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Reviewer: John B. Slater

Mathematica version 4, replacing version 3 in 1999, introduces a number of substantial enhancements to the Mathematica system. Some of these are internal efficiency enhancements to the symbolic and numerical calculations. Others are enhancements to the user interface, including improved graphics and sound support, and new language and input/output features. This substantial tome, like previous versions, is a comprehensive guide to all the features of Mathematica. To use Mathematica, it is only necessary to know the relevant part of the system. The work is organized to allow readers to dip into it where they wish—it is essentially a well-structured reference work. The software itself is open— a kernel is present in all systems that run Mathematica, and there are usually local extensions, and this is reflected in the write-up. The work claims to provide a good way of learning Mathematica for all users, from beginning students to professional scientists. A problem-solving approach is followed, with a lot of examples. Wolfram correctly suggests that the only sensible way to use the book is alongside a system, allowing interplay between the two. This agrees with reported experience. Part 1 is an introduction to Mathematica that concentrates on its use as a scientific calculating problem solver. Somewhat surprisingly, and rather datedly, the reader is assumed to know some mathematics but nothing about computers. For instance, there are lengthy sections explaining graphics and sound, and input and output from Notebooks. The part ends with an interesting short section on the internals and problems of the system. Most of this is familiar from earlier versions and is well written, clear, and easy to follow on the page. Part 2 discusses how to enhance Mathematica to add one's own expressions and functions, including recursion; modularity and transformation facilities; the handling of text, graphics, and sound; and file structures. Again the examples are rich, relevant, and well laid out. Part 3 assumes greater mathematical knowledge, of the kind to be expected from a college graduate; this may be more typical of the likely user. It covers calculus, limits, linear algebra, and algebraic manipulation and other routine calculations. Since it is written in a modular fashion, it is again possible to address only the part that users want, or for which they have the mathematical background. This is followed by a substantial set of useful formulas and generator functions and a valuable set of appendices, including a reference guide. A usable index completes the work. Inevitably, there are few references. This work continues to be a very handy, if heavy, guide to Mathematica both for the regular user and for the beginner. Any Mathematica-using organization needs this book on its shelves.