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I want a really good book of Complex Analysis, for good understanding theory. There are many complex variable books that are only list of identities and integrals, I hate it. For example Munkres is a very good book for my to learn topology and "Curso de Análise vol I" of Elon Lages Lima is the best book of Real-Analisys and the best math book that I was read with many examples, good theory and challenger exercises.

Is not very important the intuitive and introductory approach if the book is good explained and has correct proofs.

Added: If it is possible, tell me ur experience with your recommended books and if you got a really good understanding of complex analysis with a deep reading.

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See the question, "What is a good complex analysis textbook?". – Dylan Moreland Jun 19 '12 at 1:16
probably any book of this topic satisfies the kind of book of what I was looking, I don't want an intermediary book with an application approach. – Gastón Burrull Jun 19 '12 at 1:20

7 Answers

up vote 5 down vote accepted

Conway, "Functions of One Complex Variable I" http://books.google.ca/books?id=9LtfZr1snG0C

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Looks good, and it is avaible in my university library. I'll check it. – Gastón Burrull Jun 19 '12 at 1:23
Im looking Conway preview and looks very good with a topologycal and metric spaces early approachs. Probably this is my book. – Gastón Burrull Jun 19 '12 at 1:41
I second that. Excellent choice. – ncmathsadist Jun 19 '12 at 1:43
Thanks, I'll see it with Alfors in my university and I'll decide for one, probably I'll decide for Conway. – Gastón Burrull Jun 19 '12 at 2:06
up vote 9 down vote

My biggest recommendation is Tristan Needham's Visual Complex Analysis. Although not a strict textbook, all of the traditional theorems in elementary complex analysis are covered. Proofs aren't thorough, but are instead explained geometrically in general outlines. The big advantage with this book is the massive amount of pictures, nearly on every page in some sections.

Other great classics are Rudin's Real and Complex Analysis, Conway's Functions of One Complex Variable. For a thorough but relatively intuitive approach, I also heavily recommend Sarason's Complex Function Theory.

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I think that Tristan's book is not the book that I was looking. But Conways looks good. What about Rudin, which topics cover it?. – Gastón Burrull Jun 19 '12 at 1:26
up vote 6 down vote

A very classic book to learn complex analysis from is Ahlfor's book (which I used). There is also Shakarchi and Stein's book, and Bak and Newman's book.

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I've heard about Ahlfor's, is really a good book? Do you full understand the basic topics with a deep reading? Has challenger problems? Tell me about your own experience. – Gastón Burrull Jun 19 '12 at 1:56
@GastónBurrull I thought it was a good book but definitely on the tough side. It's not like Churchill's book which is more geared towards applications, this is intent on rigor. The exercises I thought were challenging but that's a relative opinion. It's pricey but there's an international edition which is a lot cheaper (about $20). – Eugene Jun 19 '12 at 2:00
@GastónBurrull Another thing is Ahlfors isn't heavy on examples. If you like lots of examples then maybe you won't like Ahlfors. In my opinion, which may be controversial, is that this is the baby Rudin of complex analysis. – Eugene Jun 19 '12 at 2:03
Ahlfors looks very good but some difficult and not many examples I'll get it in my library is avaible I'll check it with conway (both are avaible in my university library) and I'll decide for one. – Gastón Burrull Jun 19 '12 at 2:05
I strongly suggest the book by Bak and Newman. I learned complex analysis there, and I still believe it is a wonderful textbook for undergrads. – Siminore Jun 19 '12 at 7:39
up vote 5 down vote

The followings are very, very good. Note that you should start with the first one if you are a beginner.

  • Reinhold Remmert. Theory of complex functions. Springer 1991.
  • Reinhold Remmert. Classical topics in complex function theory. Springer 2010.
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Can I read the second one helping with the first one? – Gastón Burrull Jun 19 '12 at 1:30
I would recommend starting with the first one, and you will know when you need to get the second one. – timur Jun 19 '12 at 1:32
Thanks, I think that I'll start with the both and Conway and I'll decide which book I will definitely read. – Gastón Burrull Jun 19 '12 at 1:39
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In case it helps, here are some lecture notes I produced when I taught a complex analysis course: math.mcgill.ca/gantumur/math566f10/?Lecture_notes – timur Jun 19 '12 at 1:43
thanks for your proper material =) – Gastón Burrull Jun 19 '12 at 1:54
up vote 4 down vote

The books below are excellent:

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Both looks so good, for my level, thanks. – Gastón Burrull Jun 19 '12 at 1:32
up vote 2 down vote

Here's one that I love: Donald Sarason's book, Complex Function Theory. It's beautifully and economically written, so that it really flows. It was published by Henry Helson in his garage for a long time, but has been taken over by the American Math. Soc.

It covers complex analysis up to and including some advanced topics such as the Riemann mapping theorem, starting from basic real analysis.

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Thanks for your opinion and experience. – Gastón Burrull Jun 19 '12 at 13:11
up vote 1 down vote

I had really good luck with Fisher's Complex Variables and Gamelin's Complex Analysis.

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What do you mean with really good luck?, what is the approach? – Gastón Burrull Jun 19 '12 at 1:55
Books looks good and basic with many examples thanks! – Gastón Burrull Jun 19 '12 at 5:15
Really good luck means that I was taking a graduate complex analysis course that used Ahlfors. I floundered horribly until I studied from the two books mentioned. Fisher was thorough, but it wasn't comprehensive enough. Gamelin helped me make up the difference between Fisher and where I needed to be. – Nicholas Kirchner Jun 19 '12 at 21:33

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