MAA Reviews

Complex Analysis

Joseph Bak and Donald J. Newman

Publisher: Springer (1999)
Details: 304 pages, Hardcover
Edition: 2 Series: Undergraduate Texts in Mathematics
Price: $79.95
ISBN: 9780387947563
Category: Textbook
Topics: Complex Analysis, Complex Variables

[Reviewed by Allen Stenger, on 10/15/2009]

This is a clever, concise, concrete, and classical complex analysis book, aimed at undergraduates with no background beyond single-variable calculus. The book has an eclectic flavor; rather than develop any general theories, the authors work toward a number of classical results, and usually take the shortest path to get there.

The book generally takes an analytic rather than a geometric approach; the Cauchy-Riemann equations are central. It works up to analytic functions by going through polynomials and entire functions, and only then considers functions analytic on a disk and then analytic on a region. The elementary functions are developed as extensions of those functions on the reals, rather than as power series. Similarly, the book starts with polygonal paths with only horizontal or vertical segments, and works up to general curves. There are no Riemann surfaces, and multi-valued functions such as the logarithm are sidestepped by explicitly defining a useful branch and showing that it has the desired properties.

The book has a modest number of applications, including some discussion of fluid flow and the Riemann mapping theorem. Most of the applications are to other branches of mathematics rather than to other sciences, and cover fields such as combinatorics and evaluation of definite integrals and infinite series. There is even has a complete proof of the prime number theorem.

There are many exercises, and they cover a wide range of difficulty, from routine applications of techniques in the text through quite challenging problems. Answers to all exercises are given in the back of the book, although usually they are sketches of the answer in a couple of sentences rather than a detailed answer.

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.