# 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