This is a relentlessly classical look at complex analysis (or “Theory of Functions,” to use the classical term). One measure of how classical it is might be the age of items in the bibliography: 37 of the 76 works cited date from the 1800s, and 13 are older than that. Maybe this classicism accounts for its appearance in the Modern Birkhäuser Classics series despite its being only 11 years old (it is an unaltered reprint of the 1998 edition).
The selection of topics is unconventional for a modern introduction, with a lot about special functions and a quarter of the book devoted to elliptic functions. An extended example running through the book shows that the error term in the prime number theorem sometimes takes small values (this is a weak form of Skewes’s result that the error term changes sign infinitely often). The book contains a lot of clever proofs, particularly in the first chapter where it does an amazing amount without the mechanisms of complex analysis.
But I think the book’s self-description as a book of fundamentals is misleading — the author has really tried to write an accessible introduction to special functions, and to do it better than Whittaker & Watson’s A Course of Modern Analysis. The announced method is to reverse the order of presentation in Whittaker & Watson, putting the examples and applications first and the foundations and theorems last. I admire the sentiment, but I think this book is not successful, in a couple of senses.
First, the book’s solution to the perceived problem is to put a lot of material about the special functions first. But from a motivational viewpoint this merely replaces the question “What is complex analysis good for?” with another question, “What are the special functions good for?”, and the book does not answer this second question. The applications are limited to number theory (distribution of prime numbers and quadratic and biquadratic reciprocity). Although number theory (especially the prime number theorem) has been an important driver in the development of complex analysis, the field of applications of complex analysis is much wider, and this is not even hinted at in this book.
Second, the situation is not as dire as the author tries to make it seem when he writes in the Preface, “All modern introductions to complex analysis follow, more or less explicitly, the pattern laid down by Whittaker and Watson.” Whittaker & Watson is really two books in one, the first book developing complex analysis and the second book developing the special functions. Whittaker & Watson’s first book does contain numerous examples, although it is true that all material on the special functions is delayed until the second book. Most modern books weave the examples and applications into the narrative, and one not-so-modern book, Titchmarsh’s Theory of Functions, does in fact introduce the gamma function and the Riemann zeta function in the first chapter, before there is any apparatus of complex integration to deal with them thoroughly.
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.