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Ref: The Road to Reality: a complete guide to the laws of the universe, (Vintage, 2005) by Roger Penrose [Chap. 7: Complex-number calculus and Chap. 8: Riemann surfaces and complex mappings]

I'm searching for an easily readable and understandable book (or resource of any kind; but preferably textbook with many worked-out problems and solutions and problem sets) to learn complex analysis and basics of Riemann surfaces - and applications to theoretical physics. (Particularly: material geared towards / written with undergraduate-level physics / theoretical physics students in view).

Any suggestions?

My math background: I have a working knowledge of single- and multivariable calculus, linear algebra, and differential equations; also some rudimentary real analysis.

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I liked Howie's Complex Analysis, but I have yet to find a decent Riemann surfaces text... – Zhen Lin Sep 14 '11 at 0:46
Crosspost from MO – t.b. Sep 14 '11 at 0:47
I've looked through the posts with that tag. I found mostly questions similar to mine. ... I think the posts with that tag are different for the two sites. ... (Need clarification here.) – UGPhysics Sep 17 '11 at 22:56
Ref: ... Plus, the description field is empty. ... – UGPhysics Sep 17 '11 at 22:59
Ok. Thanks. - regards – UGPhysics Sep 17 '11 at 23:08
up vote 7 down vote accepted

I don't think you'll be able to read a book on Riemann surfaces before you study complex analysis. Once you've finished learning the rudiments of complex analysis, I recommend Rick Miranda's book "Algebraic Curves and Riemann Surfaces". It probably is the gentlest introduction I know.

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I've cursorily glanced through Weyl's "The Concept of a Riemann Surface". But not sure it might be the right fit for my needs- I'm searching for a work to allow me to learn about Riemann surfaces as hassle-free as possible [from a physics / theoretical physics point of view] - without the complications of pure mathematics; and possibly with substantial example questions-and-solutions and problems sets at ends of chapters. – UGPhysics Sep 14 '11 at 2:25
(Plus, Weyl's book seems to be outdated for some reason.) – UGPhysics Sep 14 '11 at 2:26
You definitely should not try to read Weyl's book. It's a great historical document (for instance, the original edition of it contained the first modern definition of a manifold in terms of charts and atlases). However, it is not an easy read. I'm not sure what you mean by a book "without the hassles of pure mathematics". Do you mean a book that doesn't prove anything? If you don't like the proofs, skip them. In any case, Miranda's book has tons and tons of examples and good (but pretty easy) exercises at the end of each section. – Adam Smith Sep 14 '11 at 2:32
If you look at the preface to Jost's book, he is not claiming that his book will give students a broad overview of "modern mathematics", whatever that is. Instead, he points out that the subject of compact Riemann surfaces is connected to a huge number of topics in mathematics while remaining relatively accessible, so it is a good first place to see the tools one would learn early in one's graduate education combined and used to prove some deep theorems. I don't think he is making any claim about "modern" vs "pre-modern" mathematics or anything like that. – Adam Smith Sep 14 '11 at 3:46
Isn't that kind of a silly question? He's important because he proved a lot of great theorems and introduced a number of concepts that subsequently became central to mathematics. The same way anyone doing anything becomes a "towering figure" -- by doing great work. – Adam Smith Sep 14 '11 at 4:41

You may enjoy Needham's Visual Complex Analysis.

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I have his text. Penrose especially recommends it, but for Riemann surfaces I'm still searching for a good starter. ... Any suggestions there? – UGPhysics Sep 14 '11 at 1:35

On the reference on Riemann surfaces: Forster's book "Lectures on Riemann Surfaces" is a great book that is easy to read and has many exercises (of course assuming knowledge of single variable complex analysis).

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Theodore Gamelien's Complex Analysis requires only basic calculus, is very geometric and covers just about everything a mathematics undergraduate or graduate student needs to know about functions of a complex variable.It includes an excellent and very basic introduction to Riemann surfaces. For all those reasons,it's a must have for any mathematics or physics student at either level.

A bit more sophisticated but equally wonderful is Singerman and Jones Complex Functions:A Geometric and Algebraic Approach-an incredibly rich and sophisticated second course for students who have already had an "epsilon-delta" type complex variables course and need to learn about the less analytic aspects of the subject. There's a terrific introduction to Riemann surfaces and meromorphic functions.

There's LOTS of others-the already mentioned book by Needham is a treasure,if you're willing to overlook its sometimes loose approach.That should be more then enough to get you started!

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I totally agree! I like Jones and Singerman book. However, Gamelin's Complex Analysismight be more readable. – yaa09d Oct 4 '11 at 3:42

As a beginning theoretical physics graduate student, incidentally I am these days going through the book "Introduction to Algebraic Curves" by Phillip Griffiths and it does seem to be a nice book for Riemann surfaces and such stuff. Also Jost's book on the same topic is great but focused in a different direction. For various things in complex analysis that I am once in a while getting stuck with, I seem to be able to pick them up from the book by Eliash Stein and Ramishakarchi. Somehow this combination of books does seem to work for me at least till now.

As a physics student one has to enter mathematics "laterally". Thats part of the challenge and thrill of physics whether or not the mathematical "purists" cringe about it :)

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