Atlas for Computing Mathematical Functions: An Illustrated Guidebook for Practitioners With Programs in Fortran 90 and Mathematica

Rating:
Summary: Not a substitute for Handbook of Mathematical Functions
Review: According to the author, the practicality of books on functions is often limited by the graphical content, and that books of tables like Abramowitz & Stegun "serve a very limited purpose" today (primarily to check function evaluating software). This "Atlas" tries to fill this void, and it does a respectable job. However, detailed function exploration and visualization is left to Mathematica users, the primary audience for this book. Without Mathematica 2.2 or later, and/or a Fortran 90 complier, a great deal in this title cannot be fully appreciated. Of course, the reader can often explore function behavior by plotting an equation using common spreadsheet software, or a graphing calculator, without using F90 or Mathematica. But much of this high-priced textbook consists of small blocks of F90 code and everlasting references to Mathematica notebook calls. Part II contains ~200 pages of Mathematica notebook descriptions and ~100 pages of F90 driver programs, not to mention the F90 code documented throughout the first 600 pages.

Most the "Atlas" graphics are small (~2") gray-scale screenshots of Mathematica plots. The quality of some graphics leaves something to be desired though (Figure 7.2.2 p. 117 for example). Many figures are obviously grainy - vertical lines and text characters often appear as broken line segments, not unlike a tilted faxed image. These gray-scale images are fairly bland; I expected at least a little color and only the highest quality graphics for a book calling itself an "atlas", especially for the asking price.

"Atlas" is no substitute for the timeless books of tables and equations such as the (inexpensive) A&S (ISBN 0486612724) or the CRC Standard Math Tables. In computing the error function (erf), for example, Dr. Thompson defines erf in terms of a function call of the gamma function, while A&S provides many, many more alternatives suitable for machine solution. The discussions here, while more generous than A&S, are often not quite as insightful as Numerical Recipes, which the author often references. Instead, pictorial surveys primarily forego a lot of the detailed explanation of the underlying function theory. A few of the functions, such as the Voight distribution, are hard to find in the classic references, but the reader will find very few new topics here. "Atlas" is a well packaged presentation but not quite the insightful, general purpose book for which I had hoped.

The contents are almost identical to an earlier C version by the same name (ISBN 0471002607). The availability in C, F90 and Mathematica is commendable, although it seems that the F90 version may now be out-of-print having been listed at a price for almost two hundred dollars for several years. Programmers of the older Fortran 77 standard will find the level of F90 programming reasonably suited for translation back to the older standard - or even C itself, if necessary. Therefore, Mathematica users in particular will find the used but now heavily discounted F90 copies the much greater bargain.

Rating:
Summary: nice reference book for practitioners
Review: Don't expect an in-depth coverage of the theory behind the calculations. Said that, the book is a "concise" presentation of the programs used to calculate special functions present on the CD. For every program, the author shows the underlying equations and the bibliographic sources, plus nice discussions on accuracy/performance when you have the option of several methods.

I would say the book is more useful to the engineer/physicist having to do certain calculations, than to the mathematician/student wanting to learn the ins and outs of special functions theory.

The only dissapointment is a rather poor produced CD: the names of the programs on the CD are in 8.3 format, and I think (not sure though) that not all the code of the driver programs shows up on the CD- there are names of program snippets that are not on the CD- this may be an unnecessary hasle for someone trying to follow the examples in the book to the letter.