Second Order Differential Equations

Gerhard Kristensson

Second Order DifferentialEquations

Special Functions and Their Classification

Springer New York Dordrecht Heidelberg London

© Springer Science+Business Media, LLC 2010 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, 233 Spring Street, New York, NY 10013, USA), 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 in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Gerhard Kristensson

and Information Technology

SE-221 00 LundSwedenGerhard.Kristensson@eit.lth.se

Department of Electrical

ISBN 978-1-4419-7019-0 e-ISBN 978-1-4419-7020-6 DOI 10.1007/978-1-4419-7020-6

Library of Congress Control Number: 2010931828

Mathematics Subject Classification (2010): 33XX, 33CXX, 34XX, 34MXX

Lund University

To the Memory of my Father, Gunnar

Preface

With modern computers and software tools, there are, in general, no problems gen-erating numerical values to various kinds of special functions. Indeed, excellentsoftware, such as Mathematica, Maple, and MATLAB, can generate a large vari-ety of special functions with high precision. Moreover, the general topic of specialfunctions is well covered in the literature, see, e.g., [2, 3, 8, 14, 20, 28]. However,the knowledge of the overall structure and relationship between the different specialfunctions is often lacking nowadays. This textbook tries to remedy that need. Thepresentation of the theory in the book does not deal with the particular properties ofvarious special functions, but rather focuses on the generic connection between thefunctions and the families they belong to.

The way the special functions are introduced and classified is the subject of thecurrent textbook. The aim is to provide a self-contained treatment of the subjectintended for the upper undergraduate, graduate student, or the researcher in mathe-matical physics who has a need to understand the underlying systematics of specialfunctions. There are many ways of approaching this subject indeed. The most pop-ular ones are:

• Classification and systematics based upon the singular behavior of the coeffi-cients of the underlying ordinary differential equation, see, e.g., [12, 23]

• Group theoretical approach, see, e.g., [17, 27, 29]• Classification and systematics based upon integral averages, see, e.g., [4]

We pursue the first, most traditional, approach in this textbook. The singular be-havior of the coefficients manifests itself by the number of singular points — polesor branch points — in the complex plane. For systems with less than three singularpoints, the solutions of the ordinary equation belong to a rather well defined classof functions. As the number of singular points becomes three or more, the solutionclass becomes “rich” in the sense that most functions encountered in mathematicalphysics are found among these solutions. The purpose of this book is to explore andunderstand the systematics of these classes of functions and their relations to themany special functions encountered in applications and in mathematical physics.

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The focus is on the overall relationship of these solutions, rather than the analysisof the particular special functions themselves.

This textbook originates from a series of seminars in the mid-1970s on spe-cial functions given by the late Professor Nils Svartholm at the Institute of The-oretical Physics, Chalmers University of Technology, Goteborg, Sweden. Inciden-tally, Professor Svartholm made several important contributions to the solution ofHeun’s equations in the late 1930s, see Section 8.4. The general outline of ProfessorSvartholm’s notes is to some extent kept, but numerous extensions have been madein order to make the text more complete.

The text intends to cover a three- to four-week upper undergraduate or graduatecourse on the subject. The prerequisites of the course are analytic function theory onthe level of, e.g., E. Hille [13] or R. Greene and S. Krantz [9]. Specifically, the readershould have basic knowledge of the method of residues and multi-valued functions.To make the textbook more self-contained and complete, a series of appendices arefound at the end of the book, which contain specific background material. At theend of each chapter, there are problems that illustrate the analysis in the chapter. Itis recommended that students solve these problems in order to get a better under-standing of the theory. Problems marked with a dagger, †, indicate problems that aremore difficult. A solution manual to all problems is available at the home page ofthe author.

I am most grateful to Professor Anders Melin, who has been very supportive andhelpful during this whole project. He has contributed numerous valuable commentsand improvements to the text. This is particularly true for Appendices A and Bin which he has given esteemed input and criticism. Writing this book has takenmost of my leisure time, and thanks to an understanding wife, the project had anhappy ending. Thank you Mona-Lisa! The author is also grateful to Martin Nor-gren, Kristin Persson, Daniel Sjoberg, and Christian Sohl for finding typos. FinallyI like to thank Springer, especially Vaishali Damle and Marcia Bunda, for very con-structive collaboration.

Lund, May 5, 2010 Gerhard Kristensson

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Basic properties of the solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 ODE of second order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Standard forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Classification of points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Solution at a regular point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 The second solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Solution at a regular singular point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.1 The indicial equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.2 Convergence of the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.3 The second solution — exceptional case . . . . . . . . . . . . . . . . . 20

2.5 Solution at a regular singular point at infinity . . . . . . . . . . . . . . . . . . . 24Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Equations of Fuchsian type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1 Regular singular point at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Regular point at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 The displacement theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Equations with one to four regular singular points . . . . . . . . . . . . . . . . . 434.1 ODE with one regular singular point . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.1 Regular singular point at infinity . . . . . . . . . . . . . . . . . . . . . . . 434.1.2 Regular point at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 ODE with two regular singular points . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.1 One regular singular point at infinity . . . . . . . . . . . . . . . . . . . . 454.2.2 Regular point at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.3 Confluence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 ODE with three regular singular points . . . . . . . . . . . . . . . . . . . . . . . . . 49

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4.3.1 One regular singular point at infinity . . . . . . . . . . . . . . . . . . . . 494.3.2 Regular point at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 ODE with four regular singular points . . . . . . . . . . . . . . . . . . . . . . . . . 564.4.1 One regular singular point at infinity . . . . . . . . . . . . . . . . . . . . 564.4.2 Regular point at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 The hypergeometric differential equation . . . . . . . . . . . . . . . . . . . . . . . . . 615.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2 Hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3 Recursion and differentiation formulae . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3.1 Gauss’ contiguous relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.4 Kummer’s solutions to hypergeometric differential equation . . . . . . . 725.5 Integral representation of F(α,β ;γ;z) . . . . . . . . . . . . . . . . . . . . . . . . . 775.6 Barnes’ integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.6.1 Relation between F(·, ·; ·;z) and F(·, ·; ·;1− z) . . . . . . . . . . . . 895.7 Quadratic transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.8 Hypergeometric polynomials (Jacobi) . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.8.1 Definition of the Jacobi polynomials . . . . . . . . . . . . . . . . . . . . 935.8.2 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.8.3 Rodrigues’ generalized function . . . . . . . . . . . . . . . . . . . . . . . . 985.8.4 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.8.5 Integral representation (Schlafli) . . . . . . . . . . . . . . . . . . . . . . . . 1005.8.6 Recursion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6 Legendre functions and related functions . . . . . . . . . . . . . . . . . . . . . . . . . 1076.1 Legendre functions of first and second kind . . . . . . . . . . . . . . . . . . . . . 1076.2 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.3 Associated Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7 Confluent hypergeometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.1 Confluent hypergeometric functions — first kind . . . . . . . . . . . . . . . . 123

7.1.1 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.1.2 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.1.3 Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307.1.4 Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.2 Confluent hypergeometric functions — second kind . . . . . . . . . . . . . . 1327.2.1 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.2.2 Bessel functions — revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.3 Solutions with three singular points — a summary . . . . . . . . . . . . . . . 1377.4 Generalized hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

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8 Heun’s differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1418.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1418.2 Power series solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.3 Polynomial solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1468.4 Solution in hypergeometric polynomials . . . . . . . . . . . . . . . . . . . . . . . 147

8.4.1 Asymptotic properties of the polynomials yn(z) . . . . . . . . . . . 1488.4.2 Asymptotic properties of the coefficients cn . . . . . . . . . . . . . . 1518.4.3 Domain of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

8.5 Confluent Heun’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1588.6 Special examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8.6.1 Lame’s differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1598.6.2 Differential equation for spheroidal functions . . . . . . . . . . . . . 1608.6.3 Mathieu’s differential equation . . . . . . . . . . . . . . . . . . . . . . . . . 161

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

A The gamma function and related functions . . . . . . . . . . . . . . . . . . . . . . . . 163A.1 The gamma function Γ (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163A.2 Estimates of the gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166A.3 The Appell symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173A.4 Psi (digamma) function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175A.5 Binomial coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175A.6 The beta function B(x,y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176A.7 Euler–Mascheroni constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

B Difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179B.1 Second order recursion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179B.2 Poincare–Perron theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182B.3 Asymptotic behavior of recursion relations . . . . . . . . . . . . . . . . . . . . . 185B.4 Estimates of some sequences and series . . . . . . . . . . . . . . . . . . . . . . . . 193

B.4.1 Riemann zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194B.4.2 The sum ∑

nk=1 k−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

B.4.3 Convergence of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

C Partial fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

D Circles and ellipses in the complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . 203D.1 Equation of the circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

D.1.1 Harmonic circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204D.2 Equation of the ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

E Elementary and special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209E.1 Hypergeometric function 2F1(α,β ;γ;z) . . . . . . . . . . . . . . . . . . . . . . . . 209E.2 Confluent functions 1F1(α;γ;z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

E.2.1 Error functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212E.3 Confluent functions 0F1(γ;z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

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F Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217