An Atlas of Functions by Keith Oldham, Jan Myland, and Jerome Spanier is a heavy book, metaphorically and literally. It contains a wealth of information and weighs in at 5.1 lbs (2.3 kg). The book is primarily concerned with special functions. The title omits the word “special” for a good reason: some of the functions in the book are not “special” in the traditional sense.
There's no rigorous definition special functions, but the following definition is in line with the general consensus: functions that are commonly used in applications, have many nice properties, and are not typically available on a calculator. Obviously this is a sloppy definition, and yet it works fairly well in practice. Most people would agree, for example, that the gamma function is included in the list of special functions and that trigonometric functions are not.
The Atlas includes chapters for several functions that are not “special.” About 20% of the book is devoted to functions such as quadratic and cubic polynomials, absolute value, square roots, etc. These functions are commonly used in applications and have many nice properties, but they are classified as “elementary” rather than “special” because they can be computed on a typical calculator. However, the chapters on “elementary” functions are not as elementary as one might expect. The book examines even simple functions in unusual depth.
Every chapter is divided into the same fourteen sections:
- Special cases
- Particular values
- Numerical values
- Limits and approximations
- Operations of the calculus
- Complex argument
- Cognate functions
- Related topics
Most of the section titles are self-explanatory. However, three sections deserve further explanation. The sections on “operations of the calculus” include not only ordinary differentiation and integration but also fractional order derivatives and integrals. (Fractional calculus is used frequently throughout the book to establish connections between different families of functions.) In each chapter, “cognate” functions are functions related to the chapter topic but not listed under special cases or generalizations. Finally, the “related topics” given in the final sections of each chapter are quite broad. For example, some of the topics included in the final sections of chapters include cubic splines, linear regression, and Laplace transforms. Some chapters actually have 15 sections because there are two separate sections on related topics.
The Atlas devotes a considerable amount of space to exploring the connections between families of functions. Having the same internal structure for every chapter helps bring out these connections. Hypergeometric functions are mentioned frequently throughout the book because the theory of such functions clarifies relationships between groups of functions. While many books on special functions emphasize recurrence relations and differential equations as unifying principles, the Atlas places more emphasis on hypergeometric functions and operations for synthesizing one such function from another.
The most popular reference for special functions over the last 40 years has been Handbook of Mathematical Functions by Abramowitz and Stegun, affectionately known as A&S, and so it is natural to ask how the Atlas compares to this classic reference.
Technology has clearly changed since A&S was first published. Whereas A&S includes tables of function values and numerical algorithms, the Atlas comes with software for computing the functions described in the book. And while A&S is printed in black and white on matte paper; the Atlas is printed in color on glossy paper.
Atlas and A&S cover essentially the same list of special functions. (Occasionally the Atlas will use a different name than A&S for the same function. However, when a function is commonly known by multiple names, the Atlas lists the alternatives.) The Atlas goes into greater theoretical depth than A&S. While the Atlas is valuable as a reference, it also contains a large amount of expository text. A&S was written to be pulled off the shelf for quick reference. The Atlas seems to have been written for the reader who is not in such a hurry and who might enjoy looking around a bit.
John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs daily at The Endeavour