Complex Variables: (Pure and Applied Undergraduate Texts)

The text covers a broad spectrum between basic and advanced complex variables on the one hand and between theoretical and applied or computational material on the other hand. With careful selection of the emphasis put on the various sections, examples, and exercises, the book can be used in a one- or two-semester course for undergraduate mathematics majors, a one-semester course for engineering or physics majors, or a one-semester course for first-year mathematics graduate students. It has been tested in all three settings at the University of Utah. The exposition is clear, concise, and lively. There is a clean and modern approach to Cauchy's theorems and Taylor series expansions, with rigourous proofs but no long and tedious arguments. This is followed by the rich harvest of easy consequences of the existence of power series expansions. Through the central portion of the text, there is a careful and extensive treatment of residue theory and its application to computation of integrals, conformal mapping and its applications to applied problems, analytic continuation, and the proofs of the Picard theorems. Chapter 8 covers material on infinite products and zeroes of entire functions. This leads to the final chapter which is devoted to the Riemann zeta function, the Riemann Hypothesis, and a proof of the Prime Number Theorem.

Table of Contents:
Preface The complex numbers Analytic functions Power series expansions The general Cauchy theorems Residue theory Conformal mappings Analytic continuation and the Picard theorems Infinite products The gamma and zeta functions Bibliography Index

About the Author :
Joseph L. Taylor, University of Utah, Salt Lake City, UT

Review :
This textbook provides a profound introduction to the classical theory of functions of one complex variable . . . [T]he present text covers a remarkably broad spectrum between basic and advanced complex analysis, on the one hand, and between purely theoretical and concrete computational aspects on the other hand . . . Every section comes with a long series of related exercises, which differ widely with respect to both their abstraction and their difficulty . . . [T]hese exercises are utmost carefully selected and supplement the main text in a very instructive and efficient manner. Furthermore, the entire text is amply interspersed with illustrating examples, most of which appear as relevant working problems followed by detailed model solutions. All in all, the rich material is presented in a fashion that stands out by its laudable clarity, rigor, functionalism, versatility, and straightforward explanations. The book appears to be tailor-made for both students and instructors, and it certainly bespeaks the author's great teaching experience, expository expertise, enthusiasm for the subject, and sympathy for the needs of students." - Werner Kleinert, Zentralblatt MATH "[This] book presents topics in a logical way that allows the reader to build their intuition about the subject. [For] example, the elementary functions are defined in the first chapter. The first chapter then provides the reader with applications of the rectangular and polar coordinate forms of complex numbers and with examples to understand the properties of analytic and meromorphic functions developed in the subsequent chapters. The proofs of the major theorems follow the same logical and intuitive approach. The exercises are thoughtful and yet accessible to anyone with a sound understanding of multivariable calculus. This is an excellent book." - Peter Trombi, University of Utah