5.0 out of 5 stars
A true classic of classics indeed..., Oct. 9 2002
I decided to purchase this title about three months ago after hearing lots of praise about it on the internet and wanting to learn the subject, and I can now see that this praise was not exaggerated. A hundred years after its first publication, this classic still remains the definitive general reference in the field of special functions and is a very solid textbook in its own right.
The book is split into two main parts: the first consists of short (but detailed) overviews of the various sub-disciplines of analysis from which results are required to develop later results, and the second part is devoted to developing the theories of the various kinds of special functions. The sheer breadth of topics and material that this book covers is utterly incredible. The major topics covered in the first part of the book are convergence theorems, integration-related theories, series expansions of functions and differential/integral equation theories, each of which are split into two or three chapters. The reader is assumed to be familiar with some of the subjects here and these chapters are intended more as a review, but they are still quite self-contained and will also appeal to those who have not encountered the subjects yet. (I am only 16 and know no more than ODEs and a little real analysis, but I learned some material from this)
The second section, which is really the heart of the book, starts off with a detailed treatment of the fundamental gamma and related functions, followed by a chapter on the famous zeta function and its unusual properties. The book then covers the hypergeometric functions - the focus is on the 1F1 and 2F1 types, being ODE solutions - which are perhaps the cornerstone of this field, followed the special cases of Bessel and Legendre functions. There are a number of ways of developing and teaching the ideas regarding these functions; this book mainly uses the differential equation approach, starting by defining these functions as solutions to ODEs and going from there. There is also a chapter on physics applications (using these functions to solve physics equations), which is sure to please the more applied math readers. The next three chapters are devoted to elliptic functions, covering the theta, Jacobi and Weierstrass types. (one chapter on each) The two remaining chapters are on Mathieu functions and ellipsoidal harmonic functions. Along the way, some additional functions are also sometimes mentioned in the problem sets. (barnes G, appell, and a few others) About the only room for improvement here would be some analyses of named integrals (EI, fresnel, etc.) and inverse functions (lambert W log, inverse elliptics, etc.), and perhaps more on multivariable hypergeometrics, but these things are not a big deal considering how much else appears in here, and I have not really seen any book out there that covers these anyway.
Each chapter has several subsections, usually one on each major theorem or property of the function in question, and these consist of the main discussion and proof, a few corollaries, and a couple of exercises that illustrate the usage of the theorem. At the end of the chapter, some more sets of problems are given; these mostly consist of proving identities and formulas involving the functions, so answers are not needed, but it would be nice if there was a showed-work solutions book available for students. The problems themselves are very well designed and some really require the use of novel methods of proof to obtain the result. The language is a bit in the older style with some unconventional spelling and usage, but it does not detract from the subject material at all (actually, I personally liked this form of writing), and the price is about right.
The only real complaint I have with this book has nothing to do with its content; it is the printing quality. The text font is simply too small in a number of places and also sometimes looks "washed out;" while it is still readable, such a classic gem as this definitely deserves a better effort on the publisher's part. (one of CUP's other works on the same subject, Special Functions by Andrews et al, has much better printing, although is not as good as this in other respects)
For those interested in the field of special functions and looking for something to start off with, A Course of Modern Analysis would be, hands down, my first recommendation. You cannot really do much better than this.