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Asymptotic Analysis at Differential Equatiau Revised Edition ,,,,, / I I I I ,,, / / I I I I \ \ \ ' _,. / / / I I I ' \ ' / I I I ' ' \ // /I l l ' ' - . . - - - / / / / Asymptotic Analysis at Differential Equations Revised Edition Roscoe B White Princeton University, USA ' . -~~~:°Of I 1'; MADRlD \ BIBUOTEC.A l OENCl>S 1 ill .__--- --------· •~~~~~~~~~~Im_p_e_ria_l_C_o_lle_g_e_P_re_ss_ Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ASYMPTOTIC ANALYSIS OF DIFFERENTIAL EQUATIONS (Revised Edition) Copyright© 2010 by Imperial College Press All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-1-84816-607-3 ISBN-10 1-84816-607-9 ISBN-13 978-1-84816-608-0 (pbk) ISBN-10 1-84816-608-7 (pbk) Printed in Singapore by World Scientific Printers. Preface The material for this book has resulted from teaching a graduate course in ordinary differential equations at Princeton for several years. Methods of asymptotic analysis covered include dominant balance, the use of diver gent asymptotic series, phase-integral methods, asymptotic evaluation of integrals, and boundary layer analysis. The construction of integral solu tions and the use of analytic continuation are used in conjunction with the asymptotic analysis, to show the interrelatedness of these methods. Some of the functions of classical analysis are used as examples, to provide an introduction to their analytic and asymptotic properties, and to give derivations of some of the important identities satisfied by them, since it is my experience that students are insufficiently familiar with Whittaker and Watson. There is no attempt to give a complete presentation of all these functions. The emphasis is on the various techniques of analysis: obtaining asymptotic limits, connecting different asymptotic solutions, and obtain ing integral representations. Less attention is paid to strict mathematical rigor. A sufficient number of different examples are chosen to demonstrate the various techniques and approaches. Prerequisite material consists of elementary calculus and a basic knowledge of the theory of functions of a complex variable. In Chapters 1-9 the material in each chapter depends on the results obtained in the previous chapters. These chapters cover: the use of dom inant balance for solving equations with a small parameter; a review of the techniques for obtaining exact solutions of soluble differential equa tions; complex variable theory and analytic continuation; classification of singular points of differential equations and the construction of local expan sions to solutions, including nonconvergent asymptotic expansions; phase integral methods; perturbation theory; asymptotic evaluation of integrals; v vi Asymptotic Analysis of Differential Equations basic properties of the Euler gamma function; and methods for finding in tegral solutions of differential equations. Some basic results concerning the gamma function are in fact used in earlier chapters, so some parts of this chapter should be previewed when necessary. Most of the relations proved concerning r(z), however, depend on results in Chapters 1-7, so it was im practical to move it to an earlier point in the book. Some of the material in the first chapters may be already known, in which case it can be quickly reviewed or skipped. The remaining chapters are independent of one another, depending only on the material in the first nine, and can be covered in any order depending on personal interest. All chapters cannot be covered in one semester, and I normally choose Chapters 11, 13, 17, and 18 after the first nine. Chapter 10 introduces the theory of expansions in series of orthogonal polynomi als and derives some of their properties, and also introduces the theory of wavelets. Chapter 11 gives the theory of the Airy function, an understand ing of which is essential for a good grasp of the WKB treatment of scattering and bound state problems. Chapter 12 continues the development of the theory of phase-integral methods. Chapter 13 further illustrates the use of integral representations and analytic continuation through an examination of the Bessel function. Chapter 14 does the same for the parabolic cylinder function, and Chapter 15 for the Whittaker function. Chapter 16 examines inhomogeneous equations, some arising from the theory of resistive recon nection of a magnetized plasma, and further explores the role of causality using the Green's function technique for the examination of the equations for a parametric instability. Chapter 17 on the Riemann zeta function pro vides a brief introduction to the distribution of prime numbers. Chapter 18 treats differential equations containing a small parameter giving rise to boundary layers. Aside from corrections of typos and other errors, this revised edition differs by the addition of Chapter 12, an extension of the techniques of WKB analysis, a rewriting and extension of Chapter 18 using the meth ods of Kruskal-Newton diagrams, providing a much more powerful way to determine boundary layer scaling, and the addition of several new sections expanding material in different chapters. Roscoe B. White Princeton 2009 Acknowledgments This text is the outcome of interactions with many students and research colleagues over many years. Some of the material presented here is also present in other texts which treat asymptotic methods. Normally not covered are the use of Kruskal-Newton graphs, the derivation of Heading's rules for analytic continuation of phase-integral solutions, the associated derivation of the Stokes constants, and the methods for obtaining integral solutions. These approaches have been integrated to show how they are interrelated. The book was written using Latex, by Leslie Lamport (Addison Wesley, 1994). The figures were made using Super Mongo, written by Robert Lup ton, Princeton University ([email protected]), and Patricia Monger ([email protected]). I am grateful to Bedros Afeyan for discussions concerning wavelets, Greg Rewoldt for help with Latex, Leonid Zakharov for help with computing, Thomas Fischaleck for valuable suggestions re garding boundary layer problems, and to Tim Stoltzfus-Dueck and Jeffrey Parker for proofreading several chapters. Finally, I am grateful to my wife Laura and my daughter Veronica for their patience and support during my many years of teaching and research, and to my brother Bob for encouragement and stimulation. vii Contents Preface ..... . v Acknowledgments . vii List of Figures . . XVll 1. Dominant Balance 1 1.1 Introduction . . . . . . . . . . . . . . . . . . 2 1.2 Solutions Using Kruskal-Newton Diagrams 3 1. 2 .1 Third order . . . . . 3 1. 2. 2 Non polynomial form 6 1. 2. 3 Higher order . 8 1.2.4 Hidden points 10 1.3 Problems 12 2. Exact Solutions .... . . 15 2.1 Introduction . . . . . 16 2.2 Constant Coefficients . 17 2.3 Inhomogeneous Linear Equations 18 2.4 The Fredholm Alternative ... . 20 2.5 The Diffusion Equation .... . 22 2.6 Exact Solutions to Nonlinear Equations 23 2. 7 Phase Plane Analysis . 24 2.8 Problems 28 3. Complex Variables ..... . 31 3.1 Analyticity ..... . 32 3.2 Cauchy Integral Theorem 33 3.3 Series Representation .. 34 ix x Asymptotic Analysis of Differential Equations 3.4 The Residue Theorem 35 3.5 Analytic Continuation 37 3.6 Inverse Functions .. . 39 3.7 Problems ...... . 41 4. Local Approximate Solutions 43 4.1 Introduction .... . 44 4.2 Classification ... . 44 4.2.1 Ordinary point . 45 4.2.2 Regular singular point 45 4.2.3 Irregular singular point 48 4.3 Asymptotic Series ...... . 51 4.3.1 Properties . . . . . . . 52 4.3.2 Truncation: A series about x = 0. 54 4.3.3 Truncation: A series about x = oo. 56 4.3.4 Truncation: A series about x = 0 57 4.3.5 Asymptotic oscillation ..... 61 4.3.6 Construction of asymptotic series 62 4.3. 7 The error function . . . 62 4.4 Origin of the Divergence . . . 63 4.5 Improving Series Convergence 66 4.5.1 Shanks transformation 67 4.5.2 Euler summation .. . 67 4.5.3 Borel summation .. . 68 4.6 The Special Functions of Mathematical Physics 68 4. 7 Problems . . . . 70 5. Phase-Integral Methods I ..... . 73 5.1 Introduction .......... . 74 5.2 Solutions far from Singularities 77 5.3 Connection Formulae: Isolated Zero 78 5.4 Derivation of Stokes Constants 81 5.5 Rules for Continuation .. 83 5.6 Causality . . . . . . . ... 85 5. 7 Bound States and Instabilities . 86 5.8 Scattering, Overdense Barrier 87 5.9 Scattering, Underdense Barrier 89 5.10 The Budden Problem. 93 5 .11 The Error Function . 96 5.12 Eigenvalue Problems 97 5.13 Problems ..... . 101