I am looking for a good relatively short textbook or lecture notes on graduate level complex analysis which does not ignore the rest of the mathematics (what I mean is that a lot of texbooks avoid mentioning fundamental group, differential forms, stokes theorem and so on).
I intend not to learn the subject but rather refresh my memory, get my knowledge in order. I want it to contain all the classical results (Cauchy's theorems, Laurent series, maximum module principle, Shwartz lemma and so on).
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For someone with at least a tangential interest in the number-theoretic (zeta function! elliptic functions, modular forms) origins of quite a bit of (classical) complex analysis, my own notes are intended to provide a reasonable introduction. http://www.math.umn.edu/~garrett/m/complex/
The "connection with the rest of mathematics" is a very good criterion to apply! My own notes don't do so much the connection with multi-variable calculus (differential forms and such), but more the number-theoretic and intro-algebraic-geometry aspects.
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Certainly enjoyed Lang's Complex Analysis as a Berkeley undergrad. It's in the GTM series, the so-called "yellow peril". Springer-Verlag I think.
Anyway I don't recall much about additional topics, though I don't think simple connectedness is avoided. Most or all of the standard topics are covered.
Incidentally it got me far enough to get a marginal pass on the complex analysis qual at Ucla before beginning graduate studies.
I like the lecture notes by William Chen a lot, here I'd go for his "Introduction to complex analysis". Also take a look at Beck et al "A first course in complex analysis".
In any case, a web search will probably give a large collection.