The appropriate audience for this book is frankly and accurately described in the preface: "The book is designed for those who, although they have to use mathematics, are not mathematicians themselves, and may not even have any great mathematical aptitude or ability." Over a dozen possible applications, mostly to physics, are mentioned in the preface, but none are worked out in the book; thus, though it is conceivable that a mathematician would find some use for it, this is not the place to go to see how special functions arise in applied problems. It could be used as a textbook in an undergraduate course on special functions, though I would not use it if I were to teach such a course. I can recommend it as an inexpensive supplementary text for an undergraduate class on mathematical methods of physics, and it may also find an audience among advanced undergraduate and beginning graduate students in physics who want to know a little more about the special functions they are seeing than their physics books can tell them. I might have found it useful when I was an undergraduate taking quantum mechanics, but even then I would have wished it were less dull.
The book requires only a good working knowledge of calculus. Complex numbers occur in a few places in chapters 3, 4, 7 and 10, but only at the level of exp ix=cos x + i sin x; complex analysis as such is never used. The first chapter is on series solutions of second order differential equations, and it lays the foundation for some of what follows. The blurb on the back cover goes farther than the author would himself: it claims that it is most natural to introduce each of the classical special functions through a(n ordinary) differential equation, and that this is what distinguishes the author's approach. Chapter 2 is on the gamma and beta functions, and of course the author is not foolish enough to use differential equations here. Even in chapters 7-9, where it would not be unreasonable to begin with a differential equation, the author chooses not to do so — though the differential equations viewpoint still has had an adverse effect on chapter 7.
Chapter 3 is on Legendre polynomials and functions, and chapter 4 on Bessel functions. These are the two longest chapters by far; together they make up about half the book. Apparently the author much prefers ordinary differential equations to partial. The only PDE in the book, except for a trivial one at the bottom of p. 158, is in section 3.11, on spherical harmonics. The preface points out that the ODEs studied in the book often come from separating variables in PDEs, but only in section 3.11 do we actually see this technique.
Hermite polynomials come up for consideration in chapter 5. The author fails to mention that a different normalization is frequently used that makes them monic. This is a surprising omission in that the most spectacular application of Hermite polynomials in physics is to the quantum mechanical harmonic oscillator, and these monic polynomials are often used there, for example in Weyl's The Theory of Groups and Quantum Mechanics. These beautiful polynomials are essentially the only ones that are both orthogonal and have the Appell property — the derivative of the n-th one is n times the (n-1)-st one — and they are the matching polynomials of complete graphs (see Godsil's Algebraic Combinatorics). Chapter 6 is on Laguerre polynomials. It is half again as long as chapter 5, and here the author does mention that a different normalization is sometimes used.
Chapter 7 is very peculiar in that the author's Chebyshev polynomials of the second kind are not polynomials. He defines Un(x) by Un(cos t) = sin nt, rather than the standard Un(cos t)=[sin((n+1)t)] /[sin t]. Presumably he does this so that Un(x) will be the other solution of his differential equation for the polynomials of the first kind, but it is highly unusual. The last three chapters contain cursory treatments of Jacobi polynomials, hypergeometric functions, the Riemann zeta function and some others. If you only looked at chapters 8 and 10, you would think this book was much worse than it is.
The author's proofs are often not the best possible, even given the constraint of real variables. In Example 1 on p. 164 he takes 2/3 of a page to prove something that is immediate on iterating part (i) of Theorem 5.6. The proof of Theorem 4.6 could be finished in one line once it has been proved for positive integers, but the author goes on for another 3/4 of a page. A shorter proof of Theorem 2.10 could be given using Theorem 2.5 and the double angle formula for the sine. The book's strongest point is that, although some derivations are omitted and others could be done better, the calculations it does are clearly and carefully explained.
The book was originally published in 1968, and a lot has happened in special functions since then. The WZ method (see A=B, by Petkovsek, Wilf and Zeilberger) now furnishes an algorithmic approach to many special functions identities, and the connections with combinatorics and with group representation theory are much stronger now than forty years ago. A far better book that discusses these developments, though still heavily concentrated on the classical theory, is Special Functions by Andrews, Askey and Roy. This is also the best source for the history of the subject; there are no historical remarks of any kind in the book under review. However, Andrews-Askey-Roy is less appropriate than this book for readers who are not so confident of their mathematical ability.
Warren Johnson (firstname.lastname@example.org) is visiting assistant professor of mathematics at Connecticut College.