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.)
2026-04-09 09:09:40.1775725780
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What is a good complex analysis textbook, barring Ahlfors's?
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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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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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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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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 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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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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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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Introduction to Complex Analysis by Hilary Priestley is excellent for self study - very clear and well-written
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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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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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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'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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"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
Visual Complex Analysis by Needham is good. There is also Complex Variables and Applications by Churchill which is geared towards engineers.