Function Theory of One Complex Variable: Third Edition (Graduate Studies in Mathematics, 40) - Hardcover

Robert E. Greene And Steven G. Krantz

Hardcover

ISBN 10: 0821839624 ISBN 13: 9780821839621

Publisher: American Mathematical Society, 2006

Synopsis

Complex analysis is one of the most central subjects in mathematics. It is compelling and rich in its own right, but it is also remarkably useful in a wide variety of other mathematical subjects, both pure and applied. This book is different from others in that it treats complex variables as a direct development from multivariable real calculus. As each new idea is introduced, it is related to the corresponding idea from real analysis and calculus. The text is rich with examples and exercises that illustrate this point. The authors have systematically separated the analysis from the topology, as can be seen in their proof of the Cauchy theorem. The book concludes with several chapters on special topics, including full treatments of special functions, the prime number theorem, and the Bergman kernel. The authors also treat $H^p$ spaces and Painlevé's theorem on smoothness to the boundary for conformal maps. This book is a text for a first-year graduate course in complex analysis. It is an engaging and modern introduction to the subject, reflecting the authors' expertise both as mathematicians and as expositors.

About this title

From the Publisher

Rather than using the traditional approach of presenting complex analysis as a self-contained subject, the authors demonstrate how it can be connected with calculus, algebra, geometry, topology, and other parts of analysis. They emphasize how complex analysis is a natural outgrowth of multivariable real calculus by comparing and contrasting complex variable theory with real variable theory. The text relates the subject matter to concepts that students already know and motivates these ideas with numerous examples. Special topics in later chapters deal with current research including the Bergman kernel function, Hp spaces, and the Bell-Ligocka approach to proving smoothness to the boundary of biholomorphic mappings. Features many examples as well as 75 illustrations, which is provided through exercise sets.

From the Back Cover

A new approach that treats complex analysis in a broad context

This book presents a new approach to one of mathematics' oldest fields. It departs from the tradition of teaching complex analysis as a self-contained subject and, instead, treats the subject as a natural development from calculus. It also shows how complex analysis is used in other areas, exploring connections with calculus, algebra, geometry, topology, and other parts of analysis.

The authors provide the ideal framework for a first-year graduate course in complex analysis--building upon ideas the student is already familiar with and simplifying the transition to advanced topics. The book is also for those using complex numbers and functions in applied fields, including engineering, physics, and other areas.

Function Theory of One Complex Variable Compares and contrasts complex variable theory with real variable theory Clarifies analytical ideas belonging to complex analysis by separating them from topological issues Discusses some of the current research in the field, including a number of interesting topics not discussed in other textbooks Features many examples as well as 75 illustrations Provides especially thorough exercise sets

Edition details

• Publisher: American Mathematical Society
• Publication date: 2006
• ISBN 10: 0821839624
• ISBN 13: 9780821839621
• Binding: Hardcover
• Language: English
• Edition number: 3
• Number of pages: 504