What is a good introductory book for complex analysis for self study?
I thought "Complex Variables and Applications" by Brown/Churchill was a fairly good book, although I thought the book was sometimes a little too easy. A lot of problems come with answers and (sometimes an overdose of) hints, so it might be a good book for self study. It is also considered to be a canonical textbook for an undergraduate introductory complex analysis course, so the materials you see on this textbook (mostly chapters 1 through 7) are something you would see in any complex analysis course. The only drawback of this book is the price, so you probably want to get it from your library, or buy a cheaper international edition.
Fisher is pretty good, dover printing and it's cheap.
I read a chapter or two of Ahlfors when I took real analysis this past year and found the text fairly impressive. It's extremely well-written and Ahlfors gives a lot of intuition, probably because he was an expert in the field. I do remember that you have the real analysis background, so this text would be appropriate. I'm taking complex in the fall, and I'm attempting to at least understand complex integration well over the summer. Our school uses Lang, but I think many previous math majors have recommended Conway over Lang.
It depends what you intend to use it for.
A good engineering/applied physics text such as Kreisig contains all you need for this purpose.
Churchill is very engineering oriented.
A classical mathematical introduction can be had from Oxford University
An introduction to the Theory of Functions of a Complex Variable by Copson
A more spicy fully modern treatment is available from Cambridge University
Complex analysis by Stewart and Tall
If you are going to be extending to much Fluid Mechanics the the Alan Jeffrey book
Complex Analysis and Applications is good.
We are using Lang's book in a course. It's decent, but not perfect. Stein and Shakarchi's book is good. I recently discovered https://www.amazon.com/Introduction-Complex-Function-Undergraduate-Mathematics/dp/038797427X, and like it very much. Very complete is https://www.amazon.com/Complex-Analysis-Universitext-Eberhard-Freitag/dp/3540257241.
More advanced is https://www.amazon.com/Complex-Analysis-Variable-Raghavan-Narasimhan/dp/0817641645. You could also read this list.
Separate names with a comma.