Good Complex Analysis Book?

[LukeD]

I've never had any complex analysis, but I'd like to teach myself. I don't know of any good books though. I learned Real Analysis with Pugh, so I'd like a Complex Analysis book on a similar level (or maybe higher).

I.e., I'm looking for a book that develops Complex Numbers and functions axiomatically (maybe with some knowledge of Real Analysis).

Does anyone have any recommendations?

[kts123]

I'm using Fundamentals of Complex Analysis: With applications to engineering and science by Staff and Snider. It starts off with a clear explanation of complex arithmetic, and smoothly introduces the Riemann Sphere (at least in the latest edition) as well as other concepts such as exponential form, complex limits, domain and range, etc, and finally moves into the core concepts.

Also, I have to add, it has VERY clear explanations. After a few chapters, complex numbers become coherent objects -- in otherwords, it feels natural to work with complex numbers. After some time with the book, looking over the Euler Identity made me feel gitty inside at how much sense it made. The identity is marvelous just to look at, but understanding it is satisfying on an entirely different level.

[BryanP]

Visual Complex Analysis by Tristan Needham

[Howers]

Pugh is honors level mathematics, and the books the other posters posted are more on the applied to science kind. Your best bet is the classic of the genre, Complex Analysis by Lars Ahlorfs. It very well known, written by the first Fields Medal recipient, and is at the level of Rudin/Pugh.

[Adeimantus]

I can't comment on whether it is a good book, not having read it myself, but you might look for Introduction to Complex Analysis by Rolf Nevanlinna and V. Paatero at your university library. It contains a proof of Picard's theorem on entire functions. I don't think even Ahlfors has that, but I might be mistaken.

Wikipedia article on Nevanlinna:
http://en.wikipedia.org/wiki/Rolf_Nevanlinna


mathwonk, who posts here, would know what's the best book for you.

[ForMyThunder]

Here's a [downloadable] good book (also free) that I used to teach myself complex analysis when I was 15. I think that a page might be "messed up" from it but it is easily deducible if you have basic knowledge of line integrals. It's very old (over a century) and I don't know if you want a recent book on the subject but this is a good book if you have no other options.

http://books.google.com/books?id=JDUNAAAAYAAJ&pg=PA20&dq=complex+variables&as_brr=1#PPA20,M1

[HallsofIvy]

I like An invitiation to Complex Analysis by Ralph Boas.

[maze]

I'm using Fundamentals of Complex Analysis: With applications to engineering and science by Staff and Snider. It starts off with a clear explanation of complex arithmetic, and smoothly introduces the Riemann Sphere (at least in the latest edition) as well as other concepts such as exponential form, complex limits, domain and range, etc, and finally moves into the core concepts.

Also, I have to add, it has VERY clear explanations. After a few chapters, complex numbers become coherent objects -- in otherwords, it feels natural to work with complex numbers. After some time with the book, looking over the Euler Identity made me feel gitty inside at how much sense it made. The identity is marvelous just to look at, but understanding it is satisfying on an entirely different level.

I learned from this book first, and it was great. However, you may wish to transition to a more advanced text after you have worked through this one.

[SCV]

I second the Ahlfors recommendation for what you are looking for. You might also try "Functions of One Complex Variable" by John B. Conway.

Another good book is "Function Theory of One Complex Variable" by Robert E. Greene and Steven G. Krantz.

These are all graduate level Complex Analysis books and that's really what you should be looking for.

Are you going to be buying the book you get or getting it from a library?

[Count Iblis]

"Real and Complex Analyses" by Rudin is also very good.

[LukeD]

Thanks for the recommendations. I'm reading through Conway's book on Amazon right now, and it seems to be at the level I'm looking for. When I get to my library, I might check out some of the others.

[Fredrik]

I really like Saff & Snider too. (It's "Saff", not "Staff"). Their explanations and proofs are very easy to understand. If you're a physics student, you probably don't need a more advanced text.