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?
Best Answer
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.