Special Functions & Their Applications

336 Pages·1972·9.47 MB·English

by  N. N. Lebedev

SPECIAL FUNCTIONS AND THEIR APPLICATIONS
N. N. LEBEDEV
Physico-Technieal Institute
Academy of Sciences, U.S.S.R.
Revised English Edition
Translated and Edited by
Richard A. Silverman
DOVER PUBLICATIONS, INC.
New York
Copyright © 1972 by Dover Publications, Inc.
Copyright © 1965 by Richard A. Silverman.
All rights reserved.
This Dover edition, first published in 1972, is an unabridged and corrected republication of the work originally
published by Prentice-Hall, Inc., in 1965.
International Standard Book Number: 0-486-60624-4
Library of Congress Catalog Card Number: 72-86228
Manufactured in the United States by Courier Corporation 60624414
AUTHOR’S PREFACE
This book deals with a branch of mathematics of utmost importance to scientists and
engineers concerned with actual mathematical calculations. Here the reader will find a
systematic treatment of the basic theory of the more important special functions, as well
as applications of this theory to specific problems of physics and engineering. In the
choice of topics, I have been guided by the goal of giving a sufficiently detailed
exposition of those problems which are of greatest practical interest. This has naturally
led to a certain curtailment of the purely theoretical part of the book. In this regard, it
should be noted that various useful properties of the special functions which do not
appear in the text proper, will be found in the problems at the end of the appropriate
chapters.
The book presupposes that the reader is familiar with the elements of the theory of
functions of a complex variable, without which one cannot go very far in the study of
special functions. However, in order to make the book more accessible to non-
mathematicians, I have made a serious attempt to keep to a minimum the required
background in complex variable theory. In particular, this has compelled me to depart
from the order of presentation found in other treatments of the subject, where the
special functions are first defined by certain convenient representations in terms of
contour integrals.
The usual elementary course in complex variable theory is adequate for an
understanding of most of the material presented here. It is also desirable, but not
necessary, to know something about the analytic theory of linear differential equations.
I occasionally draw upon other branches of mathematics and physics, but only in
connection with certain specific examples, so that lack of familiarity with the relevant
information is no obstacle to reading the book.
It is assumed that the reader already appreciates, from his own experience, the need for
using special functions. Therefore, I have not made a special point of motivating the
introduction of various functions. By the same token, I have always sought the simplest
way of defining the special functions and deriving their properties, without concern for
historical or other considerations.
The arrangement of the material in the separate chapters is dictated by the desire to
make the different parts of the book independent of each other, at least to a certain
extent, so that one can study the simplest classes of functions without becoming
involved with functions of a more general type. For example, I have separated the
theory of the Legendre polynomials and Bessel functions of integral order from the
general theory of spherical harmonics and cylinder functions, and I have also
constructed the theory of spherical harmonics without recourse to the properties of the
hypergeometric function.
The applications of the theory were selected with the aim of illustrating the different
ways in which special functions are used in problems of physics and engineering. No
attempt has been made to give a detailed treatment of the corresponding branches of
mathematical physics. In this regard, most space has been devoted to the application of
cylinder functions, and particularly, of spherical harmonics.
In preparing the present second edition of the book I have revised an earlier edition in
various ways: Chapter 4 now contains a new version of the theorem on expansions in
series of Hermite polynomials, which extends the previous theorem to a larger class of
functions. I have also increased the number of examples illustrating the technique of
expanding functions in series of Hermite and Laguerre polynomials. In Chapter 5 there
is a new section dealing with the theory of Airy functions, which are often encountered
in mathematical physics and play an important role in deriving asymptotic
representations of various special functions. Chapter 9, devoted to the theory of the
hypergeometric function, has been completely revised, and I hope that in its present
form, this chapter will be useful to theoretical physicists and others concerned with the
application of the hypergeometric function, thereby partially filling a gap in the
literature on the subject. I have added many new problems, which serve both as exercise
material and as a source of supplementary information not to be found in the text itself.
At the same time, I have removed a few problems of no particular interest. Finally, the
references have been brought up-to-date.
I would like to take this opportunity to thank I. P. Skalskaya for help in preparing the
present edition of my book.
N.N.L.
TRANSLATOR’S PREFACE
For the most part, this edition adheres closely to the revised Russian edition (Moscow,
1963). However, as always with the volumes of this series, I have not hesitated to
introduce whatever improvements occurred to me in the course of working through the
book. In the present case, two departures from the original text merit special mention:
1. The Bibliography and the references cited in the footnotes have been slanted
towards books available in English or the West European languages.
2. Chapters 6 and 8 have been equipped with problems, most of them taken from the
excellent collection by Lebedev, Skalskaya and Uflyand (Moscow, 1955).
Finally, it was deemed impractical to build in sufficiently detailed references to
numerical tables of the special functions. Here all roads eventually lead to a
consultation of the exhaustive bibliography compiled by Fletcher, Miller, Rosenhead
and Comrie, or its Russian counterpart by Lebedev and Fedorova.
R.A.S.
CONTENTS
1 THE GAMMA FUNCTION
1.1. Definition of the Gamma Function
1.2. Some Relations Satisfied by the Gamma Function
1.3. The Logarithmic Derivative of the Gamma Function
1.4. Asymptotic Representation of the Gamma Function for Large |z|
1.5. Definite Integrals Related to the Gamma Function
Problems
2 THE PROBABILITY INTEGRAL AND RELATED FUNCTIONS
2.1. The Probability Integral and Its Basic Properties
2.2. Asymptotic Representation of the Probability Integral for Large |z|
2.3. The Probability Integral of Imaginary Argument. The Function F(z)
2.4. The Probability Integral of Argument x. The Fresnel Integrals, 21.
2.5. Application to Probability Theory
2.6. Application to the Theory of Heat Conduction. Cooling of the Surface of a
Heated Object
2.7. Application to the Theory of Vibrations. Transverse Vibrations of an Infinite
Rod under the Action of a Suddenly Applied Concentrated Force
Problems
3 THE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS
3.1. The Exponential Integral and its Basic Properties
3.2. Asymptotic Representation of the Exponential Integral for Large |z|
3.3. The Exponential Integral of Imaginary Argument. The Sine and Cosine
Integrals
3.4. The Logarithmic Integral
3.5. Application to Electromagnetic Theory, Radiation of a Linear Half-Wave
Oscillator
Problems
4 ORTHOGONAL POLYNOMIALS
4.1. Introductory Remarks
4.2. Definition and Generating Function of the Legendre Polynomials
4.3. Recurrence Relations and Differential Equation for the Legendre Polynomials
4.4. Integral Representations of the Legendre Polynomials
4.5. Orthogonality of the Legendre Polynomials
4.6. Asymptotic Representation of the Legendre Polynomials for Large n
4.7. Expansion of Functions in Series of Legendre Polynomials
4.8. Examples of Expansions in Series of Legendre Polynomials
4.9. Definition and Generating Function of the Hermite Polynomials
4.10. Recurrence Relations and Differential Equation for the Hermite Polynomials
4.11. Integral Representations of the Hermite Polynomials
4.12. Integral Equations Satisfied by the Hermite Polynomials
4.13. Orthogonality of the Hermite Polynomials
4.14. Asymptotic Representation of the Hermite Polynomials for Large n
4.15. Expansion of Functions in Series of Hermite Polynomials
4.16. Examples of Expansions in Series of Hermite Polynomials
4.17. Definition and Generating Function of the Laguerre Polynomials
4.18. Recurrence Relations and Differential Equation for the Laguerre Polynomials
4.19. An Integral Representation of the Laguerre Polynomials. Relation between the
Laguerre and Hermite Polynomials
4.20. An Integral Equation Satisfied by the Laguerre Polynomials
4.21. Orthogonality of the Laguerre Polynomials
4.22. Asymptotic Representation of the Laguerre Polynomials for Large n
4.23. Expansion of Functions in Series of Laguerre Polynomials
4.24. Examples of Expansions in Series of Laguerre Polynomials
4.25. Application to the Theory of Propagation of Electromagnetic Waves.
Reflection from the End of a Long Transmission Line Terminated by a Lumped
Inductance
Problems
5 CYLINDER FUNCTIONS: THEORY
5.1. Introductory Remarks
5.2. Bessel Functions of Nonnegative Integral Order
5.3. Bessel Functions of Arbitrary Order
5.4. General Cylinder Functions. Bessel Functions of the Second Kind
5.5. Series Expansion of the Function Y (z)
n
5.6. Bessel Functions of the Third Kind
5.7. Bessel Functions of Imaginary Argument
5.8. Cylinder Functions of Half-Integral Order
5.9. Wronskians of Pairs of Solutions of Bessel’s Equation
5.10. Integral Representations of the Cylinder Functions
5.11. Asymptotic Representations of the Cylinder Functions for Large |z|
5.12. Addition Theorems for the Cylinder Functions, 124.Zeros of the Cylinder
Functions
5.13. Expansions in Series and Integrals Involving Cylinder Functions
5.14. Definite Integrals Involving Cylinder Functions
5.15. Cylinder Functions of Nonnegative Argument and Order
5.16. Airy Functions
Problems
6 CYLINDER FUNCTIONS: APPLICATIONS
6.1. Introductory Remarks
6.2. Separation of Variables in Cylindrical Coordinates
6.3. The Boundary Value Problems of Potential Theory. The Dirichlet Problem for
a Cylinder
6.4. The Dirichlet Problem for a Domain Bounded by Two Parallel Planes
6.5. The Dirichlet Problem for a Wedge
6.6. The Field of a Point Charge near the Edge of a Conducting Sheet
6.7. Cooling of a Heated Cylinder
6.8. Diffraction by a Cylinder
Problems
7 SPHERICAL HARMONICS: THEORY
7.1. Introductory Remarks
7.2. The Hypergeometric Equation and Its Series Solution
7.3. Legendre Functions
7.4. Integral Representations of the Legendre Functions
7.5. Some Relations Satisfied by the Legendre Functions
7.6. Series Representations of the Legendre Functions
7.7. Wronskians of Pairs of Solutions of Legend-re’s Equation
7.8. Recurrence Relations for the Legendre Functions
7.9. Legendre Functions of Nonnegative Integral Degree and Their Relation to
Legendre Polynomials
7.10. Legendre Functions of Half-Integral Degree
7.11. Asymptotic Representations of the Legendre Functions for Large |v|
7.12. Associated Legendre Functions
Problems
8 SPHERICAL HARMONICS: APPLICATIONS
8.1. Introductory Remarks
8.2. Solution of Laplace’s Equation in Spherical Coordinates
8.3. The Dirichlet Problem for a Sphere
8.4. The Field of a Point Charge Inside a Hollow Conducting Sphere
8.5. The Dirichlet Problem for a Cone
8.6. Solution of Laplace’s Equation in Spheroidal Coordinates
8.7. The Dirichlet Problem for a Spheroid
8.8. The Gravitational Attraction of a Homogeneous Solid Spheroid
8.9. The Dirichlet Problem for a Hyperboloid of Revolution
8.10. Solution of Laplace’s Equation in Toroidal Coordinates
8.11. The Dirichlet Problem for a Torus
8.12. The Dirichlet Problem for a Domain Bounded by Two Intersecting Spheres
8.13. Solution of Laplace’s Equation in Bipolar Coordinates
8.14. Solution of Helmholtz’s Equation in Spherical Coordinates
Problems
9 HYPERGEOMETRIC FUNCTIONS
9.1. The Hypergeometric Series and Its Analytic Continuation
9.2. Elementary Properties of the Hypergeometric Function
9.3. Evaluation of F(α, β; γ; z) for Re (γ – α – β) > 0, 243.
9.4. F(α, β; γl z) as a Function of its Parameters
9.5. Linear Transformations of the Hypergeometric Function
9.6. Quadratic Transformations of the Hypergeometric Function
9.7. Formulas for Analytic Continuation of F(α, β; γ; z) in Exceptional Cases
9.8. Representation of Various Functions in Terms of the Hypergeometric Function
9.9. The Confluent Hypergeometric Function
9.10. The Differential Equation for the Confluent Hypergeometric Function and Its
Solution. The Confluent Hypergeometric Function of the Second Kind
9.11. Integral Representations of the Confluent Hypergeometric Functions
9.12. Asymptotic Representations of the Confluent Hypergeometric Functions for
Large |z|
9.13. Representation of Various Functions in Terms of the Confluent
Hypergeometric Functions
9.14. Generalized Hypergeometric Functions
Problems
10 PARABOLIC CYLINDER FUNCTIONS
10.1. Separation of Variables in Laplace’s Equation in Parabolic Coordinates
10.2. Hermite Functions
10.3. Some Relations Satisfied by the Hermite Functions
10.4. Recurrence Relations for the Hermite Functions
10.5. Integral Representations of the Hermite Functions
10.6. Asymptotic Representations of the Hermite Functions for Large |z|
10.7. The Dirichlet Problem for a Parabolic Cylinder
10.8. Application to Quantum Mechanics