Rigorous Textbook for Introduction to Complex Numbers/Analysis?

Question

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.

The classic text by A. I. Markushevich is very good and easy to read. (Which is all the more impressive given that it is translated from Russian!) — Andrew Poelstra, Aug 05 '13 at 20:56
Possible duplicate of http://math.stackexchange.com/questions/160132/complex-analysis-book and http://math.stackexchange.com/questions/30749/what-is-a-good-complex-analysis-textbook. — lhf, Aug 06 '13 at 00:15

score 5 · Answer 1 · answered Aug 05 '13 at 19:44

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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.

answered Aug 05 '13 at 19:44

Jonathan Y.

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score 5 · Answer 2 · answered Aug 05 '13 at 21:14

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

answered Aug 05 '13 at 21:14

Emily

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score 4 · Answer 3 · answered Aug 05 '13 at 20:26

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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).

answered Aug 05 '13 at 20:26

adhalanay

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I used Shabat's excepts on my complex analysis class, but although I searched I couldn't find the whole book. Is there any reference to it around? – Daniel Teixeira Sep 05 '20 at 16:45

James S. Cook · Answer 4 · 2016-01-13T22:51:47.130

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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.

edited Jan 13 '16 at 22:51

answered Aug 05 '13 at 22:18

James S. Cook

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This is quite strange to see Gamelin's textbook as an answer for a request for a rigorous book. – Artem Jan 13 '16 at 20:37
@Artem care to elaborate? I am curious, what is the criticism. Style, or substance? Both? A link to a critical review would suffice. Thanks! – James S. Cook Jan 13 '16 at 22:52
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) – Artem Jan 13 '16 at 23:08
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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. – Artem Jan 13 '16 at 23:22

score 2 · Answer 5 · answered Aug 05 '13 at 22:06

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

score 0 · Answer 6 · answered Jan 13 '16 at 17:43

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.

Rigorous Textbook for Introduction to Complex Numbers/Analysis?

6 Answers6