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Atlas for computing mathematical functions. An illustrated guide for practitioners with programs in C and Mathematica. Incl. 1 CD-ROM. (English) Zbl 0873.68100

Chichester: Wiley. xiv, 903 p. (1997).
The Handbook of Mathematical Functions, edited by Abramowitz and Stegun and written by an authoritative team of applied mathematicians, is a standard work on special functions with many graphs and extensive tabular data of the functions covered. Unfortunately there does not exist an equally authoritative and comprehensive treatise of the numerical evaluation of these functions with freely available computer programs in FORTRAN and C. Such a treatise would probably unify and standardize the computation, testing, and documentation of numerical evaluation of mathematical functions and perhaps gain the same popularity as the aforementioned Handbook. For a very extensive survey, with hundreds of references, of the mathematical special functions software, see D. W. Lozier and F. W. J. Olver [Proc. Symp. Appl. Math. 48, 79-125 (1994; Zbl 0815.65030)]. A recent book dealing with numerical computation of special functions in FORTRAN is Zhang Shajie and Jin Jianming [Computation of special functions (1996; Zbl 0865.33001)]. The comprehensive book/CD-ROM under review provides a guided tour through the realm of special functions. The author addresses the presentation mainly for scientists using special functions as a means to express solutions of their research problems. Throughout, a visual approach is emphasized and there are more than 700 computer graphics in the book. Such an abundant collection of graphs and surfaces is certainly of great value for the readers of the book. The graphics are produced with the help of the Mathematica program and the notebooks generating these graphics are on the accompanying CD-ROM. The CD-ROM also contains algorithms in the C language for the numerical computation of the values of these functions. (A FORTRAN version is under preparation.) As a rule the behavior of each function is illustrated by graphs, surfaces, and tables of sample values. All programs are available in the DOS, Mac, and Unix file formats. The focus is on the numerical computation of the values of the 150 functions considered. In addition to the graphics features of Mathematica, its symbolic and numerical computation capabilities are also used. The book is organized into 21 chapters. Chapter 20 deals with the Mathematica notebooks and Chapter 21 the C-programs on the CD-ROM. These two chapters form a documentation for the software and occupy about a third of the entire book. The other 19 chapters deal with the theory of special functions. All the commonly used functions are covered, and the author mentions the Handbook as the most important single reference. At the end of each chapter there is a bibliography for that chapter. The graphics in the book together with the Mathematica notebooks generating them are superb - we can see that the author has found a very impressive and illuminating approach to special functions. The user of the CD-ROM can also easily modify the parameters of each graphics so as to produce another graphics with a new set of parameters. The book has a subtitle ”An Illustrated Guide for Practitioners With Programs in C and Mathematica”. The target readership of this book ”practitioners” (physicists, engineers, scientists) will certainly find this book very useful.
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MSC:

68W30 Symbolic computation and algebraic computation
68-04 Software, source code, etc. for problems pertaining to computer science
33-04 Software, source code, etc. for problems pertaining to special functions

Software:

Mathematica
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