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

{{Book|Special Functions for Scientists and Engineers|1968|D. Van Nostrand||W.W. Bell}}

===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
:::[[Gamma|$(2.1)$]]
:::[[Beta|$(2.2)$]]
::2.2 Properties of the beta and gamma functions
:::[[Gamma(1)=1|Theorem 2.1]]
:::[[Gamma(z+1)=zGamma(z)|Theorem 2.2]]
:::[[Gamma(n+1)=n!|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
:::[[Laguerre L|$(6.3)$]]
::6.2 Generating function
:::[[Generating function for Laguerre L|Theorem 6.1]]
::6.3 Alternative expression for the Laguerre polynomials
:::[[L n(x)=(e^x/n!)d^n/dx^n(x^n e^(-x))|Theorem 6.2]]
::6.4 Explicit expressions for, and special values of, the Laguerre polynomials
:::[[L n(0)=1|Theorem 6.3 (i)]]
:::[[L n'(0)=-n|Theorem 6.3 (ii)]]
::6.5 Orthogonality properties of the Laguerre polynomials
:::[[Orthogonality of Laguerre L|Theorem 6.4]]
:::[[Orthogonality of Laguerre L|Theorem 6.4]]
::6.6 Relations between Laguerre polynomials and their derivatives; recurrence relations
:::[[(n+1)L (n+1)(x) = (2n+1-x)L n(x)-nL (n-1)(x)|Theorem 6.5 (i)]]
:::[[XL n'(x)=nL n(x)-n L (n-1)(x)|Theorem 6.5 (ii)]]
:::[[L n'(x)=-Sum L k(x)|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
:::[[Chebyshev T|$(7.1)$]]
:::[[Chebyshev U|$(7.2)$]]
:::[[T n(x)=(1/2)(x+i sqrt(1-x^2))^n+(1/2)(x-i sqrt(1-x^2))^n|Theorem 7.1 (i)]]
:::[[U n(x)=(-i/2)(x+i sqrt(1-x^2))^n+(-i/2)(x-i sqrt(1-x^2))^n|Theorem 7.1 (ii)]]
:::[[T n(x)=Sum (-1)^k n!/((2k)! (n-2k)!) (1-x^2)^k x^(n-2k)|Theorem 7.2 (i)]]
:::[[U n(x)=Sum (-1)^k n!/((2k+1)!(n-2k-1)!)(1-x^2)^(k+1/2)x^(n-2k-1)|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
:::[[Hypergeometric pFq|$(9.1)$]]
::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

[[Category:Book]]