Armitage, Eberlein - elliptic functions

This page intentionally left blank Elliptic Functions In its \ufb01rst six chapters this text seeks to present the basic ideas and properties of the Jacobi elliptic functions as an historical essay , an attempt to answer the fascinating question: \u2018what would the treatment of elliptic functions ha ve been like if Abel had dev eloped the ideas, rather than Jacobi?\u2019 Accordingly , it is based on the idea of in verting integrals which arise in the theory of dif ferential equations and, in particular , the differential equation that describes the motion of a simple pendulum. The later chapters present a more con ventional approach to the W eierstrass functions and to elliptic integrals, and then the reader is introduced to the richly v aried applications of the elliptic and related functions. Applications spanning arithmetic (solution of the general quintic, the representation of an integer as a sum of three squares, the functional equation of the Riemann zeta function), dynamics (orbits, Euler\u2019 s equations, Green\u2019 s functions), and also probability and statistics, are discussed. Elliptic Functions J. V . ARMIT A GE Department of Mathematical Sciences University of Durham and the late W . F . EBERLEIN cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru , UK First published in print format isbn-13 978-0-521-78078-0 isbn-13 978-0-521-78563-1 isbn-13 978-0-511-23974-8 © Cambridge University Press 2006 2006 Information on this title: www.cambrid g e.or g /9780521780780 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. isbn-10 0-511-23974-2 isbn-10 0-521-78078-0 isbn-10 0-521-78563-4 Cambridge University Press has no responsibility for the persistence or accuracy of url s for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org hardback p a p erback p a p erback eBook (Adobe Reader) eBook (Adobe Reader) hardback W . F . Eberlein\u2019 s original was dedicated to P atrick, Kathryn, Michael, Sarah, Robert, Mary and Kristen; I should like to add: T o Sarah, Mark and Nicholas (J. V . Armitage). For some minutes Alice stood without speaking, looking out in all directions ov er the country \u2013 and a most curious country it was. There were a number of tiny little brooks, running straight across it from side to side, and the ground between was de vided up into squares by a number of little green hedges, that reached from brook to brook. \u2018I declare it\u2019 s marked out just like a lar ge chess- board!\u2019 Alice said at last. Lewis Carroll, Thr ough the looking Glass Contents P r eface p ag e ix Original partial p r eface x A c knowled g ements xii 1 The \u2018simple\u2019 pendulum 1 2 J acobian elliptic functions of a complex v ariable 25 3 General p r operties of elliptic functions 62 4 Theta functions 75 5 The J acobian elliptic functions f or complex k 107 6 Int r oduction to trans f ormation theory 136 7 The W eierstrass elliptic functions 156 8 Elliptic integrals 210 9 A pplications of elliptic functions in geometry 232 10 An application of elliptic functions in algebra \u2013 solution of the general quintic equation 276 11 An arithmetic application of elliptic functions: the repr esentation of a positive integer as a sum of three squa r es 318 vii viii Contents 12 A pplications in mechanics, statistics and other topics 338 Appendix 370 Refe r ences 378 Further r eading 383 Ind e x 385

Armitage, Eberlein - elliptic functions

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