MAA Reviews

A Second Course in Complex Analysis

Publisher:

[Reviewed by Allen Stenger, on 04/24/2008]

This is a very careful treatment of several topics in complex analysis. It covers all the details in the proofs and doesn't do any hand-waving over the tricky parts, It's not in a hurry to get anywhere, and takes a number of digressions. This new Dover edition is a slightly-corrected version of the 1967 work published by W. A. Benjamin.

The first half of the book has a strong geometrical flavor, dealing with analytic continuation, the Riemann mapping theorem, the Koebe uniformization theorem, and the Koebe-Faber distortion theorem. It then makes a smooth transition to the world of special functions, with chapters on the elliptic modular function λ(z) (used to prove Picard's two theorems on exceptional values of entire functions and at essential singularities), the Hadamard product theorem (with a detailed study of the Gamma function), and the Riemann zeta function (including Ikehara's proof of the Prime Number Theorem).

Is this the book for you? That depends on whether you like the choice of topics and the careful and geometric treatment. The topics were already classical in 1967 (as Veech notes in the Preface), and you should not shy away from this book just because it is old, especially since it is bargain-priced. But there are numerous other sources for these results:

• Much of the material is also in Titchmarsh's classic Theory of Functions, although Titchmarsh takes a much less geometric approach. Titchmarsh gives a more direct proof of Picard's theorems that does not depend on the modular function.
• The Prime Number Theorem is proved in many number theory books, usually with Ikehara's method, although it is rarely treated in complex analysis books. An exception is Bak and Newman's Complex Analysis (Springer, 2nd edition, 1997) which gives the even simpler proof by D. J. Newman that had not been discovered when Veech's book was written.
• There are several specialized texts on modular functions, for example Apostol's Modular Functions and Dirichlet Series in Number Theory.
• The material on the mapping theorems and the distortion theorem is the hardest to find. There was a whole industry of this subject (called schlicht functions) in the early 1900s but it has disappeared today. There is a good treatment as a series of problems in Pólya and Szegö's Problems and Theorems in Analysis , Part IV, Chapter 2.

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.