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Function Theory of One Complex Variable (Graduate Studies in Mathematics, 40) 2nd Edition

3.1 out of 5 stars 8 customer reviews
ISBN-13: 978-0821829059
ISBN-10: 082182905X
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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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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: 1.2 x 7.5 x 10.5 inches
  • Shipping Weight: 2.6 pounds
  • Average Customer Review: 3.1 out of 5 stars  See all reviews (8 customer reviews)
  • Amazon Best Sellers Rank: #4,511,868 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

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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Format: Hardcover Verified Purchase
I recently used this book, as a student, for a graduate complex analysis course. It seems that it is becoming more popular, and ranks as one of the standard complex analysis texts, and it's not hard to see why: a casual perusal makes it seem like a sensible, gentle-yet-rigorous introductory text which, unlike some of its competitors, doesn't presume any but the most basic knowledge of topology, which some beginning graduate students are weak in. It starts out with the most elementary fundamentals of the subject, not even presuming the reader knows what a complex number is, and in the first few chapters the exposition seems very thorough. It also has some fairly modern material in the later chapters. At the start, I myself was very optimistic about it.

However, after reading a bit further I was disappointed to find it severely flawed. It seems that the beginning was written very carefully, but by the middle chapters it becomes unforgivably sloppy. It is riddled with errors of a kind that don't belong in the second edition of anything, and more importantly, the presentation is disorganized and uneven. Most of the proofs are inelegant and could be shortened considerably; many of them contain completely unnecessary statements that serve no apparent purpose, formal or intuitive, and it seems as though the authors simply didn't put much thought into cleaning them up. The prose similarly contains a lot of hemming and hawing and little material of use. The presence of elementary material alongside advanced topics is incongruous.
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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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