Complex Analysis (Graduate Texts in Mathematics, 103)

Prior to Lang's book under review several excellent texts existed but none of them, in my opinion, possessed, what I have come to understand about Lang's books, the combination of rich pedagogical skill coupled with an excellent understanding about what research topics are - as of the writing of the book - currently important. I think Lang, whom I knew and had correspondence about this matter with, wrote to educate the next generation of mathematicians. So his books are not compendiums of everything that is known but rather guide books into the research frontiers. This book was a delight for me to read and reconnect with a subject, admittedly not my area of research while a professor at the University of California, Irvine, that I always found especially beautiful. Lang goes to gerat lengths to actually teach the subject in a detailed and leisurely way. Of course, given the series in which the book appears the reader is expected to make some effort as the material is not spoon-fed, but readers who make the effort will find themselves richly rewarded with a deep knowledge of parts of the theory of complex analysis.

A well written treatment by the OG, what more can be said? Good reference to have and nice emphasis on analytic ideas and principles.

good and fast

Lang's Complex Analysis is an very good text for anyone wanting to move beyond introductory complex analysis.

good

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 Heine-Borel 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.

This book is the best book on Complex Analysis that I have seen in a long time. It is well written and the proofs in the book are layed out nicely. I especially appriciate the section on conformal mapping. Complex Analysis is one of the most beautiful branches of mathematics which deserves a lot of attention. Lang has done a great job in his exposition of the subject. I highly recommend this book to any professor planning to teach this subject. It is important that the student have at least an undergraduate course in Real Analysis. I completed a year of Complex Analysis back in graduate school and had the misfortune of using Alfors book which is a terrible book. I had to purchase a supplementary book by Speigal to understand the basics. I wish I had got a hold of Langs book then.

I dont like lang's algebra, ugrad linear algebra, or diff/riemannian manifolds books all that much, but i LOVED this one. I think an undergrad with calculus and patience can read it. there are characteristic lang-style things like research-oriented material, and he actually has examples. He covers topics towards the end of the book which arent common elsewhere, so i've never put it down. I am not a mathematician and I like this book. It's in one of my standard 8 books that I dont leave home without (4 physics 4 math)