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I am considering complex analysis as my next area of study. There are already a few threads asking about complex analysis texts (see Complex Analysis Book and What is a good complex analysis textbook?). However, I'm looking for something a little more specific, if such a thing exists.

Is there a nice, slow-paced introductory complex analysis text that features at least some (introductory) material on Riemann surfaces?

A look through texts mentioned in the pages linked above did not yield any. I am not big on analysis and tend to favor more algebraic, topological, and geometric-flavored areas of mathematics. I am however trying to learn at least at a basic level the core disciplines of mathematics, and I feel I would be amiss if I did not study complex analysis. For background: I have a basic knowledge of real analysis, algebra (group, ring, and field theory), linear algebra, and will have knowledge of topology.

In addition to my above desire in a complex analysis text: is there one you would recommend for its view toward algebraic, topological, or geometric applications of complex analysis?

Any online lecture notes (or inexpensive book) on Riemann surfaces that would be accessible after or along with an introductory look at complex analysis would be welcome as well.

EDIT: After what has developed, I feel this question is now appropriate: Is there a complex analysis text that would be particularly recommended if one wishes to study Riemann surfaces? What topics in particular is it important to develop a good grasp of?

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You will need to learn a fair amount of complex analysis before studying Riemann Surfaces. Once you have done that, the book by Miranda mentioned below is probably your best option, given (how I perceive) your level of mathematical maturity. Another option is Griffith's "Introduction to Algebraic Curves." –  Potato Sep 22 '12 at 1:25
    
I had the feeling that might be the case and that might be why I wasn't seeing introductions to it in complex analysis books. That's what I wanted to know. Thanks. –  Alex Petzke Sep 22 '12 at 1:35
    
Added a new question above. –  Alex Petzke Sep 22 '12 at 1:44
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Regarding your new question: The material in a usual one-semester undergraduate course should be sufficient. Cauchy's theorem, residue theory, the maximum principle, the open mapping theorem, and the fact that holomorphic functions have power series expansions, at the bare minimum. Although it will probably be helpful to know more. For example, Miranda uses Mittag-Leffler's theorem to motivate trying to solve Mittag-Leffler problems on Riemann surfaces and the resulting cohomology theory (but that isn't until the middle of the book). –  Potato Sep 22 '12 at 1:57
    
Very helpful, thanks. –  Alex Petzke Sep 22 '12 at 2:01

2 Answers 2

up vote 6 down vote accepted

Narasimhan-Nievergelt's Complex Analysis in One Variable is exactly the book you want.

It is completely geometric and will introduce you, starting from scratch, not only to Riemann surfaces but also to the theory or holomorphic functions of several variables, covering spaces, cohomology,...
This unique book emphasizes how little you have to know of the classical function of one complex variable: just the forty pages of Chapter 1, aptly named Elementary Theory of Holomorphic Functions.
A book with a similar philosophy is Analyse Complexe by Dolbeault, he of the Dolbeault cohomology, which has the drawback of being in French (albeit in mathematical French, which is a far cry from Mallarmé or Proust French...)

It is an underappreciated fact, displayed in both these books, that most of the material found in books on complex analysis of one variable is useless for the study of Riemann surfaces and more generally complex manifolds.
For example all the clever computations of real integrals by residue calculus, evaluation of convergence radius of power series, asymptotic methods, Weierstraß products, Schwarz-Christoffel transformations, ... are irrelevant in complex analytic geometry: I challenge anyone to find the slightest trace of these in the work of the recently deceased H. Grauert, arguably the greatest 20th century specialist in the geometry of complex analytic spaces.

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Do you think the first book is accessible to someone with my background (see question)? Does it really introduce things like covering spaces and cohomology without presupposing a knowledge of algebraic topology? Do you feel it covers the basics of complex analysis that one should know? I may want to work toward Riemann surfaces, but not to the sacrifice of topics that I'd learn in a more typical text. –  Alex Petzke Sep 22 '12 at 14:07
    
Yes, yes and yes. You don't have to sacrifice anything: my point is that you need little complex analysis to understand the geometry of Riemann surfaces. Holomorphic functions are a marvelous subject, intrisically beautiful and well worth studying,with many applications in mathematics and physics. But that is not what your question was about. –  Georges Elencwajg Sep 22 '12 at 17:24
    
Got it. Thanks. –  Alex Petzke Sep 22 '12 at 18:16

I highly recommend Algebraic Curves and Riemann Surfaces by Prof. Rick Miranda.

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Would this have to come after I learned some complex analysis though? That's the impression I'm getting from the description. –  Alex Petzke Sep 22 '12 at 1:25
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Yes, but such is the nature of the subject. A solid knowledge of complex analysis is required for any study of Riemann surfaces. –  Potato Sep 22 '12 at 1:31
    
For example, you need to be able to answer the following question to read the Miranda's book. The question is related to the Forster's book on Riemann surfaces, but Miranda also uses the result without a proof. math.stackexchange.com/questions/200012/… –  Makoto Kato Sep 22 '12 at 3:03

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