Functions of one complex variable: geometric theory Can someone recommend a good textbook on functions of one complex variable which have good chapters on  geometric theory, in English?
When I studied complex analysis, I used two
textbooks: 


*

*An  excellent texbook  by  A.  Hurwitz (with  collaboration  of R.  Courant),  "Vorlesungen   über  allgemeine  funktionentheorie  und elliptische  funktionen",  ("Lectures  on  general  theory
of functions and  elliptic functions"), in German,  and 

*An excellent [though smaller] textbook by B.  Shabat, "Introduction to complex  analysis, vol. 1", in Russian, (Б.В.  Шабат, "Введение  в комплексный анализ"). 
Both books were never  translated into English,  besides a few chapters  from the 2nd book, which were translated by Lenya Ryzhik.
For example, here are some topic in the geometric theory in a book by Hurwitz (and Courant): 


*

*Riemann sphere; its automorphisms;

*conformal mappings;

*geometry of the maps $z^n$, $1/z$, $\exp(z)$ and $\log(z)$,

*algebraic functions given by $w^n = G(z)$, where $G(z)$ is a polynomial;

*Riemann surfaces of algebraic functions; examples thereof; Riemann-Hurwitz formula;

*analytic continuation;

*Schwarz' symmetry principle,

*Weierstrass' funcion $\wp(z,\tau)$ and the embedding of
the complex torus $\mathbb{C}/L$ as a cubic curve into $\mathbb{P}^2$; 

*the modular function $j(\tau)$.


What are the good English textbooks? Is there one textbook covering this, like one by Hurwitz?
Thank you
 A: As I said in the remark there is no book comparable to the Russian edition of
Hurwitz-Courant
(Evgrafov was the editor of the Russian translation who improved the original very much).
There are 2 comprehensive Russian books covering much of geometric theory; both exist in
English translation: Markushevich and Goluzin.
Another book which covers a lot of geometric theory is Caratheodory (2 vols).
None of these has the theory of compact Riemann surfaces, 
but Shabat (which you like)
also does not have it. I would say that Markushevich is a good replacement of Shabat.
Goluzin can serve as a source of graduate courses.
Exposition of compact Riemann surfaces in Courant is unique, on my opinion.
A: Wilhelm Schlag turned his notes from a first-year graduate course into a book of sorts. It covers a lot of the topics you listed and it has a bunch of good problems. A manuscript appears on his website here: http://www.math.uchicago.edu/~schlag/bookweb.pdf; the published version of the book can be found here: http://www.ams.org/bookstore-getitem/item=gsm-154
A: Gamelin T.W. Complex analysis. Springer, 2001.
A: I suggest "Functions of One Complex Variable" I and II, by John B. Conway. It is especially well-written and cover a lot of topics, from elementary ones (volume I) to more advanced (volume II).
A: There are many wonderful books on complex analysis in English.  In  fact it is the one subject on which I used to say one could pick almost any book and be well served.  Probably this is because it was well treated in the beginning and most books simply recast the classical theory.
My favorite is by Henri Cartan, superb and cheap.  Another nice more elementary book is by Frederick Greenleaf.  The book of Lang is also excellent.
As to geometry the book by Jones and Singerman: Complex functions, an
algebraic and geometric viewpoint, is very well done.  The classic book by Ford on Automorphic functions is also recommended.
I second the recommendation of the book by Rick Miranda, a book that is just a joy to read.
Finally, the original papers of Riemann are highly recommended.
Probably everything is in the two volumes of Einar Hille, the only basic reference I know to include the big Picard theorem.
A: I said this before, but the best book ever (in any subject) is Ahlfors'. The chapter on the Weierstrass function (e.g.) is impeccable.
I have always claimed that the best way to study Complex Analysis correctly is to read Ahlfors for the theory, and Marsden for the worked examples and long list of computational exercises. 
BTW, you will also find much interesting material in "Visual complex analysis" by T. Needham. It is full of geometric insights, although on the elementary side (and it takes loooong to get to the harder material).
A: In addition to many excellent texts already mentioned, take a look at the following:
Remmert, Reinhold
Classical topics in complex function theory. 
Translated from the German by Leslie Kay. Graduate Texts in Mathematics, 172. Springer-Verlag, New York, 1998. xx+349 pp. ISBN: 0-387-98221-3 
It is divided into three parts: A. Infinite product and partial fraction series. B. Mapping theory. C. Selecta (including e.g the theorems of Bloch, Picard and Schottky, as well as theory of Runge, related to interpolation and approximation by polynomials). 
A: A lot of this stuff is in Rick Miranda's book.  
A: I suggest McMullen's lectures: complex analysis and advanced complex analysis (in McMullen's homepage), also there is a book: Hand book of complex analysis (geometric function theory) in 2 vols by R. Kühnau, I think can be Useful for you.
