- Series: Graduate Studies in Mathematics
- Hardcover: 502 pages
- Publisher: American Mathematical Society; 2nd edition (January 1, 2002)
- Language: English
- ISBN-10: 082182905X
- ISBN-13: 978-0821829059
- Product Dimensions: 1.2 x 7.5 x 10.5 inches
- Shipping Weight: 2.6 pounds
- Average Customer Review: 1 customer review
- Amazon Best Sellers Rank: #5,910,586 in Books (See Top 100 in Books)
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Function Theory of One Complex Variable (Graduate Studies in Mathematics, 40) 2nd Edition
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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. --This text refers to an out of print or unavailable edition of this title.
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 --This text refers to an out of print or unavailable edition of this title.
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I thought it was strange that the author discusses the Cauchy integral formula for a disk, develops more aspects of the theory, and then later comes back to deal with homotopy theory and topology insofar as integration is concerned. In this aspect, Conway's treatise on the subject is superior, in my opinion.
I also prefer Conway's proof of Mittag-Leffler's theorem which is eloquent and a good application of Runge's Theorem. Additionally, I prefer Conway's proof of the Picard theorems as well (Conway uses Montel-Caratheodory which in and of itself is interesting while Greene and Krantz use the modular function and there are a few choice spots when Krantz is a bit vague).
Finally, some of the proofs and exercises contain errors (most of them minor, some of them not so minor) and a few of the proofs are quite difficult to follow at times while Conway's book seems more readable in these areas. This comment mainly applies to the 2nd edition and it is quite conceivable that the author has remedied these errors in the 3rd edition.
Overall, this book has some value. I believe that this book, coupled with Conway's book is a good combination. There are many things that Greene and Krantz do that I prefer over Conway and vice versa although if I had to compare the two, I would prefer Conway.