This book, whose first edition was written by Ralph P. Boas and published by Random House in 1987, reveals both the power of complex analysis as a tool for applications and the intrinsic beauty of the subject as a fundamental part of pure mathematics. This revised edition by the author's son, Harold P. Boas, himself an award-winning mathematical expositor, is an ideal choice for a first course in complex analysis, as a classroom text, and for independent study.

Distilling the subject into a lucid, engaging, rigorous account of the subject, the authors go beyond the standard material of power series, Cauchy's theorem, residues, conformal mapping, and harmonic functions. Included are accessible discussions of less well-known but intriguing topics ranging from Landau's notation and overconvergent series to the Phragmén-Lindelöf theorems.

Nearly 70 exercises, with detailed solutions, serve as models for students, and supplementary exercises provide even more material for the classroom.

Highlights:

- Exercises interspersed in the text, with detailed solutions, allow students to test their understanding of the material.
- Text covers the standard material on complex analysis while also discussing intriguing additional topics.
- Topics are discussed in commonly encountered terms, rather than generally and abstractly.

### Table of Contents

Preface to the Second Edition

Preface to the First Edition

To the Student

1. From Complex Numbers to Cauchy's Theorem

2. Applications of Cauchy's Theorem

3. Analytic Continuation

4. Harmonic Functions and Conformal Mapping

5. Miscellaneous Topics

Solutions of Exercises

References

Index

### Excerpt: Overconvergence (p. 126)

When a function is defined by a power series with a finite radius of convergence, one can sometimes extend the function beyond the disk of convergence by grouping the terms of the series, that is, by considering only a subsequence of the partial terms. A power series with this property is said to be **overconvergent**. We are going to construct an example of an overconvergent power series; actually it is more convenient to obtain the power series by constructing the grouped series first and then removing the parentheses.

The key to the construction is a simple lemma. . . . I present it as an exercise.

### About the Authors

**Ralph P. Boas** (1912-1992) served as MAA President (1973-1974); edited the *American Mathematical Monthly* (1976-81); received the Lester R. Ford Award for expository excellence in 1978 for his article "Partial Sums of Infinite Series, and How They Grow;" and garnered the MAA's Award for Distinguished Service to Mathematics in 1981.

**Harold P. Boas** (Texas A & M University) won the Lester R. Ford Award in 2007 for his article, "Reflections on the Arbelos," and the MAA's Chauvenet Prize in 2009 for the same article. He revised his father's *Primer of Real Functions* (4th edition, 1996).