TITLE: Primer on complex analysis and Riemann surfaces for undergraduate physics / theoretical physics majors
QUESTION [5 upvotes]: 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.

REPLY [4 votes]: 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!