Complex Analysis (Graduate Texts in Mathematics) by Serge Lang (20100219)
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Review Text
There are about as many opinions on this book as there are different books that Lang wrote, but there is a reason for this: this is one strange book, even among Lang's.I will start out by saying what I like about this book: most of it. This book provides a lot of topological flavour to complex variables, which I find very helpful. To someone who thinks topologically, many of the proofs in this book will seem more intuitive than in other texts. This is particularly true when you get into more advanced material.Overall, the writing is very clear. Lang is excellent at providing motivation, especially as you get farther along in this book. Unlike some of his other books, he can't be criticized as moving too fast in this book.Now the bad: the book starts out very slow, painfully so. It seems the first chunk of the book is aimed at teaching rigorous complex analysis to someone whose background in analysis is weak. Lang repeats all of the basic theorems about limits, differentiation, convergence, etc. in full detail. However, the material picks up eventually, and by the end of the book it's moving fast enough that anyone who enjoyed the first part will have trouble understanding the later material. This book covers a lot more material than most undergrad books on the subject, so I suppose it lives up to the GTM title.Bottom line: I don't like the choice or order of topics in initial chapters. Some of the "new" material specific to complex variables is mixed in with old results common to basic analysis on the real line. Anyone with a good background in analysis will be frustrated trying to find what they need to learn. Also, Lang confuses the logic of the subject by working with the terms "analytic" and "holomorphic" separately for a great deal of time before showing their equivalence. His definitions, terminology, and development don't line up with many other authors, and he has not convinced me that his choice of development was justified...because most of the stuff I like in this book comes after the first few chapters. However, if you can get past these hurdles, you'll find that this is a pretty great book that has a lot to offer.
Lang's book isn't bad. I used it a lot in the last year of my undergraduate degree when I was learning about modular forms, as a reference for complex analysis. But there are better books on complex analysis. Instead of proving that "if a sequence of holomorphic functions on a domain converges uniformly on every compact subset of the domain to a particular function then that function is holomorphic", a more systematic way of talking about this is to give a topology to the vector space of continuous functions on the domain with the topology induced by a family of seminorms corresponding to the compact sets, and to show that the holomorphic functions are a closed subspace. This is an instance of a Fréchet space, and Montel's theorem can then be given a clean statement: a subset of this Fréchet space is compact if and only if it is closed and bounded (namely, the space has the HeineBorel property). This point of view is taken in Chapter V of Cartan's "Elementary Theory of Analytic Functions of One or Several Complex Variables".If one is willing to take the whole dose of medicine, I think that the connected presentation of measure theory and complex analysis in Rudin's "Real and Complex Analysis" would be the best way to learn complex analysis. One would need a decent background in analysis before trying this, but would not need to know anything about holomorphic functions beforehand.Both of the books that I have recommended so far are at a higher level than Lang. A book that I think is more approachable is Stein and Shakarchi's "Complex Analysis". Finally, if you are looking for other books on complex analysis, try Conway's "Functions of One Complex Variable". I haven't read this but I have spent time with his "A Course in Functional Analysis", which I find clear.
Lang's Complex Analysis is an very good text for anyone wanting to move beyond introductory complex analysis.
if you want an introduction to complex analysis, I advise you to pass on this book, and read Churchill and Brown's introductory book. Having said this, part I of Lang's book will seem mostly review if you follow my advice. Part II, on Geometric Function Theory, is more advance material that is presented reasonably well.
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I really have two big gripes with this book: (1) it is too slow and (2) there are not enough exercises and the ones that are given could have been better.I bought this book to study for my quals, but found myself abandoning it after encountering tedious and unnecessary proofs and insufficient/trivial exercises.It might, repeat *might* be good for beginners, but anyone whose taken advanced calculus (as just about any grad student would have) will be frustrated with the slow pace first few chapters. Frustration will continue after getting to the end of a complicated chapter and finding no excercises (!).I also thought the orginization of the material was really unnatural (e.g., why are fractional linear transformations discussed so late in the book?).I don't know... If you understand theory with out any problems to practice on, then you might be able to use this book.