This book attempts to present the theory of functions of a complex variable to readers in engineering and the sciences — with mixed results. One success is that prerequisites have been minimized. The only course prerequisite mentioned is “calculus” though it is unclear how much calculus is intended. Contained within this text are many bits of the calculus of two real independent variables along with other bits of real analysis and point set topology. Another success is clear writing, and another is a large quantity of exercises. These exercises range from simple computations to proofs, and solutions to many are provided — often more than half the non-proof exercises.
What may not be as successful is the authors’ attempts to interest their intended audience. Most of the book is written in definition-theorem-proof style, which will require a leap in maturity for many readers. The authors contribute many interesting asides, but these are usually about matters of interest to mathematicians rather than scientists or engineers. Perhaps the biggest stumbling block for that audience is that there is no hint any of the mathematics might be useful until page 109. There begins a 50-page chapter on applications, almost entirely to vector fields in two dimensions. That is well done, but one has to wonder how impressed students will be. If they took the entire calculus sequence, or two years of physics, they have probably already seen all of this in three dimensions, which is where their interests lie.
At the end of the applications chapter, applications take another break until page 459, where there is a single section continuing the earlier full chapter with additional material on the same applications.
One way to characterize a book under review is to try to describe the reader for whom the book might be a best fit. For this book, that might be a motivated reader with minimal prerequisites who wants to learn functions of a complex variable on their own. Such a reader would benefit from the clear prose and the exercises that start at a low level and are usually provided with answers. The reader’s motivation for studying the subject might have to come from outside this text. Alternatively, the fact that applications are separated here from the main development means that these self-contained sections could supplement another text, or make interesting reading for a mathematician who already knows the theory.
After a few years in industry, Robert W. Hayden (firstname.lastname@example.org) taught mathematics at colleges and universities for 32 years and statistics for 20 years. In 2005 he retired from full-time classroom work. He now teaches statistics online at statistics.com and does summer workshops for high school teachers of Advanced Placement Statistics. He contributed the chapter on evaluating introductory statistics textbooks to the MAA's Teaching Statistics.