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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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    $\begingroup$ Thank you so much for all the recommendations. It's a wonderful annoyance that there are lots of excellent texts out there, and it's hard to choose between them. $\endgroup$ – MBP Apr 3 '11 at 23:04
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    $\begingroup$ MBP: While Ahlfors's book may be a bit on the more difficult side, it's definitely worth spending time with it, this book is so packed with treasures! Ahlfors himself is undoubtedly one of the outstanding figures in complex analysis and his elegance, precision and concision are hard if not impossible to surpass. You should definitely revisit the book again after reading some of the other books that were suggested below. It is one of those very rare books I keep taking out of my shelf whenever I'm in the mood of reading some beautiful mathematics. $\endgroup$ – t.b. Apr 4 '11 at 10:26
  • $\begingroup$ Thank you. I'll try to gain some confidence with one of the other texts before taking on Ahlfors's book again. $\endgroup$ – MBP Apr 4 '11 at 18:38
  • $\begingroup$ To be honest I think you overestimated the difficulty of Ahlfors's textbook. I learned it in the second year of high school and I remember at that time it does not appear so difficult. I only know some real analysis at that time. I know it is intimidating when you found parts you could not go through, arguments without proper motivation, or exercises sounds too difficult, but maybe you can post these problems in here and we can help you solve them. I do not mean to demean your mathematical capability or anything. $\endgroup$ – Kerry May 26 '11 at 13:24
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    $\begingroup$ @ Changwei Zhou If you could honestly read and learn complex analysis from Ahlfors in the second year of high school, you're far more brilliant then most of us. Most of us would find it quite difficult to slog through as beginners. $\endgroup$ – Mathemagician1234 Oct 17 '11 at 1:41

26 Answers 26

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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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    $\begingroup$ I strongly agree with referring Needham's but personally feel Marsden/Hoffman's Basic Complex Analysis is much better than Churchill's text $\endgroup$ – WWright Apr 4 '11 at 1:08
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    $\begingroup$ +1 for Needham and -1/2 for Churchill (which I found very dry when I was an engineering student). ;-) $\endgroup$ – Hans Lundmark Apr 4 '11 at 6:59
  • $\begingroup$ I second Stein and Shakarchi as a friendly beginner's introduction. It's not very rigorous (many of the proofs are incomplete), but this may be an asset if you haven't seen much analysis as the book allows you to fly over the really neat aspects of complex. The exercises are quite good, if you care about that. I also second Tristan Needham, who really started my love for complex analysis. $\endgroup$ – snar Feb 12 '12 at 6:07
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    $\begingroup$ Needham's complex analysis is about as rigorous as, well it's not rigorous. $\endgroup$ – user85798 Jul 3 '14 at 17:29
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    $\begingroup$ 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. $\endgroup$ – Jack M Mar 21 '15 at 12:09
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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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    $\begingroup$ I would go witn Stein, Shakarchi for a start. $\endgroup$ – Beni Bogosel May 27 '11 at 7:02
  • $\begingroup$ Palka's ok,but if you're going to buy that one,may as well go with Gamelin's text-it's much more comprehensive. $\endgroup$ – Mathemagician1234 Oct 17 '11 at 1:51
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    $\begingroup$ +1 for Lang's book; it's a wonderful resource for more advanced complex analysis. $\endgroup$ – Alex Nelson Jun 19 '12 at 2:02
  • $\begingroup$ +1 for Freitag, Busam; It covers a great choice of topics and I found it nice to read when I first studied Complex Analysis. $\endgroup$ – Qi Zhu May 6 at 5:35
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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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    $\begingroup$ Conway is quite dry and abstract,but if you like that kind of text,it's solid. $\endgroup$ – Mathemagician1234 Feb 12 '12 at 7:34
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    $\begingroup$ Is it a good book to study for grad school entrance? $\endgroup$ – galois Nov 4 '15 at 22:25
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You may like Stein and Shakarchi's book on Complex Analysis.

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    $\begingroup$ I second Stein and Shakarchi as a friendly beginner's introduction. It's not very rigorous (many of the proofs are incomplete), but this may be an asset if you haven't seen much analysis as the book allows you to fly over the really neat aspects of complex. The exercises are quite good, if you care about that. I also second Tristan Needham, who really started my love for complex analysis. $\endgroup$ – snar Feb 12 '12 at 6:07
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    $\begingroup$ Stein and Shakarchi is much more difficult than ahlfors in my opinion, especially the exercises. $\endgroup$ – qbert Jun 19 '16 at 22:48
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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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    $\begingroup$ Which has a good quantity of exercises. $\endgroup$ – Did Oct 17 '11 at 14:36
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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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    $\begingroup$ I love this book, but I really would not describe it as being at an "intermediate sophistication level for an undergrad." $\endgroup$ – Jesse Madnick Apr 4 '11 at 7:08
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    $\begingroup$ -1 for adult Rudin for undergraduates. Are you serious? $\endgroup$ – Mathemagician1234 Nov 9 '11 at 23:33
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    $\begingroup$ No way one begins with Rudin's book. $\endgroup$ – Grown pains Jun 25 '17 at 4:25
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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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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 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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  • $\begingroup$ I'm very curious to see both of these books-they've come up in recommendations by several people who's opinion I respect lately. $\endgroup$ – Mathemagician1234 Feb 12 '12 at 7:36
  • $\begingroup$ You mean the prime number theorem, not the fundamental theorem of arithmetic? $\endgroup$ – timur Sep 10 '13 at 0:46
  • $\begingroup$ @timur: certainly! silly mistake, thanks for telling me. $\endgroup$ – Javier Álvarez Sep 10 '13 at 6:57
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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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  • $\begingroup$ I had Ablowitz as an undergraduate and his ability to explain things comes across well in this book. $\endgroup$ – GEL May 26 '11 at 1:08
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    $\begingroup$ I used Ablowitz as a supplement to Marsden's Basic Complex Analysis (not that it really needed a supplement) $\endgroup$ – nomen Oct 28 '13 at 3:02
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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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    $\begingroup$ Having had the time to look at this book recently, I second the recommendation. Priestley's book is very good. $\endgroup$ – Willie Wong Apr 30 '12 at 11:32
  • $\begingroup$ Good book. Contains a lot of mistakes though.. a lot $\endgroup$ – user85798 Mar 15 '18 at 14:46
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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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Whittaker and Watson. Hardy, Wright, and Hardy and Wright learned complex analysis from it.

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  • $\begingroup$ Do you know which mathematics books Alan Turing ( likely ) studied during his education? $\endgroup$ – nilo de roock Mar 21 '15 at 11:53
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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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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 think Using the Mathematics Literature may be helpful to answer your question.

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    $\begingroup$ For a list of great books, see The Mathematics Autodidact’s Aid. $\endgroup$ – lhf May 25 '11 at 23:02
  • $\begingroup$ This link is dead. $\endgroup$ – nilo de roock Mar 21 '15 at 11:43
  • $\begingroup$ @ndroock1, fixed now. Thanks. $\endgroup$ – Jack Mar 21 '15 at 12:09
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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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  • $\begingroup$ Gamelin's book is quite good, but it is not at all accurate to say that it "reaches very near the boundaries of research". It contains almost nothing that was not known by the mid 20th century at the latest. $\endgroup$ – Adam Smith Oct 17 '11 at 19:30
  • $\begingroup$ @Adam I seriously doubt you could name a complex analysis textbook-TEXTBOOK,not research monograph or paper collection-that reaches closer to the boundaries of current research. It takes several decades for research material to filter down to the textbook level, even texts by prominent researchers. $\endgroup$ – Mathemagician1234 Nov 10 '11 at 3:43
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    $\begingroup$ There are a number of textbooks that get closer to contemporary research on complex analysis than Gamelin. My favorite is Narasimhan and Nievergelt's book, whose exposition is heavily influenced by ideas from several complex variables and differential geometry. It also contains Wolff's elementary proof of the Corona theorem, which is one of the gems of post-1960's complex analysis. But given your taste in books, I suspect that you would find it too austere and difficult. It's definitely intended for (well prepared) graduate students rather than undergraduates... $\endgroup$ – Adam Smith Nov 10 '11 at 4:19
  • $\begingroup$ @Adam And you consider this an appropriate book to recommend to a BEGINNER in complex analysis, why exactly? I actually have Narasimhan and Nievergelt, as well as the quite different in content book by Greene and Krantz. N and N is a very good book indeed, particularly for graduate students who have an analytic bent and some background in measure theory and functional analysis. But no,I certainly wouldn't recommend it to a beginner,particularly one trying to learn on his or her own. You need to learn to come down from that mountain peak of yours and consider the target audience. $\endgroup$ – Mathemagician1234 Nov 23 '11 at 6:51
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    $\begingroup$ It depends on the background of the student. N-N does start at the very beginning. I've taught a first year graduate course using it, and the course went reasonably well. But you didn't ask about a book for beginners -- you simply asked for a textbook (as opposed to a research monograph) that got closer to the boundaries of current research than Gamelin. N-N most definitely fits that bill. $\endgroup$ – Adam Smith Nov 23 '11 at 16:01
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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.

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