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Function Theory of One Complex Variable (Graduate Studies in Mathematics, 40) Hardcover – January 1, 2002

ISBN-13: 978-0821829059 ISBN-10: 082182905X Edition: 2nd

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Function Theory of One Complex Variable (Graduate Studies in Mathematics, 40) + Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) + Abstract Algebra, 3rd Edition
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Product Details

  • 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: 10.2 x 7.1 x 1.3 inches
  • Shipping Weight: 2.6 pounds
  • Average Customer Review: 2.9 out of 5 stars  See all reviews (7 customer reviews)
  • Amazon Best Sellers Rank: #4,512,860 in Books (See Top 100 in Books)

Editorial Reviews

Review

"I can say that I have read this book with great pleasure and I do recommend it for those who are interested in complex analysis." ---- Zentralblatt MATH --This text refers to an alternate Hardcover edition.

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.

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Customer Reviews

I found the present book to be rather disappointing in this respect.
Glitzer
The authors make too many assumptions of the readers, leave many proofs to exercises, and constantly say that missing steps are "obvious".
C. J. Fessler
This book is reasonably accessible to those who may not have had any previous exposure to complex analysis.
ikantspel

Most Helpful Customer Reviews

10 of 11 people found the following review helpful By Alexander C. Zorach on January 11, 2007
Format: Hardcover
This book is rather unorthodox in a number of respects, but it has become one of my favourite texts in complex analysis. The authors claim that their motivation for their presentation of the subject is to emphasize the interconnectedness of complex function theory with multivariable calculus, and de-emphasize the connection with topology. While I do not exactly agree with these goals, I think they do an excellent job of acheiving them. My only complaint about the book is that a few proofs in early chapters result in a sea of differential operators that is resolved by a plug-and-chug computation, something I'd always rather avoid.

The level of the book is elementary, especially for a graduate text, and I appreciate the authors for making honest and reasonable claims about the accessibility of their book. This book would probably even work well for someone who has not had a prior course in complex analysis, such as senior undergraduates. Some of the more advanced topics are presented in clearer ways in this book than I have seen elsewhere.

This book has a wealth of exercises, and the difficulty level is somewhat inconsistent. Some of the exercises are outright inane--possibly inappropriate for a graduate-level text, but useful for rote practice. Others are more interesting. I appreciate, however, the inclusion of more elementary exercises: many graduate texts have the problem of not including enough such exercises, which can make it hard for students to master the fundamentals. This book avoids this pitfall.

The best part about this book is the prose. This book is well-written and is a pleasure to read. Theorems and results are well-motivated, and necessary nuances are effectively communicated through the text.
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8 of 9 people found the following review helpful By Glitzer on May 20, 2011
Format: Hardcover Verified Purchase
I used this book as the text in a two semester graduate course on complex analysis that I taught recently. I found this to be a rather traditional introduction to a very classical core area of pure mathematics. (It is true that the authors originally define analyticity via the Cauchy-Riemann equations, but this is a very minor aspect of the book, and of course the connection with the existence of complex derivatives is quickly made.) Greene-Krantz move at a rather gentle pace, especially when compared to other, more classical texts (take Ahlfors, for example); this can be a major advantage, depending on your tastes.

However, if it was ever true that the devil is in the details, then this certainly applies to a mathematics book. I found the present book to be rather disappointing in this respect. I probably shouldn't have been surprised as Krantz has acquired some notoriety as a mass producer of math books. Few of the proofs can be called polished, and occasionally there are minor gaps (usually easy to fix, though) in the arguments. Cross references are often done awkwardly; sometimes, essentially the same argument is presented several times at different places without clarifying comment, setting the reader's head spinning unnecessarily.

The quality of the writing gets successively worse as we approach the end of the book. For example, I cannot shake off the suspicion that the treatment of the analytic continuation of Riemann's zeta function in Ch. 15 was hastily copied, with errors, from some other book; definitely, very little can be taken at face value here and the authors manage to completely obscure the main point behind the procedure used (which is essentially Riemann's original argument). Curiously, the following (final) Ch. 16 is independent of Ch.
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8 of 11 people found the following review helpful By ikantspel on May 2, 2007
Format: Hardcover
This book is reasonably accessible to those who may not have had any previous exposure to complex analysis. Many parts of the text are well written and easy to read and I really enjoyed the exposition on harmonic functions. With that being said, however, there are some things that I did not particularly like.

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.
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