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Complex Analysis

Eberhard Freitag and Rolf Busam
Springer Verlag
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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At first glance, this doesn't seem like a huge book, and the table of contents supplied by Springer doesn't really change this impression. So let's start by clarifying that. Though it does not seem much larger in the hand than your average "Universitext", this book actually has almost 550 pages. (It must be something about the paper.) It is, in fact, a massive introduction to complex analysis, covering a very wide range of topics.

The book is a translation of Funktionentheorie I, based on the fourth edition (Springer, 2005). (There is no indication of what might be in the second volume, nor any hint that a translation of that might be in the works.) The first 175 pages contain an account of the basic ideas of complex analysis: the complex plane, the Cauchy-Riemann equations, line integrals, the Cauchy Integral Theorem and Formula, uniform approximation results, analytic functions as power series, mapping properties, singularities, Laurent series, residues. This is the material that I like to cover in an undergraduate course.

The rest of the book, however, is less standard… and for that very reason much more interesting. Chapter IV considers ways of constructing nontrivial examples of analytic functions. It begins with the Gamma function, then considers Weierstrass product representations, the Mittag-Leffler representation using partial fractions, and the Riemann Mapping Theorem. Then come chapters on Elliptic Functions, Elliptic Modular Forms (including some material on theta functions), and applications to Analytic Number Theory. As a number theorist who has done work on modular forms and related subjects, this, of course, warms my heart. It also seems, to my eyes, to offer a good example of the power of the material just developed. Spending 300 pages on elliptic functions, modular forms, and analytic number theory may, however, strike some readers as a little too much in an introductory book.

The exposition is terse, but clear. Theorems and proofs are clearly delimited, which many students find helpful. There is little hand-holding, however, so the book may be difficult for all but the best undergraduates. There are problems at the end of each section, and sketches of solutions are given in chapter VIII. These are the sort of answer one might get from a professor if one asks how to do a problem. One solution, for example, just says "See Proposition I.1.7"; another says "Start with the double sum... [long formula]... and split it into 4 double sums, which can then be written as products of simpler sums." Thus, the fact that solutions are included in the book need not mean that the book's problems cannot be assigned for credit.

One gripe: the book follows the German tradition of using small caps whenever it refers to a mathematician or author. I find this very distracting.

Overall, this is quite an attractive book. It makes high demands on the reader's "mathematical maturity", but it does deliver the goods.

Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME.

Differential Calculus in the Complex Plane C

Integral Calculus in the Complex Plane

Sequences and Series of Analytic Functions, the Residue Theorem

Construction of Analytic Functions

Elliptic Functions

Elliptic Modular Forms

Analytic Number Theory

Solutions to the Exercises