Book:W.W. Bell/Special Functions for Scientists and Engineers

W.W. Bell: Special Functions for Scientists and Engineers

Published $1968$, D. Van Nostrand.

Contents

PREFACE

LIST OF SYMBOLS

Chapter 1 Series Solution of Differential Equations

1.1 Method of Frobenius

1.2 Examples

Problems

Chapter 2 Gamma and Beta Functions

2.1 Definitions

$(2.1)$

$(2.2)$

2.2 Properties of the beta and gamma functions

Theorem 2.1

Theorem 2.2

Theorem 2.3

2.3 Definition of the gamma function for negative values of the argument

2.4 Examples

Problems

Chapter 3 Legendre Polynomials and Functions

3.1 Legendre's equation and its solution

3.2 Generating function for the Legendre polynomials

3.3 Further expressions for the Legendre polynomials

3.4 Explicit expressions for and special values of the Legendre polynomials

3.5 Orthogonality properties of the Legendre polynomials

3.6 Legendre series

3.7 Relations between the Legendre polynomials and their derivatives; recurrence relations

3.8 Associated Legendre functions

3.9 Properties of the associated Legendre functions

3.10 Legendre functions of the second kind

3.11 Spherical harmonics

3.12 Graphs of the Legendre functions

3.13 Examples

Problems

Chapter 4 Bessel Functions

4.1 Bessel's equation and its solutions; Bessel functions of the first and second kind

4.2 Generating function for the Bessel functions

4.3 Integral representations for Bessel functions

4.4 Recurrence relations

4.5 Hankel functions

4.6 Equations reducible to Bessel's equation

4.7 Modified Bessel functions

4.8 Recurrence relations for the modified Bessel functions

4.9 Integral representations for the modified Bessel functions

4.10 Kelvin's functions

4.11 Spherical Bessel functions

4.12 Behaviour of the Bessel functions for large and small values of the argument

4.13 Graphs of the Bessel functions

4.14 Orthonormality of the Bessel functions; Bessel series

4.15 Integrals involving Bessel functions

4.16 Examples

Problems

Chapter 5 Hermite Polynomials

5.1 Hermite's equation and its solution

5.2 Generating function

5.3 Other expressions for the Hermite polynomials

5.4 Explicit expressions for, and special values of, the Hermite polynomials

5.5 Orthogonality properties of the Hermite polynomials

5.6 Relations between Hermite polynomials and their derivatives

5.7 Weber-Hermite functions

5.8 Examples

Problems

Chapter 6 Laguerre Polynomials

6.1 Laguerre's equation and its solution

$(6.3)$

6.2 Generating function

Theorem 6.1

6.3 Alternative expression for the Laguerre polynomials

Theorem 6.2

6.4 Explicit expressions for, and special values of, the Laguerre polynomials

Theorem 6.3 (i)

Theorem 6.3 (ii)

6.5 Orthogonality properties of the Laguerre polynomials

Theorem 6.4

Theorem 6.4

6.6 Relations between Laguerre polynomials and their derivatives; recurrence relations

Theorem 6.5 (i)

Theorem 6.5 (ii)

Theorem 6.5 (iii)

6.7 Associated Laguerre polynomials

6.8 Properties of the associated Laguerre polynomials

6.9 Notation

6.10 Examples

Problems

Chapter 7 Chebyshev Polynomials

7.1 Definition of Chebyshev polynomials; Chebyshev's equation

$(7.1)$

$(7.2)$

Theorem 7.1 (i)

Theorem 7.1 (ii)

Theorem 7.2 (i)

Theorem 7.2 (ii)

7.2 Gneerating function

7.3 Orthogonality properties

7.4 Recurrence relations

7.5 Examples

Problems

Chapter 8 Gegenbauer and Jacobi Polynomials

8.1 Gegenbauer polynomials

8.2 Jacobi polynomials

8.3 Examples

Problems

Chapter 9 Hypergeometric Functions

9.1 Definition of hypergeometric functions

$(9.1)$

9.2 Properties of the hypergeometric function

9.3 Properties of the confluent hypergeometric function

9.4 Examples

Problems

Chapter 10 Other Special Functions

10.1 Incomplete gamma functions

10.2 Exponential integral and related functions

10.3 The error function and related functions

10.4 Riemann's zeta function

10.5 Debye functions

10.6 Elliptic integrals

10.7 Examples

Problems

Appendices

1 Convergence of Legendre series

2 Euler's constant

3 Differential equations

4 Orthogonality relations

5 Generating functions

Hints and Solutions to Problems

Bibliography

Index