Never before have I began reading a book more predisposed to hate it. Generally, I like to read math books that are slim because I feel that it forces the author to get right to the heart of the material as quickly as possible. I also like my math books to have a rigid structure of formal proofs surrounded by expositional paragraphs. This book, on the other hand, sits at an intimidating 478 pages and has no proofs that are set aside in the proof environment in LaTeX. The proofs are blended together with the general commentary paragraphs in the flow of this book's exposition. So it was a huge surprise for me when I actually found that I enjoyed reading this book.
The first thing that I think should be noted is that this book is written in an informal language. I know many reviewers have stated that this bothers them and have hinted at the fact that they think some of the proofs here are less-than-rigorous (perhaps implying wrong). I don't think this is the case at all and there's a good reason for this. This book is about conveying the essence of a proof to the reader much more than the gritty technical details (which there is little of in basic complex analysis in the first place). This is a good thing. Whatever can be said about the books by Rudin or some others, you cannot possibly say that a newcomer to the subject would walk away with a good intuition for the subject on a first go. Here is a book entirely devoted to teaching the reader how to think about the material so that they will see the results as being natural, and the book does a marvelous job at it. Now, with this in mind, I thought that all of the proofs were very rigorously stated, and the fact that he used English instead of mathematical symbols for everything greatly enhanced the readability of the proofs. The only proof that was less-than-rigorous in my eyes was the proof of Green's Theorem, but Gamelin comes right out and states that it won't be rigorous, and I don't think you can bash a complex analysis book for not proving a multivariable result. Another thing worth noting is that this book takes a more geometric approach to the material in that it focuses on how FLTs behave and how regions are deformed/not deformed by certain types of mappings. This is the style that seems most popular for introductory texts and was seemingly popularized by Ahlfors's text.
Another thing that I think should be mentioned is that by the time a student can reasonably begin looking for books on complex analysis, they should have the ability and mathematical maturity to take a description or an explanation and digest it and turn it into formal mathematics need be. When you converse with your friends or colleagues about math, you don't speak in epsilons and deltas, but rather in general terms which cut to the heart of the argument. Gamelin is doing this, but in a much more structured format than what you would expect, and so I think this style is beneficial. For example, when I came to the proof of the Reside Theorem, I instantly knew what he was going to do for the proof before I read it. I had an intuition for the material that made the results appear natural, and for a first book to do this is quite impressive.
So what about the problems. As staying in the first book on the subject category, this is a book which has a wide variety of problems and difficulty levels. If you attempt most of them (which you should with this book) then I think you would walk away with a very good understanding of how to use the results in both simple and complicated scenarios. However, you must do most of the harder problems. It would be very easy to simply do the easy problems and then be lured into a false sense of security about your depth of understanding.
Another thing that needs to be mentioned is that a first book in complex analysis should teach computational techniques as well as theoretical ones. Complex analysis was made, in part, to compute definite and indefinite integrals. So having a book that does not teach you how to use the residue theorem to compute an integral is doing you a disservice. This is another reason why books like Rudin's Real and Complex Analysis are not good first choices for textbooks. Here is a book that teaches the student how to do computations when they are needed, and gives plenty of clear examples and practice problems so that the student can become proficient.
The breadth of information that this book covers is also impressive. This book covers all of the standard material (with a pinch of not-so-standard material) in the first eleven chapters (out of sixteen). I took a graduate course at UMich which covered this material exactly and it was a very solid course. After this material, any student could go on to further topics in other books and would be completely comfortable recognizing and implementing complex analytic techniques in more advanced analysis books. I have not read the last five chapters (which cover special functions, approximation theorems, Riemann surfaces, and solutions to the Dirichlet problem), but they appear to carry the same style as the rest of the book, which in my mind means they should be good as well.
All in all, I think this book is as good of a book as any other for a first exposure to the material. Obviously more advanced books will be needed by graduate students later in their studies, but I would not look past this book as a first course (especially when Ahlfors is almost two-hundred dollars now).
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Equations format wrong
I wouldn't post this review except that I cannot for the life of me figure out how to report a problem so hopefully someone will contact me with help. The formatting for this textbook is all messed up causing the equations to display over the text.
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Abhishek Roy
Product is in good condition
Reviewed in India on April 14, 2021
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Apollionus
Well regarded text on Complex Analysis by Theodore Gamelin
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Contents are great, but so so printing quality
Reviewed in Canada on October 6, 2013