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

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## 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
- 2.2 Properties of the beta and gamma functions
- 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
- 6.2 Generating function
- 6.3 Alternative expression for the Laguerre polynomials
- 6.4 Explicit expressions for, and special values of, the Laguerre polynomials
- 6.5 Orthogonality properties of the Laguerre polynomials
- 6.6 Relations between Laguerre polynomials and their derivatives; recurrence relations
- 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
- 7.2 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
- 9.2 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