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I am currently using Gamelin's textbook for studying Complex Analysis. (I just finished first seven chapter, I'm still a beginner on this subject.)

This is definitely a nice book, it covers everything I need, its every comprehensive.

However, Gamelin's writing style is really killing me.

For example, how he uses the language of differential form to prove Cauchy's theorem, how he talks about homotopy without using the word homotopy.

There are also a lot of minor problems in the text. such as he defines analyticity for a function on a set, but later he uses analyticity for a function on a point.

So I want to find a textbook that is about the level of Gamelin, could be deeper, on Complex Analysis, that is more rigorous or formal than Gamelin's book. (I really enjoy the style of Rudin, Munkres, and Hungerford) (Also I have enough background from analysis and topology to discuss things like forms and homotopy.)

I heard that Ahlfors is a great book, but it is also really old. I'm worried about the notation maybe a little bit outdated.

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  • $\begingroup$ Have you looked at Stein and Shakarchi? $\endgroup$
    – user169852
    Jan 25 '21 at 2:33
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Serge Lang wrote more books than just about any one I know of. He was pretty good at it. His Algebra is very highly regarded, for one example.

Professor Hung-Hsi Wu chose Lang's Complex Analysis as the text when I took $185$ up at Berkeley. I believe it was a very good choice, and enjoyed it alot. I dare say it's a beautiful treatment of a beautiful subject (but then so was Wu's teaching). Maybe have a look.


Btw I sat in on Gamelin's Complex Analysis course as a grad student at UCLA, and, while I may be mistaken, it seems to me that he used Ahlfors' as the text, rather than his own book, for what it's worth.

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  • $\begingroup$ I also sat in on Gamlin's complex analysis class at UCLA. He is a great teacher and helped me get in graduate school. $\endgroup$ Jan 25 '21 at 4:09
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    $\begingroup$ Yes, he is gifted. @MustafaSaid $\endgroup$
    – user403337
    Jan 25 '21 at 4:10
  • $\begingroup$ Thanks for your comments, I think I'm going to try Ahlfors after I finish Gamelin, since most people on internet are recommending this book. $\endgroup$
    – Andy
    Jan 25 '21 at 12:48
  • $\begingroup$ Ahlfors won a Fields medal once upon a time $\endgroup$
    – user403337
    Jan 26 '21 at 6:17
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Ahlfors book may be old but it is one of the best books on the subject and I highly recommend that you struggle through it. If you are looking for a more modern book then I suggest Robert Greene's Function Theory of One Complex Variable.

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  • $\begingroup$ How was your experience with Ahlfors? seems like most people on the internet are recommending this book, can you please give me some brief description about the book? such as Ahlfors writing style and his selection of the topics and notations? any thoughts and comments about his book would be really helpful. $\endgroup$
    – Andy
    Jan 25 '21 at 12:46

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