Special Functions: An Introduction to the Classical Functions of Mathematical PhysicsThis book gives an introduction to the classical, well-known special functions which play a role in mathematical physics, especially in boundary value problems. Calculus and complex function theory form the basis of the book and numerous formulas are given. Particular attention is given to asymptomatic and numerical aspects of special functions, with numerous references to recent literature provided. |
Contents
Bernoulli Euler and Stirling Numbers | 1 |
Useful Methods and Techniques | 29 |
The Gamma Function | 41 |
29 | 66 |
Differential Equations | 79 |
Hypergeometric Functions | 107 |
Orthogonal Polynomials | 133 |
Confluent Hypergeometric Functions | 171 |
Separating the Wave Equation | 257 |
Special Statistical Distribution Functions | 275 |
Elliptic Integrals and Elliptic Functions | 315 |
Numerical Aspects of Special Functions | 333 |
Bibliography | 349 |
107 | 355 |
| 360 | |
Notations and Symbols | 361 |
Common terms and phrases
algorithm assume asymptotic behavior asymptotic expansion Chapter Chebyshev coefficients complex values compute confluent hypergeometric functions consider continued fraction contour converges cosh cylinder functions defined derive differential equation dominant elliptic function elliptic integrals erfc error function Euler Exercise exponential follows formula Fourier Gauss GAUTSCHI given gives Hankel functions Helmholtz equation Hence incomplete gamma function initial values integral representation Jacobi polynomials Laguerre polynomials Laplace Legendre functions Legendre polynomials linear Math method minimal solution modified Bessel functions obtain OLVER orthogonal polynomials pair parameters Pn(x pole power series proof Q-function Qu(x Qu(z recurrence relation respect result right-hand side saddle point satisfies second kind series expansions Show sin² sinh special functions substituting theorem theory theta functions transformation uniform asymptotic variable Verify write