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Special functions have important applications in almost every field of engineering and science. Their computation is often required when numerical results of analyses are desired. Since most of these functions have the form of infinite series or infinite integrals, the computation is by no means a trivial task. Many mathematicians and scientists have worked on this subject during the past several decades. Their fruitful efforts were reflected partly in the famous Handbook of Mathematical Functions (edited by M. Abramowitz and I. Stegun with contributions from 28 scientists, first published by the National Bureau of Standards in 1964) and in several well-known commercial software packages.
With the book by Abramowitz and Stegun and the software packages referenced above, one naturally asks: "Why write another book on special functions?" Our answer is twofold. First, the book by Abramowitz and Stegun is basically a collection of formulas, graphs, and tables for special functions. It is an ideal handbook and research reference but does not emphasize the issue of numerical computation using computers. Our book, as its name suggests, concerns primarily the automated computation of special functions. Moreover, our book contains many computer programs (over 100 in total and all written by the authors). Second, most software packages do not contain programs for all special functions, although they do include many of them. The special functions not contained in some of those packages include the Bessel functions and modified Bessel functions with arbitrary orders and complex arguments, the Mathieu functions, the modified Mathieu functions, parabolic cylinder functions, and various prolate and oblate spheroidal wave functions. All of these functions are important despite their complicated computation. These functions, along with many others and their derivatives (and in some cases their integrals), are all addressed in this book. Furthermore, the programs in the software packages can only be used as black boxes although this is usually sufficient for most applications, some sophisticated problems require users to know the program's internal structure to make necessary modifications. This book describes the algorithms and thus provides the reader with specific information about each individual program. It is our hope that the publication of this book can provide scientists in engineering, physics, chemistry, and other fields a necessary, convenient, and affordable tool for computing a variety of special functions. As an addition, we make the computer programs (over 100 in total) available in the diskette accompanying the book.
Bernoulli and Euler Numbers
Orthogonal Polynomials
Gamma, Beta, and Psi Functions
Legendre Functions
Bessel Functions
Modified Bessel Functions
ntegrals of Bessel Functions
Spherical Bessel Functions
Kelvin Functions
Airy Functions
Struve Functions
Hypergeometric and Confluent Hypergeometric Functions
Parabolic Cylinder Functions
Mathieu Functions
Spheroidal Wave Functions
Error Function and Fresnel Integrals
Cosine and Sine Integrals
Elliptic Integrals and Jacobian Elliptic Functions
Exponential Integrals
Summary of Methods for Computing Special Functions
A: Derivation of Some Special Differential Equations
B: Root-Finding Methods
C: About the Software