Complex Variables

by Robert B. Ash

Complex Variables deals with complex variables and covers topics ranging from Cauchy's theorem to entire functions, families of analytic functions, and the prime number theorem. Major applications of the basic principles, such as residue theory, the Poisson integral, and analytic continuation are given. Comprised of seven chapters, this book begins with an introduction to the basic definitions and concepts in complex variables such as the extended plane, analytic and elementary functions, and Cauchy-Riemann equations. The first chapter defines the integral of a complex function on a path in the complex plane and develops the machinery to prove an elementary version of Cauchy's theorem. Some applications, including the basic properties of power series, are then presented. Subsequent chapters focus on the general Cauchy theorem and its applications; entire functions; families of analytic functions; and the prime number theorem. The geometric intuition underlying the concept of winding number is emphasized. The linear space viewpoint is also discussed, along with analytic number theory, residue theory, and the Poisson integral. This book is intended primarily for students who are just beginning their professional training in mathematics.

Genres: Mathematics

255 pages, Kindle Edition

First published January 1, 1971

About the author

Professor Emeritus, Mathematics
University of Illinois at Urbana-Champaign

Reviews

[John Hammond · Rating 4 out of 5]

This is a good reference book. The dover edition is cheap, and the text is entirely available online for free.
That said, we used it for an introduction, and it wasn't great for that. I recommend Asmar's text for a solid intro.

[Liora Sophie · Rating 5 out of 5]

readable, precise, organized. There are also solutions to all the exercises on their website. This was an excellent guide to the course. Highly recommended.