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I want a really good book on Complex Analysis, for a good understanding of theory. There are many complex variable books that are only a list of identities and integrals and I hate it. For example, I found Munkres to be a very good book for learning topology, and "Curso de Análise vol I" by Elon Lages Lima is the best Real Analysis book (and the best math book) that I have read with many examples, good theory and challenging exercises.

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

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

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6  
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

11 Answers 11

up vote 7 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
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I've always felt Conway was overrated. It's ok,but pretty dry and it misses a lot of the beautiful geometry of the complex plane. It's probably the second volume that really impressed most people-the material in THAT book isn't readily available in a lot of sources. –  Mathemagician1234 Aug 27 '13 at 6:46

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

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

A very classic book to learn complex analysis from is Ahlfors's book (which I used). There is also Stein and Shakarchi'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
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@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

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
    
I haven't seen either,but I'm told Ulrich is outstanding. –  Mathemagician1234 Aug 27 '13 at 6:56

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
    
It's a bit terse for my tastes, but I agree,this is quite a nice book if that's your cup of potion. –  Mathemagician1234 Aug 27 '13 at 6:52

Ahlfors, Complex Analysis. It is an absolute classic and, while spartan-seeming, is a fantastic introduction to the course. It was actually my second introduction to the subject (I had it at an earlier undergraduate level using Churchill & Brown, which isn't bad, but no classic.) That book, coupled with an amazing instructor, made a huge impact.

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What is it with people and Ahlfors?!? The book SUCKS.I'm sorry,I just think there are so many better choices at the graduate level now that there's no reason to suffer through this monstrosity anymore! –  Mathemagician1234 Aug 27 '13 at 6:48
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@Mathemagician1234: different strokes, etc. I love it and still have it, 22 years after having taken the course. I also still have my notes. I guess the book in concert with an excellent lecturer helped instill in me a love for the subject. BTW Ahlfors is a Fields medalist and one of the most prominent names in complex analysis in the 20th century. That said, I am very sorry you dislike the book; as long as something else works for you. –  Ron Gordon Aug 27 '13 at 18:07
    
US $ 244.39 !!!. We need a magician to buy it. –  Felix Marin Jun 5 at 7:13

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
    
+1 for 2 great books. The former is a classic in a wonderful cheap Dover edition and the latter is on it's way to becoming a classic. My one quibble with Gamelin is that it tends to be a little TOO soft for a graduate course at times, especially early in the book. But that's what makes it ideal for self study. It really starts from jump,unlike most books-even honors high school students could use the early chapters! –  Mathemagician1234 Aug 27 '13 at 6:55

Elias Wegert's book: Visual Complex Functions: An Introduction Using Phase Portraits might not be so good for analytic techniques, but I've found it to be really good for honing one's intuition. (it's the first of a planned two volume set).

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As my opinion s.ponnusamy ''foundation of complex analysis'' is the best book. same concepts are taught in simple and different way. amit soni india

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Look up this free complex analysis book by Shabbat http://math.stanford.edu/~ryzhik/shabat-all.pdf

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