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Is there any book on $1$-dimensional complex analysis, where all is written in the language of sheaf theory? It's clear, that a lot of constructions can be formulated in simplier way using it. There are a lot of such books of n-dimensional complex analysis. And what about 1-dimensional?

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What is 1D complex analysis? Complex numbers have a real and an imaginary part, so that is 2D. If you have a complex valued function, then it becomes 4D. – glebovg Dec 4 '12 at 19:13
@glebovg Probably 1D over $\mathbb{C}$... – Cocopuffs Dec 4 '12 at 19:22
up vote 2 down vote accepted

Robert Gunning's "old" Princeton-Yellow-Series book "Intro to Riemann Surfaces" (the first in a sequence of several books he wrote about Riemann surfaces and related matters...) systematically uses sheaf theory (albeit not the derived-functor version, but Cech). In my opinion, it wonderfully illustrates how sheaf theory can be used, in a very tangible example.

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And what about ramification of analytical functions, Grothendieck's 'children's drawings'? – user46336 Dec 4 '12 at 19:34
The Gunning book is mostly "global", and "algebraic" in the sense that it addresses compact (connected) Riemann surfaces. In one complex variable, there's not a lot to worry about "ramification", of course. He does Riemann-Roch, Serre duality, things like that. – paul garrett Dec 4 '12 at 19:47
Thank you! I know Riemann-Roch and Serre duality only in algebraic case. It would be interesting to study it in complex case. – user46336 Dec 4 '12 at 19:52

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