# Download Complex Analysis 2: Riemann Surfaces, Several Complex by Eberhard Freitag PDF By Eberhard Freitag

The booklet offers an entire presentation of complicated research, beginning with the idea of Riemann surfaces, together with uniformization idea and a close therapy of the idea of compact Riemann surfaces, the Riemann-Roch theorem, Abel's theorem and Jacobi's inversion theorem. This motivates a quick advent into the idea of numerous complicated variables, by means of the speculation of Abelian features as much as the theta theorem. The final a part of the e-book offers an creation into the idea of upper modular functions.

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Additional info for Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions

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This construction gives an arbitrarily small U in the sense that, for a given neighborhood W of a, we can ﬁnd U such that a ∈ U ⊂ W . 17 Remark. Let f : X → Y be a nonconstant analytic map of a connected Riemann surface X into a Riemann surface Y . Let a ∈ X be a point and let b = f (a) be its image. There exist analytic charts ϕ : U −→ E, a ∈ U ⊂ X, ψ : V −→ E, b ∈ V ⊂ Y, f (U ) = V, and a natural number n such that the diagram U ϕ f  V ψ /E _q  /E  qn commutes (ψ(f (x)) = ϕ(x)n ). For the proof, we can assume that X and Y are open subsets of C and that a = b = 0.

Simple Examples of Riemann Surfaces To deﬁne a Riemann surface, one has to ﬁnd an analytic atlas A on a topological surface X. Let U ⊂ X be an open subset of a Riemann surface X = (X, A) and let ϕ : Uϕ → Vφ be a chart on X. We can then consider the restricted chart ∼ ϕ|U : U ∩ Uϕ −→ ϕ(U ∩ Uϕ ). 18 I. Riemann Surfaces Obviously, the set A|U := {ϕ|U ; ϕ ∈ A} is an analytic atlas on U and thus provides U with a structure in the form of a Riemann surface. Of course, the class [A|U ] depends only on the class [A].

6 Proposition. 5. The ε-neighborhoods Uε (a, P ) are open. The natural projection p : R → C gives a topological map Uε (a, P ) −→ Uε (a). Corollary. The natural projection p : R −→ C is locally topological. ) 30 I. 6). The space R is Hausdorﬀ. Proof. Let (a, P ), (b, Q) be function elements. First case. a = b. We choose ε > 0 smaller than |a − b| and smaller than the radii of convergence of P and Q. Then, trivially, Uε (a, P ) ∩ Uε (b, Q) = ∅. Second case. a = b but P = Q. We choose ε smaller than the radii of convergence of P and Q and again obtain disjoint neighborhoods.