Customer Reviews

Complex Analysis

5 star:		(7)
4 star:		(4)
3 star:		(1)
2 star:		(1)
1 star:		(1)

 

 

Gamelin's book covers an interesting and wide range of topics in a somewhat unorthodox manner. Examples: Riemann surfaces are introduced in the first chapter, whereas winding numbers don't make an appearance until halfway into the book. Cauchy's theorem and its kin are instead developed in the context of piecewise-smooth boundaries of domains (in particular, simple closed curves) and only later generalized to arbitrary closed paths, almost as an afterthought.

In general, the author successfully conveys the spirit of the subject, and manages to do so quite efficiently. It's not the most painstakingly rigorous text out there, and the reader is expected to fill in some of the details himself, but the payoff is that a lot of ground is covered without getting bogged down in technicalities. In many books on this subject it can be tough to see the forest for the trees. This one is a pleasant exception.

There are a lot of good complex analysis books out there: Conway, Ahlfors, Remmert, Palka, Narasimhan, the second half of big Rudin, and of course Needham's "Visual Complex Analysis." (And many others that are well-regarded but that I have not looked at, such as Lang and Jones/Singerman, as well as the old classics by Hille, Knopp, Cartan, Saks and Zygmund.) Every one of these has its own perspective, and complex analysis is a big, multifaceted subject that is perhaps best studied from multiple points of view. Anyone wanting to learn this subject well will benefit from having several books at hand.

Gamelin's contribution to the pantheon is not revolutionary, but it does collect between its pages a wide assortment of topics not generally found in a single text. The reader is whisked from the basics to the Riemann mapping theorem in 300 pages with surprising ease. The ensuing "topics" chapters include a dynamical systems-flavored section on Julia sets and fractals; special functions (gamma, zeta, etc.); the prime number theorem; and an introduction to abstract Riemann surfaces.

Overall a fun text. Certainly not the only complex analysis book one should read, but then again the the same can be said of any complex analysis book. My only real complaint is that the selection of exercises is somewhat small in some chapters.

Gamelin's 'Complex Analysis' is purported to be a text that, while it falls in the UTM series, can really be used for anything up through the Ph.D. qualifying exam level. This is true, but there are some problems with this text that would keep it from true brilliance.

The text covers a superb variety of topics, from the basic arithmetic up through graduate level complex analysis. The exercises to be found at the end of each section are likewise excellently chosen, and give students some great 'hands-on' practice using complex analysis. The exercises are often a little too easy, at least early on in the text, and can lull a reader into a great false sense of security with the field.

The approach that Gamelin takes makes for a very readable book, one that can easily give you an idea of what is happening. The problem however is that there is a level of generality--as well as rigor--that are sorely lacking from this text. The results that Gamelin presents can (here and there) be generalized without too much work, which really should be done for a graduate course. Similarly, his writing rather often seems to lack any semblence of the rigor that students of analysis would normally expect. Justifications would be a better word than proofs for many of the ways he convinces a reader a theorem must be true.

This does not detract from the value of the book however, but merely shift it to a different role in one's study of complex analysis. This is a great companion book--one that should find a well worn home on your reference shelf. It is an excellent book to go to when you want to get an idea of what a concept means, and then get a variety of doable problems that relate to that idea.

Although I can see what others might like in this book, I did not care for it. (To be fair, I am not sure how much of this is the book's fault and how much is the fault of the subject.) I was looking for something a bit more mathematical, and more along the lines of (say) Rudin's real analysis, and instead this book was less formal than I would have liked, seemed geared toward applications of the material in physics and engineering, and was more calculus-oriented. (Maybe the latter is inherent in the subject, I don't know.)

I prefer mathematics textbooks written in the definition-theorem-proof (followed by examples) style, and this book is not that. Terms are sometimes defined only intuitively (I don't mind intuition in addition to a formal definition, but I do mind when it is in place of a formal definition), and there are no "marked" proofs (instead, proofs are supposed to follow from the surrounding discussion, which is sometimes formal and sometimes less so).

The index was awful. I was looking for a proof of the fundamental theorem of algebra and found only one reference: to page 4, where the author promises that we will see many proofs of this theorem. (Where? Who knows!)

This book was used for my undergraduate complex analysis course. I struggled with the author's lack of clarity and ended up getting a B. Then just last month I picked up a more formal treatment of the subject--Serge Lange's Complex Analysis--and felt like I learned more in one week than I did in a whole semester with Gamelin.

Short of providing formal proofs, the text does not even provide definitions. For example, the meaning of "homotopy", which can be made mathematically rigorous, is left totally to intuition. For a student accustomed to the definition-theorem-proof structure, this text can be extremely frustrating.

This is the closest I come to a favourite book on Complex Analysis. It wins on clarity, amount of material covered, and the order in which topics are presented.

Gamelin's writing is very clear and he provides a lot of motivation and discussion; his proofs are easy to follow, and the book has a healthy dose of geometry that clarifies and enriches the subject. In spite of being very easy to read, this book manages to cover a lot of ground, and gets into some more advanced topics. I have not been able to find any book that is as accessible as this book is while also being as comprehensive. Also, this is one of the few books that explores the connections of complex analysis to applied mathematics and pure mathematics equally well.

This book's greatest asset is that it address the differences in background that students inevitably have when approaching the subject of complex analysis. This book covers all the necessary ground thoroughly, but in neat sections which are easy to skip, and it introduces more advanced topics (again in neat sections) very early so that students with a strong background will not be bored. The early introduction to Riemann surfaces is outstanding and greatly enriches the study of the material; the book's final chapter presents the theory rigorously. The two chapters on integration move slowly, but develop the subject in a manner that explores the rich interplay between the theory of analytical functions and the general theory of differentiable functions of two variables.

The exercises are outstanding. They are fairly diverse in difficulty level, and very interesting and informative. They range from simple computations up through interesting tangential theorems.

I think this would make an outstanding text for an undergraduate complex analysis course, and it might make a good text for a graduate course as well. It is also useful for self-study or as a reference. The best part about this book is that it can be used for a first course, and motivated students can plow through the rest of the book, getting into some more advanced and interesting material. After this book one should have no trouble tackling a denser text like the Ahlfors.

There are plenty of Complex Analysis books to choose from, but I really like this one. The exercises are very interesting and there hints for most of the more complicated ones. I've used this book in both undergraduate and the first year graduate courses, and it's been pretty consistently enjoyed.

The explanations were clear, and the exercises are usefull. It's fairly rigorous, but avoids getting so tangled up in rigor that it obscures the conceptual development of the book. Also, it avoids the persistent (and dreadfull) habit of presenting in the proposition, lemma, corollary, theorem, format which finds its way into a lot of analysis books. Also, I though it was nice that the book develops some concepts from real analysis (continuity, convergence of sequences and series, etc...) so that the book was fairly self contained. Finally, I liked the two sections on applications to fluid dynamics, but I wish the book would have included some further applications. Overall, a good introduction to the subject. (Although it contains way more material than one can cover in a semester)

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).

This is a neat textbook on complex analysis. Covers the basic undergraduate curriculum, plus it has some surprisingly refreshing material thrown in towards the end.
The exercises are really helpful and there is a nice variety of problems. The good part is that there are partial hints to problems in the back. With the hints and the answers, one feels that one is heading in the right direction.
I would definitely recommend it.

There were many typos and wrong answers at the back of the book which confused me sometimes, but besides this, I've never seen a more complete and readable book on complex analysis. He proves every theorem (although sometimes the proof comes several chapters after he uses the result) and provides plenty of examples for each new concept. Gamelin even has graduate-level complex analysis in this book too.