If you're reading this, you're probably already somewhat familiar with the Handbook of Mathematical Functions, lovingly known to techies as "AMS 55" or "Abramowitz and Stegun" (after its editors), the thick, brick-red bestiary of special functions, the sometimes ugly solutions to everyday differential equations.

If anything, you probably at least took it down from next to the Table of Integrals, Series, and Products in a dusty corner of the library once as a student, maybe to use the tables to do homework involving Gaussian quadrature or to reassure yourself figure out that there's no easy simplification for an elliptic integral.

If so, you probably know that it's exactly as the title says: a guide to important properties of special functions, from elementary transcendentals like circular and hyperbolic trigonometric functions, to harmonics such as Bessel and Legendre functions, to the one-size-almost-fits-all hypergeometric function, to non-analytic nasties like the gamma function. Most sections begin with a differential equation or two, followed by sections on representations of the function, identities, integration, differentiation, and the like, illustrated by graphs where appropriate. At the end of most sections is a series of computer-generated tables of the functions discussed in the section for common values of the parameters.

This is all augmented nicely by a section on numerical integration, differentiation, and interpolation, which of course expands the utility of the tables provided, and another on the Laplace transform. The Fourier transform, strangely, is missing completely. Sections on combinatorial analysis and probability and inferential statistics are present, but are short and not too useful, although their respective tables are well done.

Presentation is mostly well-structured, although it's not systematic enough to find anything quickly, and it doesn't even come close to the level of organization of Gradshteyn and Rhyzik . Inclusion of a symbolic index for those trying to find functions for which they don't know the name, or to identify a differential equation, would've been nice; the reader must instead rely on guesses, blind luck, persistence, and use of other references. Using this book is still preferable to the shot-in-the-dark that is typing a function or equation into a computer algebra system, but a little rearrangement would make it much easier to extract useful information.

In general, although this has aged far better than the rest of the National Bureau of Standards's Applied Mathematics Series, it's in need of revision. The typesetting is sometimes hard on the eye and complicates finding useful formulae. The section end-notes could be more helpful were they not so out-of-date. I don't think it would be a good idea to get rid of the tables, but a CD-ROM of text files would be good for those who like to incorporate them into computer programs, and discussion of various numerical approximations of the respective special functions--a more subtle topic than it would seem--would also be of great help to the modern reader.

Despite being a bit dated, due to its comprehensive treatment of special functions common in applied mathematics, this is still a must-have desk reference for scientists and engineers. The Dover edition, currently the only one in print, is a bright green and somewhat flimsy paperback--any paperback of 1000 pages is flimsy. If you can find a copy I recommend buying a used hardcover instead. The content is the same, and although it may be a dull brick red and come with an old-library smell, it'll last much longer. If you can't find the hardcover at a good price, the paperback is cheap, and by buying it you'll save yourself midnight trips to the library or minutes to hours spent thumbing through Internet references, most of which don't nearly match old standards such as this in depth or ease-of-use.

Recommended: Yes