,g3)' . . . . . . . . . .. 328

Bibliography. . 330 Index . . . . . . 331