Complex Analysis, Elias M. Stein and Rami Shakarchi, Princeton Lectures in Analysis II, Princeton University Press (April 7, 2003). ISBN-10: 0691113858
Acquaintance with analytic arguments at the level of Rudin's Principles of Mathematical Analysis is necessary. Some knowledge of algebra and point-set topology is useful.
The study of differentiable functions of one complex variable has applications and extensions to many other areas of mathematics, so that it is a basic tool in diverse situations. Topics to be covered include: differentiability of complex functions, complex integration and Cauchy's theorem, series expansions, calculus of residues, maximal principle, conformal mapping, analytic continuation and time permitting, the Prime Number Theorem.