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

Robert E. Greene, Steven G. Krantz

Hardcover

ISBN 10: 082182905X ISBN 13: 9780821829059

Publisher: American Mathematical Society, 2002

Synopsis

Complex analysis is one of the most beautiful subjects that we learn as graduate students. Part of the joy comes from being able to arrive quickly at some "real theorems". The fundamental techniques of complex variables are also used to solve real problems in neighboring subjects, such as number theory or PDEs. 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. All the material usually treated in such a course is covered here, but following somewhat different principles. To begin with, the authors emphasize how this subject is a natural outgrowth of multivariable real analysis. Complex function theory has long been a flourishing independent field. However, an efficient path into the subject is to observe how its rudiments arise directly from familiar ideas in calculus. The authors pursue this point of view by comparing and contrasting complex analysis with its real variable counterpart. Explanations of certain topics in complex analysis can sometimes become complicated by the intermingling of the analysis and the topology. Here, the authors have collected the primary topological issues in a separate chapter, leaving the way open for a more direct and less ambiguous approach to the analytic material. 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. A large number of exercises are included. Some are simply drills to hone the students' skills, but many others are further developments of the ideas in the main text. The exercises are also used to explore the striking interconnectedness of the topics that constitute complex analysis.

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: 2002
• ISBN 10: 082182905X
• ISBN 13: 9780821829059
• Binding: Hardcover
• Language: English
• Edition number: 2
• Number of pages: 502