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Does anybody know where I can find a rigorous textbook on developing complex numbers/analysis? I'm currently working through Needham's Visual Complex Analysis, which is interesting but non-rigorous. I would like to supplement that text. Thanks.

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L. Ahlorfs's Complex Analysis is widely accepted as a good textbook. Also, Rudin's Real and Complex Analysis, though a bit dated, is certainly rigorous enough.

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The book of Lavrent'ev and Shabat (Methods of complex functions) is very good but it seems there is no English translation. A book by Shabat, Introduction to complex analysis has been translated and published by AMS (it has a second volume on functions of several variables).

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I love-hate Complex Analysis by Bak and Newman, in the same way that a cat love-hates its human.

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This is the clearest presentation I have seen. And since it's Dover, it is a bargain. The proofs are detailed, replete with pictures, and very accessible. Plus numerous examples. Flanigan's "Complex Variables."

http://www.amazon.com/Complex-Variables-Dover-Books-Mathematics/dp/0486613887

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For rigor and extensive discussion it is hard to beat Remmert's classic Theory of Complex Functions (Graduate Texts in Mathematics)(v. 122).

For clarity of concept Gamelin's text is hard to beat. Although, he seems to think we teach harmonic analysis in calculus III. Sadness. It's nicely written. I found the introduction of complex-valued line-integrals and complex-Green's theorem a bit different than what I saw elsewhere. It provides a nice path to interesting results. Moreover, the whole reason I used the text to start with was that a practicing analyst recommended it. So, the criticism of lacking rigor rings hollow in my estimation. I'm sure there is a more rigorous book, but, surely there are less as well. I thought it struck a nice balance.

Another deep text that is a good second book in complex analysis is by Freitag and Busam. I emphasize, second book you read, not the first. His second volume is on modular forms and there is a truckload of gory details on Gamma functions and other sundry analytic number theory bent topics.

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  • $\begingroup$ This is quite strange to see Gamelin's textbook as an answer for a request for a rigorous book. $\endgroup$ – Artem Jan 13 '16 at 20:37
  • $\begingroup$ @Artem care to elaborate? I am curious, what is the criticism. Style, or substance? Both? A link to a critical review would suffice. Thanks! $\endgroup$ – James S. Cook Jan 13 '16 at 22:52
  • $\begingroup$ Gamelin's book appeals too much to intuition and uses too much hand waving to be advised as a rigorous textbook. This is, of course, my personal opinion (which is shared by many, e.g., reviewers on Amazon) $\endgroup$ – Artem Jan 13 '16 at 23:08
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    $\begingroup$ Or, putting unnecessary politeness aside, Gamelin's book simply lacks any rigour and can be superconfusing for those who see complex functions for the first time. $\endgroup$ – Artem Jan 13 '16 at 23:22
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To add to @adhalanay's recommendation, unfortunately, Shabat's Volume 1 has not been translated. As if on purpose, nothing in English comes close to those books in terms of the beautiful balance of theory and applications. For just theory, Markushevich is by far the best I've seen, and luckily has been translated into English. However, it is pretty long and slow and probably a little too technical for applications. Lavrentiev&Shabat has been translated into several languages like German and Spanish though.

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