Complex analysis prerequisites for Apostol's Modular Functions

Question

I have studied Apostol's Introduction to Analytic Number Theory and it didn't much required Complex Analysis. However, I have studied Complex Analysis from the greatest book Churchill's Complex Variables but unfortunately it doesn't cover prerequisites for Apostol's second book Modular Functions and Dirichlet Series in Number Theory. It uses some knowledge of Complex Analysis which I couldn't find any good book for that.

Not a duplicate question, I am definitely looking for a very readable (esp for self-learning) advance book in Complex Analysis which covers ALL the prerequisites for Apostol's second book.

Any right suggestion would be much appreciated.

PS I tried Remmert's two volumes books on the subject as they looked nice at first (with good amazon reviews (why?!)) but instead of giving detailed rigorous proofs they were obsessed with history and the German quotes!

Added - the book gives an quick review or none at all to the following subjects as if the reader already knows about them:

doubly periodic functions in Complex Space

The Fourier expansions in Complex Space

Proof of "If is modular and not identically zero, then in the closure of the fundamental region $R_{\Gamma}$, the number of zeros of f is equal to the number of poles" is not understandable.

Answer 1

I'm quite sure that with these introductory lecture notes (they starts from the the definition of holomorphic function and arrive to residue theorem/argument principle) and with these more advanced lecture notes, you will be able to tackle any topic involving complex analysis.