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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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    $\begingroup$ See the question, "What is a good complex analysis textbook?". $\endgroup$
    – Dylan Moreland
    Jun 19, 2012 at 1:16
  • $\begingroup$ 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. $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 1:20
  • $\begingroup$ @GastónBurrull The analysis book by Lima that you have mentioned, does it have an english version? $\endgroup$
    – Bhaskar Vashishth
    Nov 4, 2014 at 20:33
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    $\begingroup$ @BhaskarVashishth No, only in portuguese. But you can read "courant and john introduction to calculus and analysis" it is excellent and it is very similar to lima. $\endgroup$
    – Gaston Burrull
    Nov 5, 2014 at 1:58
  • $\begingroup$ @GastónBurrull I do not like analysis. I am past under graduation. But I want to give it a try if I can find some gem of a book, otherwise I guess its too late for me.. $\endgroup$
    – Bhaskar Vashishth
    Nov 5, 2014 at 2:18

16 Answers 16

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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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  • $\begingroup$ 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?. $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 1:26
  • $\begingroup$ Rudin is interested in showing you how clever he is. Conway gets you to believe that you can do this yourself. Sarason is an insightful man of few words, and every one of them counts. $\endgroup$
    – Disintegrating By Parts
    Dec 26, 2023 at 7:37
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Conway, "Functions of One Complex Variable I" http://books.google.ca/books?id=9LtfZr1snG0C

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  • $\begingroup$ Looks good, and it is avaible in my university library. I'll check it. $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 1:23
  • $\begingroup$ Im looking Conway preview and looks very good with a topologycal and metric spaces early approachs. Probably this is my book. $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 1:41
  • $\begingroup$ I second that. Excellent choice. $\endgroup$
    – ncmathsadist
    Jun 19, 2012 at 1:43
  • $\begingroup$ Thanks, I'll see it with Alfors in my university and I'll decide for one, probably I'll decide for Conway. $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 2:06
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    $\begingroup$ 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. $\endgroup$
    – Mathemagician1234
    Aug 27, 2013 at 6:46
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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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  • $\begingroup$ Can I read the second one helping with the first one? $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 1:30
  • $\begingroup$ I would recommend starting with the first one, and you will know when you need to get the second one. $\endgroup$
    – timur
    Jun 19, 2012 at 1:32
  • $\begingroup$ Thanks, I think that I'll start with the both and Conway and I'll decide which book I will definitely read. $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 1:39
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    $\begingroup$ 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 $\endgroup$
    – timur
    Jun 19, 2012 at 1:43
  • $\begingroup$ thanks for your proper material =) $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 1:54
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The books below are excellent:

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  • $\begingroup$ Both looks so good, for my level, thanks. $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 1:32
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    $\begingroup$ I haven't seen either,but I'm told Ulrich is outstanding. $\endgroup$
    – Mathemagician1234
    Aug 27, 2013 at 6:56
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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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  • $\begingroup$ 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. $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 1:56
  • $\begingroup$ @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). $\endgroup$
    – Eugene
    Jun 19, 2012 at 2:00
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    $\begingroup$ @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. $\endgroup$
    – Eugene
    Jun 19, 2012 at 2:03
  • $\begingroup$ 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. $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 2:05
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    $\begingroup$ 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. $\endgroup$
    – Siminore
    Jun 19, 2012 at 7:39
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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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  • $\begingroup$ Thanks for your opinion and experience. $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 13:11
  • $\begingroup$ It's a bit terse for my tastes, but I agree,this is quite a nice book if that's your cup of potion. $\endgroup$
    – Mathemagician1234
    Aug 27, 2013 at 6:52
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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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I had really good luck with Fisher's Complex Variables and Gamelin's Complex Analysis.

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    $\begingroup$ What do you mean with really good luck?, what is the approach? $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 1:55
  • $\begingroup$ Books looks good and basic with many examples thanks! $\endgroup$
    – Gaston Burrull
    Jun 19, 2012 at 5:15
  • $\begingroup$ 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. $\endgroup$
    – Nicholas Kirchner
    Jun 19, 2012 at 21:33
  • $\begingroup$ +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! $\endgroup$
    – Mathemagician1234
    Aug 27, 2013 at 6:55
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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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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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I think Serge Lang's book on Complex Analysis is a good a one to go with.

http://www.amazon.com/Complex-Analysis-Graduate-Texts-Mathematics/dp/0387985921

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No one has mentioned here, but 'A First Course in Complex Analysis with Applications' by zill is my favourite book. It is so clear and comprehensive, and much simpler and intutive explainantions.

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    $\begingroup$ From the authors taste in Munkres, I expect he was looking for a graduate book. However, I did use this book as an undergraduate. I do still have Zills book on DE too. $\endgroup$
    – dustin
    Nov 4, 2014 at 20:41
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Complex Analysis, by Beardon is unmatched in quality. I put it above Ahlfors, because in the preface of Ahlfors CA, Allen F Beardon is mentioned with gratitude.

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  • $\begingroup$ I've never seen that book! How could Beardon have written a complex analysis text and I didn't know? $\endgroup$
    – Mathemagician1234
    Jan 14, 2016 at 19:46
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Complex Analysis by Kunihiko Kodaira

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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.

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An Introduction to Complex Analysis by Ravi P. Agarwal , Kanishka Perera , Sandra Pinelas is a fantastic book!

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  • $\begingroup$ I found this book to be the most infuriating. Eventually got tired of filling in the gaps in the steps. Does not develop any intuition. Produces key concepts like the exponential form like a rabbit out of the hat instead of motivating them. The only redeeming quality is solutions to exercises. $\endgroup$
    – user643160
    Apr 13, 2019 at 22:41

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