Advanced complex function theory book recommendation

Question

I would like to have some recommendations in order to self study the above topic.

I have already studied some complex function theory, covering some of the more 'classical' theorems (the Bloch-Landau theorem, the Little & Big Picard theorems, Riemann mapping theorem) and some introductry to analytic continuation ideas. I would like to further study this, and more specifically: Gamma&Zetta functions, elliptic functions, harmonic functions, and further study of holomorphic and meromorphic functions.

Also, What books out there have a proof of Zalcman's lemma?

Answer 1

It sounds like you've already had a healthy dose of complex analysis. I would strongly recommend that you have a look at "Classical topics in Complex Function Theory" by Remmert (Springer-Verlag).

It is very well written and contains a delightful choice of interesting results, many of which can't be found in most textbooks on complex analysis, but deserve to be better known.

I don't have my copy at hand, so I'm not sure it contains a proof of Zalcman's lemma, but it does cover several of the topics you are interested in, such as the Gamma and Beta functions, and a very extensive coverage of the "deeper" theory for holomorphic and meromorphic functions, including topics such as Mittag-Leffler's theorem, Carathéodory's theorem of boundary regularity of conformal mappings, Radó's theorem, Schottky's theorem and a very extensive treatment of Runge's theorem with various generalizations compared to what can be found in most texts.

Answer 2

Have you tried Conway's books on Complex Analysis in the Graduate Texts in Mathematics?

Answer 3

You can check out the textbook suggestions in the following closely related MSE question - What is a good complex analysis textbook?.