Graduate Texts in Mathematics

Complex Analysis

In the Spirit of Lipman Bers

Authors: Gilman, Jane P., Kra, Irwin, Rodriguez, Rubi

  • Comprehensive work that covers the vast majority of the material needed for a beginning graduate level course on complex analysis
  • Wonderfully elegant and economical treatment of complex analysis
  • Provides a real variety of alternative ways of understanding the concept of analyticity
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eBook $54.99
price for USA (gross)
  • ISBN 978-0-387-74715-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
About this Textbook

This book is intended for a graduate course on complex analysis, also known as function theory. The main focus is the theory of complex-valued functions of a single complex variable. This theory is a prerequisite for the study of many current and rapidly developing areas of mathematics including the theory of several and infinitely many complex variables, the theory of groups, hyperbolic geometry and three-manifolds, and number theory. Complex analysis has connections and applications to many other subjects in mathematics and to other sciences. It is an area where the classic and the modern techniques meet and benefit from each other. This material should be part of the education of every practicing mathematician, and it will also be of interest to computer scientists, physicists, and engineers.

The first part of the book is a study of the many equivalent ways of understanding the concept of analyticity. The many ways of formulating the concept of an analytic function are summarized in what is termed the Fundamental Theorem for functions of a complex variable. The organization of these conditions into a single unifying theorem with an emphasis on clarity and elegance is a hallmark of Lipman Bers's mathematical style. Here it provides a conceptual framework for results that are highly technical and often computational. The framework comes from an insight that, once articulated, will drive the subsequent mathematics and lead to new results.

 

In the second part, the text proceeds to a leisurely exploration of interesting ramifications of the main concepts.

 

The book covers most, if not all, of the material contained in Bers’s courses on first year complex analysis. In addition, topics of current interest such as zeros of holomorphic functions and the connection between hyperbolic geometry and complex analysis are explored.

Reviews

From the reviews:

"This is a fairly conventional text for a first course in complex analysis. It is an interesting mix of the concrete and the abstract, and of the formulaic and the geometric. It has good exercises … . It is nominally a graduate text (it is in Springer’s series of Graduate Texts in Mathematics) … . The book covers all the usual topics for a first course and includes a lot of advanced topics … ." (Allen Stenger, MathDL, February, 2008)

"This book is based on the original courses of Complex Analysis that was delivered by the well-known American specialist and lecturer in Analysis, Professor Lipman Bers. … the book is written in a clear and easily readable manner. … this book is useful for all specialists and lecturers in Complex Analysis … and also all specialists who deal with applications of Complex Analysis. I also recommend this book to beginners who want to study Complex Analysis … ." (Peter Zabreiko, Zentralblatt MATH, Vol. 1139 (17), 2008)

“An introduction to complex analysis based on original lectures by Lipman Bers … . the whole text is clearly suitable as a first course in complex analysis. … In addition, it contains many exercises and a brief guide to the literature.” (G. Teschl, Monatshefte für Mathematik, Vol. 156 (4), April, 2009)


Table of contents (10 chapters)

  • The Fundamental Theorem in Complex Function Theory

    Gilman, Jane P. (et al.)

    Pages 1-5

  • Foundations

    Gilman, Jane P. (et al.)

    Pages 7-21

  • Power Series

    Gilman, Jane P. (et al.)

    Pages 23-57

  • The Cauchy Theory–A Fundamental Theorem

    Gilman, Jane P. (et al.)

    Pages 59-82

  • The Cauchy Theory–Key Consequences

    Gilman, Jane P. (et al.)

    Pages 83-99

Buy this book

eBook $54.99
price for USA (gross)
  • ISBN 978-0-387-74715-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
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Bibliographic Information

Bibliographic Information
Book Title
Complex Analysis
Book Subtitle
In the Spirit of Lipman Bers
Authors
Series Title
Graduate Texts in Mathematics
Series Volume
245
Copyright
2007
Publisher
Springer-Verlag New York
Copyright Holder
Springer-Verlag New York
eBook ISBN
978-0-387-74715-6
DOI
10.1007/978-0-387-74715-6
Series ISSN
0072-5285
Edition Number
1
Number of Pages
XIV, 220
Number of Illustrations and Tables
20 b/w illustrations
Topics