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I'm out of college, and trying to self-learn complex analysis. I'm finding Ahlfors' text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann Zeta function and the Prime Number Theorem.)

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Visual Complex Analysis by Needham is good. There is also Complex Variables and Applications by Churchill which is geared towards engineers.

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    I found Visual Complex Analysis to be utterly incomprehensible when I was trying to learn Complex Analysis. It's not just non-rigorous, it's barely even a textbook: theorems are indirectly hinted at rather than explicitly stated, definitions are non-existent and there didn't seem to be any proofs at all.2015-03-21
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My favorites, in order:

Freitag, Busam - Complex Analysis (The last three chapters are called Elliptic Functions, Elliptic Modular Forms, Analytic Number Theory)

Stein, Shakarchi - Complex Analysis (clear and economic introduction)

Palka - An Introduction to Complex Function Theory (quite verbal, but covers a lot in great detail)

Lang - Complex Analysis (typical Lang style with concise proofs, altough it starts quite slowly, a nice coverage of topological aspects of contour integration, and some advanced topics with applications to analysis and number theory in the end)

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I like Conway's Functions of one complex variable I a lot. It is very well written and gives a thorough account of the basics of complex analysis. And a section on Riemann's $\zeta$-function is also included.

There is also Functions of one complex variable II featuring for instance a proof of the Bieberbach Conjecture, harmonic functions and potential theory.

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    Is it a good book to study for grad school entrance?2015-11-04
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You may like Stein and Shakarchi's book on Complex Analysis.

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    Stein and Shakarchi is much more difficult than ahlfors in my opinion, especially the exercises.2016-06-19
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Elementary theory of analytic functions of one or several complex variables by Henri Cartan.

(The Prime Number Theorem is not proved in this book.)

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    Which has a good quantity of exercises.2011-10-17
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Complex Analysis by Joseph Bak and Donald J. Newman has a proof of the Prime Number Theorem.

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Rudin's Real and Complex Analysis is always a nice way to go, but may be difficult due to the terseness.

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    No way one begins with Rudin's book.2017-06-25
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I second the answer by "wildildildlife" but specially the book by Freitag - "Complex Analysis" and the recently translated second volume to be published this summer. It is the most complete, well-developed, motivated and thorough advanced level introduction to complex analysis I know. The first volume starts out with complex numbers and holomorphic functions but builds the theory up to elliptic and modular functions, finishing with applications to analytic number theorem proving the prime number theorem. The second volume develops the theory of Riemann surfaces and introduces several complex variables and more modular forms (of huge importance to modern number theory). They are filled with interesting exercises and problems most of which are solved in detail at the end!

You just need a good background in undergraduate analysis to manage. Moreover, I think they should be your next step after a softer introduction to complex analysis if you are interested in deepening your knowledge and getting a good grasp at the different aspects and advanced topics of the whole subject.

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    @timur: certainly! silly mistake, thanks for telling me.2013-09-10
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The followings are very, very good. Note that they form a set.

  • Reinhold Remmert. Theory of complex functions. Springer 1991.
  • Reinhold Remmert. Classical topics in complex function theory. Springer 2010.
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I agree with @WWright. Marsden/Hoffman is (one of) the best of the undergraduate complex analysis books in my opinion, although it does not mention the PNT or RZ equation at all.

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Yet another good one: Complex Variables: Introduction and Applications by Ablowitz & Fokas.

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    I used Ablowitz as a supplement to Marsden's Basic Complex Analysis (not that it *really* needed a supplement)2013-10-28
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Introduction to Complex Analysis by Hilary Priestley is excellent for self study - very clear and well-written

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    Good book. Contains a lot of mistakes though.. a lot2018-03-15
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The little Dover books by Knopp are great. They get to the integral fast -- and that's where the fun really begins. Get 'em.

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Complex variables: An introduction, by Carlos A. Berenstein and Roger Gay (Springer, 1991).

An underrated masterpiece.

This is a self-contained, very accessible, comprehensive, and masterfully written textbook that I do find very suitable for the serious self-taught possessing the rare mathematical maturity, and being in command of a quite modest (but non-negligible) background.

Among its many competitors, this work distinguishes itself by being, by far, the most modern in scope and means, since it introduces in a very harmonious way and from the very beginning, mainly from scratch, key ideas from homological algebra, algebraic topology, sheaf theory, and the theory of distributions, together with a systematic use of the Cauchy-Riemann $\bar{\partial}$-operator. So for instance, once you're going to tackle Cauchy's integral theorem, you'll be fully equipped to prove it in its full generality, and without the typical "hand-weaving" most texts rely on and hide behind.

A following up by the same authors is Complex analysis and special topics in harmonic analysis (Springer, 1995).

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Whittaker and Watson. Hardy, Wright, and Hardy and Wright learned complex analysis from it.

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    Do you know which mathematics books Alan Turing ( likely ) studied during his education?2015-03-21
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I think Using the Mathematics Literature may be helpful to answer your question.

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    @ndroock1, fixed now. Thanks.2015-03-21
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A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka is a well written free online textbook. It is available in PDF format from San Francisco State University at this authors website.

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I don't think it has the zeta function or the PNT (I could be wrong, it has been a long time since I looked at it), but "Invitation to Complex Analysis" by Ralph P. Boas is really nice, and suitable for self study because it has about 60 pages of solutions to the texts problems.

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You might like Functions of a Complex Variable by E.G. Phillips. It is slightly dated, but you can't argue with the price! I personally think this is a wonderful book.

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Concise Complex Analysis, by Sheng Gong and Youhong Gong. That's a really excellent textbook! Trust me!

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"Complex Analysis with Applications" by Richard Silverman is a gentle introduction to the subject. Only covers the basics, but explains them in a crystal clear style. http://store.doverpublications.com/0486647625.html

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I've taught a few times from Churchill's book, and used it as an undergrad. I'm liking it less all the time. I would probably switch to Marsden/Hoffman next time. At a more advanced level, I like Nevanlinna and Paatero, "Introduction to Complex Analysis." It has a chapter on the Riemann zeta function within which there is a discussion of the distribution of primes. I used this in the beginning grad course in complex, along with Hille's "Analytic Function Theory," which I liked very much.

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Some free and very very good references are:

Saying that here all is explained really properly, wouldn't be enough.

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For a good introduction i referred "A First Course in complex Analysis by Dennis G.Zill" and for little advanced case i would like to refer "Complex Analysis by Dennis G. Zill and Patrick Shanahan".

Also many good books by Churchill & Brown , another by Ponnusamy are also there . Hope this helps!

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Lots of good recommendations here-but for self study,you can't beat Complex analysis by Theodore W. Gamelin. It's highly geometric, has very few prerequisites and reaches very near the boundaries of research by the end.

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    @Adan You must have been fortunate enough to have strong students,then. Krantz and Greene also delves close to research level aspects of complex analysis with strict emphasis on the analytic side,as opposed to N and N,where the subject matter is more diverse.I'd take a look at it the next time you teach that course-I think you'll find it to your liking as well.2011-11-23
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You may find the following references useful:

  • "Schaum's Outline of Complex Variables, Second Edition" by Murray Spiegel. This has plenty of solved and unsolved exercises ranging from the basics on complex numbers, to special functions and conformal mappings. This has a note on the zeta function.

  • "Geometric Function Theory: Explorations in Complex Analysis" by Steven Krantz. This is good for more advanced topics in classic function theory, probably suitable for advanced UG/PG. It covers classic topics, such as the Schwarz lemma and Riemann mapping theorem, and moves onto topics in harmonic analysis and abstract algebra.

  • "Complex Analysis in Number Theory" by Anatoly Karatsuba. This book contains a detailed analysis of complex analysis and number theory (especially the zeta function). Topics covered include complex integration in number theory, the Zeta function and L-functions.