This is a book about complex variables that gives the reader a quick and accessible introduction to the key topics. While the coverage is not comprehensive, it certainly gives the reader a solid grounding in this fundamental area. There are plenty of figures and examples to illustrate the principal ideas, and the exposition is lively and inviting. An undergraduate wanting to have a first look at this subject, or a graduate student preparing for the qualifying exams, will both find this book to be a useful resource.
In addition to important ideas from the Cauchy theory, the book also includes the Riemann mapping theorem, harmonic functions, the argument principle, general conformal mapping, and dozens of other central topics. The book is brief but not terse.
Readers will find this book to be a useful companion to more exhaustive texts in the field. We believe that it will be a resource for mathematicians and non-mathematicians alike.
Table of Contents
1. The Complex Plane
2. Complex Line Integrals
3. Applications of the Cauchy Theory
4. Isolated Singularities and Laurent
5. The Argument Principle
6. The Geometric Theory of Holomorphic
7. Harmonic Functions
8. Infinite Series and Products
9. Analytic Continuation
About the Author
Steven G. Krantz was born in San Francisco, California and grew up in Redwood City, California. He received his undergraduate degree from the University of California at Santa Cruz and the PhD from Princeton University. Krantz has held faculty positions at UCLA, Princeton University, Penn State University, and Washington University in St. Louis. He is currently Deputy Director of the American Institute of Mathematics.
Krantz has written 160 scholarly papers and over 50 books. At least five of the latter are about aspects of complex analysis. Krantz is the holder of the Chauvenet Prize and the Beckenbach Book Award, both awarded by the Mathematical Association of America. He won the UCLA Alumni Association Distinguished Teaching Award. He is the author of How to Teach Mathematics. He has directed 16 PhD students. Krantz serves on the editorial boards of six journals and is Editor-in-Chief of two.
This text is an entry in the MAA Guides series, each volume of which is intended to provide a short, concentrated summary of a primary mathematical subject. This volume on real analysis focuses on the material in a standard first graduate course. It is ideally suited to serve as a quick look for someone new to the subject, for review, or as preparation for qualifying exams. It is concise, very clearly written, and full of little nuggets of insight. Continued...