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Computation of special functions: with over 100 computer programs in FORTRAN. (English) Zbl 0865.33001
New York, NY: John Wiley & Sons, xxv, 717 p. $ 89.95 (1996).
Since its publication more than 30 years ago, the Handbook of mathematical functions, edited by Abramowitz and Stegun and written by an authoritative team of special function theorists, has become the standard reference in the field. Though this book continues to be the most valuable reference, there is a need for books with an emphasis on computer algorithms for evaluation of special functions. It would be very fortunate if a second volume of the Handbook were written by an authoritative team of special function theorists and computer scientists with freely usable state-of-the-art computer algorithms in FORTRAN and C for all the functions of the Handbook. Such a second volume could also serve as a significant step towards establishing a standard for the documentation of mathematical special functions’ algorithms. This need for computer algorithms is particularly urgent among “computational scientists”, applied mathematicians, physicists, and engineers who can express the solutions of their research problems in terms of such functions and then want to treat the solutions numerically. The authors of the book under review are specialists in the field of electromagnetic field computations. They point out that the existing software packages fail to provide source code for all the functions needed in their research, although many are provided, and that they have written this book with more than 130 FORTRAN programs on the accompanying disk to fill this gap. Obviously a book written in this spirit is a very welcome addition to the mathematical literature. In fact, this book/disk is probably the first comprehensive treatment of the field in book form with computer programs. Most of the functions in the Handbook are implemented in this book. (For the computation of special functions in the C language, there is a smaller book by {\it S. L. Moshier}, “Methods and programs for mathematical functions” (1989; Zbl 0701.65011), programs separately available.) A very extensive survey, with hundreds of references, of the mathematical special functions software was published in 1994 by {\it D. Lozier} an {\it F. W. O. Olver} [“Numerical evaluation of special functions” (1994; Zbl 0815.65030)]. The book contains 20 chapters, with each chapter (except Chapter 20) consisting of three major parts. The first part presents an introduction to the special functions to be dealt with. The second part presents algorithms used for computing and lists the corresponding FORTRAN-77 programs. The third part presents numerical tables with representative results provided by the programs. The chapters are as follows: 1. Bernoulli and Euler polynomials; 2. Orthogonal polynomials (Chebyshev polynomials, Laguerre polynomials, Hermite polynomials); 3. Gamma, beta, and psi-functions; 4.Legendre functions; 5. Bessel functions; 6. Modified Bessel functions; 7. Integrals of Bessel functions; 8. Spherical Bessel functions; 9. Kelvin functions; 10. Airy functions; 11. Struve functions; 12. Hypergeometric and confluent hypergeometric functions; 13. Parabolic cylinder functions; 14. Mathieu functions; 15. Spheroidal wave functions; 16. Error function and Fresnel integrals; 17. Cosine and sine integrals; 18. Elliptic integrals and Jacobian elliptic functions; 19. Exponential integrals; 20. Summary of methods for computing special functions; The FORTRAN programs are written in double precision, and for each program there is a minimal main program illuminating the use of the functions implemented. The authors have written a book which will be very valuable for its target readership, engineers, applied mathematicians, and computational scientists. Many potential users of the book would like to have a parallel version of the book in the C language. On the other hand, there are extensive repositories of public domain algorithms, freely accessible by ftp, such as the SLATEC special function package (see the survey referred to above). The book would have been even better if the authors had given a review of what is nowadays available elsewhere, provided detailed justification for the choice of each algorithm, and given a more comprehensive collection of test programs (for instance those programs that were used to create the numerical tables of each chapter.

33-02Research monographs (special functions)
65D20Computation of special functions, construction of tables
33-04Machine computation, programs (special functions)