Textbook of complex variables function

Brown and Churchill, "Complex Variables and Applications" is a good starting point. Consider also taking a look at Needham, "Visual Complex Analysis," which is not a traditional presentation of the material but rather a geometric way of looking at the subject, which can often be very insightful.

 

At the graduate level, Ahlfors Complex Analysis is a standard. I also liked the complex portion of Rudin's Real and Complex Analysis.

 

All of the books mentioned are really good. Here are a few more.

Markushevich's "Theory of Functions of a Complex Variable"
This is my all time favorite complex variable book. Get the
three volume set not the abridged version (also good but get
the whoe thing). The translation by the way was done by Silverman.


Zeev Nejari's "Conformal Mapping" -- another great book and I believe is
published by Dover so it is cheap.

One more note, my complex variable advisor had Ahlfors for a thesis advisor.
In addition to to the Ahlfors text the above two were recommended for the course.