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I have already taken a course on Complex Variable. The course focused mainly on the analytical approach of the subject (power series, etc). Now, I want to study a more geometric view of the subject, specially regarding the work of the functions on the Riemann Sphere, and all the formalities behind that approach.

I've been searching for a book in this line, but haven't found many good recomendations.

Any recomendations on what books or what material may be helpful?

I'm trying to get into complex dynamics through Milnor's book, and I wanted to get more familiarity on working with the Riemann Sphere...

Jarana
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I would recommend Complex Analysis: The Geometric Viewpoint by Steven G. Krantz

Also, Algebraic Curves and Riemann Surfaces by Rick Miranda is nice, and gives a bigger picture

user0820
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    I'm not a big fan of Krantz's book. The problem with it is dimesional: In complex dimension 1 there's so little space that all the different curvature tensors and exterior forms one sees in higher dimensions collapse into one another. This causes endless confusion when learning the subject for the first time. Plus, Krantz's style is analytic and not geometric, so (for example) his metrics are smooth functions on an open set instead of smooth real $(1,1)$-forms, which is again a major source of confusion. – Gunnar Þór Magnússon Dec 03 '13 at 16:43
  • That said, the most natural statement of the Schwarz lemma (which certainly has content in dimension 1) is differential-geometric. For such an approach I'd rather recommend getting a book on the higher-dimensional case, setting $n = 1$ everywhere and figuring out what happens. A good book for this is Zhang's "Complex differential geometry". (Also, Miranda's book is very nice, but I remember it being completely algebraic (which may or may not be what the OP meant by "geometric").) – Gunnar Þór Magnússon Dec 03 '13 at 16:47
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I would strongly recommend "Complex Functions - An algebraic and geometric viewpoint" by Jones and Singerman.

It has lots of coverage of the Riemann sphere, Möbius transformations and elliptic functions.

If you are aiming for complex dynamics, then another recommendation would be:

"Iteration of Rational Functions" by Alan Beardon.

He is an excellent writer, and the book would be an good introduction to dynamics in the complex plane.

azimut
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Old John
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There's a wonderful book by Tristan Needham. It should be easy to find. Ask again if you have trouble.

It's called "Visual Complex Analysis". Home page is here.

bubba
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    Needham's book is indeed wonderful, but I would say it is too elementary for what Jarana is interested in. – user0820 Nov 29 '13 at 15:09
  • Maybe. I found that it gave me new insights into things that I thought I already understood. Anyway, the web site indicates the flavor of the book, so he (or she) can decide. – bubba Nov 30 '13 at 01:48
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Algebraic Curves and Riemann Surfaces by Rick Miranda is an excellent introduction to the world of algebraic geometry. With a first complex analysis course under your belt, you should definitely consider it.

Bruno Joyal
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