What is a good complex analysis textbook, barring Ahlfors's?

Solution 1:

Visual Complex Analysis by Needham is good. There is also Complex Variables and Applications by Churchill which is geared towards engineers.

Solution 2:

My favorites, in order:

Freitag, Busam - Complex Analysis (The last three chapters are called Elliptic Functions, Elliptic Modular Forms, Analytic Number Theory)

Stein, Shakarchi - Complex Analysis (clear and economic introduction)

Palka - An Introduction to Complex Function Theory (quite verbal, but covers a lot in great detail)

Lang - Complex Analysis (typical Lang style with concise proofs, altough it starts quite slowly, a nice coverage of topological aspects of contour integration, and some advanced topics with applications to analysis and number theory in the end)

Solution 3:

I like Conway's Functions of one complex variable I a lot. It is very well written and gives a thorough account of the basics of complex analysis. And a section on Riemann's $\zeta$-function is also included.

There is also Functions of one complex variable II featuring for instance a proof of the Bieberbach Conjecture, harmonic functions and potential theory.