SPECIAL FUNCTIONS AN INTRODUCTION TO THE CLASSICAL FUNCTIONS OF MATHEMATICAL PHYSICS

SPECIAL FUNCTIONS AN INTRODUCTION TO THE CLASSICAL FUNCTIONS OF MATHEMATICAL PHYSICS

SPECIAL FUNCTIONS AN INTRODUCTION TO THE CLASSICAL FUNCTIONS OF MATHEMATICAL PHYSICS NICO M.TEMME Centrum voor Wiskunde en Informatica Center for Mathematics and Computer Science Amsterdam, The Netherlands A Wiley-Interscience Publication JOHN WILEY & SONS, lac. New York Chichester Brisbane Toronto Singapore

This text is printed on acid-free paper. Copyright 1996 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012 Library of Congress Cataloging in Publication Data: Temme, Ν. M. Special functions: an introduction to the classical functions of mathematical physics / Nico M. Temme p. cm. Includes bibliographical references and index. ISBN 0-471-11313-1 (cloth : alk. paper) 1. Functions, Special. 2. Boundary value problems. 3. Mathematical physics. I. Title. QC20.7.F87T46 1996 515.5 dc20 95-42939 CIP

CONTENTS 1 Bernoulli, Euler and Stirling Numbers 1.1. Bernoulli Numbers and Polynomials, 2 1.1.1. Definitions and Properties, 3 1.1.2. A Simple Difference Equation, 6 1.1.3. Euler's Summation Formula, 9 1.2. Euler Numbers and Polynomials, 14 1.2.1. Definitions and Properties, 15 1.2.2. Boole's Summation Formula, 17 1.3. Stirling Numbers, 18 1.4. Remarks and Comments for Further Reading, 21 1.5. Exercises and Further Examples, 22 2 Useful Methods and Techniques 2.1. Some Theorems from Analysis, 29 2.2. Asymptotic Expansions of Integrals, 31 2.2.1. Watson's Lemma, 32 2.2.2. The Saddle Point Method, 34 2.2.3. Other Asymptotic Methods, 38 2.3. Exercises and Further Examples, 39 3 The Gamma Function 3.1. Introduction, 41 3.1.1. The Fundamental Recursion Property, 42 3.1.2. Another Look at the Gamma Function, 42 3.2. Important Properties, 43 3.2.1. Prym's Decomposition, 43 3.2.2. The Cauchy-Saalschiitz Representation, 44

νί CONTENTS 3.2.3. The Beta Integral, 45 3.2.4. The Multiplication Formula, 46 3.2.5. The Reflection Formula, 46 3.2.6. The Reciprocal Gamma Function, 48 3.2.7. A Complex Contour for the Beta Integral, 49 3.3. Infinite Products, 50 3.3.1. Gauss'Multiplication Formula, 52 3.4. Logarithmic Derivative of the Gamma Function, 53 3.5. Riemann's Zeta Function, 57 3.6. Asymptotic Expansions, 61 3.6.1. Estimations of the Remainder, 64 3.6.2. Ratio of Two Gamma Functions, 66 3.6.3. Application of the Saddle Point Method, 69 3.7. Remarks and Comments for Further Reading, 71 3.8. Exercises and Further Examples, 72 4 Differential Equations 79 4.1. Separating the Wave Equation, 79 4.1.1. Separating the Variables, 81 4.2. Differential Equations in the Complex Plane, 83 4.2.1. Singular Points, 83 4.2.2. Transformation of the Point at Infinity, 84 4.2.3. The Solution Near a Regular Point, 85 4.2.4. Power Series Expansions Around a Regular Point, 90 4.2.5. Power Series Expansions Around a Regular Singular Point, 92 4.3. Sturm's Comparison Theorem, 97 4.4. Integrals as Solutions of Differential Equations, 98 4.5. The Liouville Transformation, 103 4.6. Remarks and Comments for Further Reading, 104 4.7. Exercises and Further Examples, 104 5 Hypergeometric Functions 107 5.1. Definitions and Simple Relations, 107 5.2. Analytic Continuation, 109 5.2.1. Three Functional Relations, 110 5.2.2. A Contour Integral Representation, 111 5.3. The Hypergeometric Differential Equation, 112 5.4. The Singular Points of the Differential Equation, 114 5.5. The Riemann-Papperitz Equation, 116 5.6. Barnes' Contour Integral for F(a, b; c; z), 119 5.7. Recurrence Relations, 121 5.8. Quadratic Transformations, 122 5.9. Generalized Hypergeometric Functions, 124 5.9.1. A First Introduction to ^-functions, 125

CONTENTS νϋ 5.10. Remarks and Comments for Further Reading, 127 5.11. Exercises and Further Examples, 128 6 Orthogonal Polynomials 133 6.1. General Orthogonal Polynomials, 133 6.1.1. Zeros of Orthogonal Polynomials, 137 6.1.2. Gauss Quadrature, 138 6.2. Classical Orthogonal Polynomials, 141 6.3. Definitions by the Rodrigues Formula, 142 6.4. Recurrence Relations, 146 6.5. Differential Equations, 149 6.6. Explicit Representations, 151 6.7. Generating Functions, 154 6.8. Legendre Polynomials, 156 6.8.1. The Norm of the Legendre Polynomials, 156 6.8.2. Integral Expressions for the Legendre Polynomials, 156 6.8.3. Some Bounds on Legendre Polynomials, 157 6.8.4. An Asymptotic Expansion as η is Large, 158 6.9. Expansions in Terms of Orthogonal Polynomials, 160 6.9.1. An Optimal Result in Connection with Legendre Polynomials, 160 6.9.2. Numerical Aspects of Chebyshev Polynomials, 162 6.10. Remarks and Comments for Further Reading, 164 6.11. Exercises and Further Examples, 164 7 Confluent Hypergeometric Functions 171 7.1. The M-function, 172 7.2. The [/-function, 175 7.3. Special Cases and Further Relations, 177 7.3.1. Whittaker Functions, 178 7.3.2. Coulomb Wave Functions, 178 7 3.3. Parabolic Cylinder Functions, 179 7 3.4. Error Functions, 180 7.3.5. Exponential Integrals, 180 7.3.6. Fresnel Integrals, 182 7.3.7. Incomplete Gamma Functions, 185 7.3.8. Bessel Functions, 186 7.3.9. Orthogonal Polynomials, 186 7.4. Remarks and Comments for Further Reading, 186 7.5. Exercises and Further Examples, 187 8 Legendre Functions 193 8.1. The Legendre Differential Equation, 194 8.2. Ordinary Legendre Functions, 194

viii CONTENTS 8.3. Other Solutions of the Differential Equation, 196 8.4. A Few More Series Expansions, 198 8.5. The function Q n (z), 200 8.6. Integral Representations, 202 8.7. Associated Legendre Functions, 209 8.8. Remarks and Comments for Further Reading, 213 8.9. Exercises and Further Examples, 214 9 Bessel Functions 219 9.1. Introduction, 219 9.2. Integral Representations, 220 9.3. Cylinder Functions, 223 9.4. Further Properties, 227 9.5. Modified Bessel Functions, 232 9.6. Integral Representations for the /- and ΛΓ-Functions, 234 9.7. Asymptotic Expansions, 238 9.8. Zeros of Bessel Functions, 241 9.9. Orthogonality Relations, Fourier-Bessel Series, 244 9.10. Remarks and Comments for Further Reading, 247 9.11. Exercises and Further Examples, 247 10 Separating the Wave Equation 257 10.1. General Transformations, 258 10.2. Special Coordinate Systems, 259 10.2.1. Cylindrical Coordinates, 259 10.2.2. Spherical Coordinates, 261 10.2.3. Elliptic Cylinder Coordinates, 263 10.2.4. Parabolic Cylinder Coordinates, 264 10.2.5. Oblate Spheroidal Coordinates, 266 10.3. Boundary Value Problems, 268 10.3.1. Heat Conduction in a Cylinder, 268 10.3.2. Diffraction of a Plane Wave Due to a Sphere, 270 10.4. Remarks and Comments for Further Reading, 271 10.5. Exercises and Further Examples, 272 11 Special Statistical Distribution Functions 275 11.1. Error Functions, 275 11.1.1. The Error Function and Asymptotic Expansions, 276 11.2. Incomplete Gamma Functions, 277 11.2.1. Series Expansions, 279 11.2.2. Continued Fraction for Γ(α, ζ), 280 11.2.3. Contour Integral for the Incomplete Gamma Functions, 282 11.2.4. Uniform Asymptotic Expansions, 283

CONTENTS ix 11.2.5. Numerical Aspects, 286 11.3. Incomplete Beta Functions, 288 11.3.1. Recurrence Relations, 289 11.3.2. Contour Integral for the Incomplete Beta Function, 290 11.3.3. Asymptotic Expansions, 291 11.3.4. Numerical Aspects, 297 11.4. Non-Central Chi-Squared Distribution, 298 11.4.1. A Few More Integral Representations, 300 11.4.2. Asymptotic Expansion; m Fixed, j Large, 302 11.4.3. Asymptotic Expansion; j Large, m Arbitrary, 303 11.4.4. Numerical Aspects, 305 11.5. An Incomplete Bessel Function, 308 11.6. Remarks and Comments for Further Reading, 309 11.7. Exercises and Further Examples, 310 12 Elliptic Integrals and Elliptic Functions 319 12.1. Complete Integrals of the First and Second Kind, 315 12.1.1. The Simple Pendulum, 316 12.1.2. Arithmetic Geometric Mean, 318 12.2. Incomplete Elliptic Integrals, 321 12.3. Elliptic Functions and Theta Functions, 322 12.3.1. Elliptic Functions, 323 12.3.2. Theta Functions, 324 12.4. Numerical Aspects, 328 12.5. Remarks and Comments for Further Reading, 329 12.6. Exercises and Further Examples, 330 13 Numerical Aspects of Special Functions 333 13.1. A Simple Recurrence Relation, 334 13.2. Introduction to the General Theory, 335 13.3. Examples, 338 13.4. Miller's Algorithm, 343 13.5. How to Compute a Continued Fraction, 347 Bibliography 349 Notations and Symbols 361 Index 365

PREFACE This book gives an introduction to the classical well-known special functions which play a role in mathematical physics, especially in boundary value problems. Usually we call a function "special" when the function, just as the logarithm, the exponential and trigonometric functions (the elementary transcendental functions), belongs to the toolbox of the applied mathematician, the physicist or engineer. Usually there is a particular notation, and a number of properties of the function are known. This branch of mathematics has a respectable history with great names such as Gauss, Euler, Fourier, Legendre, Bessel and Riemann. They all have spent much time on this subject. A great part of their work was inspired by physics and the resulting differential equations. About 70 years ago these activities culminated in the standard work A Course of Modern Analysis by Whittaker and Watson, which has had great influence and is still important. This book has been written with students of mathematics, physics and engineering in mind, and also researchers in these areas who meet special functions in their work, and for whom the results are too scattered in the general literature. Calculus and complex function theory are the basis for all this: integrals, series, residue calculus, contour integration in the complex plane, and so on. The selection of topics is based on my own preferences, and of course, on what in general is needed for working with special functions in applied mathematics, physics and engineering. This book gives more than a selection of formulas. In the many exercises hints for solutions are often given. In order to keep the book to a modest size, no attention is paid to special functions which are solutions of periodic differential equations such as Mathieu and Lame functions; these functions are only mentioned when separating the wave equation. The current interest in 4-hypergeometric functions would justify an extensive treatment of this topic. It falls outside the scope of the present work, but a short introduction is given nevertheless. xi

χίϊ PREFACE Today students and researchers have computers with formula processors at their disposal. For instance, MATLAB and MATHEMATICA are powerful packages, with possibilities of computing and manipulating special functions. It is very useful to exploit this software, but often extra analysis and knowledge of special functions are needed to obtain optimal results. At several occasions in the book I have paid attention to the asymptotic and numerical aspects of special functions. When this becomes too specialistic in nature the references to recent literature are given. A separate chapter discusses the stability aspects of recurrence relations for several special functions are discussed. It is explained that a given recursion cannot always be used for computations. Much of this information is available in the literature, but it is difficult for beginners to locate. Part of the material for this book is collected from well-known books, such as from HOCHSTADT, LEBEDEV, OLVER, RAINVILLE, SZEGO and WHITTAKER & WATSON. In addition to these I have used Dutch university lecture notes, in particular those by Prof. H.A. Lauwerier (University of Amsterdam) and Prof. J. Boersma (Technical University Eindhoven). The enriching and supporting comments of Dick Askey, Johannes Boersma, Tom Koornwinder, Adri Olde Daalhuis, Frank Olver, and Richard Paris on earlier versions of the manuscript are much appreciated. When there are still errors in the many formulas I have myself to blame. But I hope that the extreme standpoint of Dick Askey, who once advised me: never trust a formula from a book or table; it only gives you an idea how the exact result looks like, is not applicable to the set of formulas in this book. However, this is a useful warning. Amsterdam, The Netherlands Nico M. TEMME

SPECIAL FUNCTIONS AN INTRODUCTION TO THE CLASSICAL FUNCTIONS OF MATHEMATICAL PHYSICS