The NIST Handbook of Mathematical Functions is the product of a massive ten-year effort by the National Institute of Standards and Technology to update the original Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables from 1964 (commonly known simply as “Abramowitz and Stegun,” after the editors). The objective of the book is to offer a comprehensive reference for researchers and others in mathematics and adjacent disciplines. This print edition is a companion to the Digital Library of Mathematical Functions found on the NIST’s website at http://dlmf.nist.gov/.
The book is divided into thirty-six chapters, with most chapters discussing a function or class of functions. Each chapter has a different author, but great care has been taken to ensure uniformity in the structure and depth of the content. It is a very easy book to navigate, which in a book of this scope is no small point.
The first three chapters (Algebraic and Analytic Methods, Asymptotic Approximations, and Numerical Methods) are about establishing notation and methods used for defining and working with functions and not about the functions themselves. Each remaining chapter is structured as follows:
Notation: Notational conventions for the chapter are succinctly established.
Properties: This is where the main definitions and discussions of the functions occur. To get a sense of what you get, we’ll have a look at the coverage of the familiar trigonometric functions, found in Chapter 4, “Elementary Functions:”
A list of definitions and periodicity of the six standard trigonometric functions
Color plots of the functions
A picture and description of the sine function as a conformal mapping
Color surface plots of f(x+iy) for sine, tangent, cosecant, and their inverses, with notes on the simple transformations to get the others
Tables of elementary properties (positivity, interrelations, etc.)
A table of special values
Some special limits and inequalities
Maclaurin and Laurent Series for the functions and some related compositions
Derivatives, integrals, and differential equations
A huge list of identities, sensibly organized
Infinite products and continued fraction expansions
The Gudermannian Function
Extended discussion of the development of the inverse functions
References for even more extensive summaries
That’s all in about ten pages (and considering that there are about 800 pages of content, you can imagine just how much this book covers). The other elementary functions in the chapter include similar discussions of exponential, logarithmic, and hyperbolic functions. The other functions covered in the book are, of course, considerably more advanced.
Of course, different chapters will feature different material depending on what is appropriate for the topic. Standard inventories include basic definitions, identities, special values, limits and differential equations, and series representations. Each chapter also features richly printed and thoughtfully prepared color graphics.
Applications: This section is generally very brief and makes no attempt to be comprehensive. The applications are meant more as a short reminder of where the functions come from. In the case of trigonometry, for example, we get some standard formulas for Euclidean and spherical triangles and cubic equations.
Computation: This section features notes on the numerical computation of the chapter’s functions, with a web address for the associated link in the DLMF (described below) for any software.
References: Each chapter ends with a list of references, including notes on where and how they are used.
The book closes with an 80-page bibliography and an extensive index. It includes a CD with a digital copy in Adobe’s portable document format (PDF). This digital copy is fully searchable and internally hyperlinked.
It is worth keeping in mind that this is a reference text and is written as such; there is almost no narrative text. An incredible amount of content is squeezed into these pages, but it’s probably not a source for learning a new field.
It is impossible to meaningfully discuss the breadth of the content in this short review, but the table of contents tells much of the story. The vast majority of the functions included lie in the domain of mathematical analysis, but fields like combinatorics and number theory get their due (albeit with an analytic emphasis). Special functions are the stars of the show, but the organization keeps its purpose as general reference well in mind. Each chapter stands on its own to the extent that it does not depend on earlier chapters. For example, one does not need to be an expert in the hypergeometric function to look up its various special cases by name. For what it’s worth, this reviewer was able to find his way through everything he thought to check and noticed no fatal omissions. Anything the book cannot list explicitly at least gets a good reference.
This print edition is, however, the sidekick to the real hero of our story, NIST’s Digital Library of Mathematical Functions. This website freely distributes the same content as the text. Like the CD edition, the site is fully searchable and hyperlinked. One excellent feature is that the massive bibliography includes external links directly to the papers or their reviews. Many pictures and formulas may be downloaded in a variety of formats such as TeX, png, and pdf.
The website features some excellent links for software. Convenient tables will identify implementations in several open source and commercial software packages. There are also plenty of links to mathematical software repositories and/or directly to computer code (most linked programs are in FORTRAN, but more recent implementations may be in C++, MATLAB, or other languages).
Those familiar with the original book will find expanded coverage and a much more streamlined presentation. Most of the tables appear to have found their ways in tact into the new edition, although a lot of material has been reorganized into different sections and a handful of minor topics did not survive. The many lists of approximate values have been mercifully dropped in favor of links to numerical implementations. The deeply concerned can also find the older edition freely online, but there is little reason to go back.
The NIST has offered a great service to the scientific community by compiling this material efficiently and thoughtfully, and by making it accessible in a variety of ways. The Handbook has caught up to the times in its content, presentation, and dissemination online and will continue to be a standard for analysts, applied mathematicians, physicists, and other researchers until the next major update, presumably sometime in the 2050s.
Bill Wood lives in Arkansas and does a variety of mathematical things.