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

Posted by

By Eberhard Freitag

The publication offers an entire presentation of advanced research, beginning with the speculation of Riemann surfaces, together with uniformization thought and an in depth remedy of the idea of compact Riemann surfaces, the Riemann-Roch theorem, Abel's theorem and Jacobi's inversion theorem. This motivates a brief creation into the idea of numerous complicated variables, via the idea of Abelian capabilities as much as the theta theorem. The final a part of the booklet presents an advent into the speculation of upper modular services.

Read Online or Download Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions (Universitext) PDF

Best mathematical analysis books

Understanding the fast Fourier transform: applications

It is a instructional at the FFT set of rules (fast Fourier remodel) together with an creation to the DFT (discrete Fourier transform). it truly is written for the non-specialist during this box. It concentrates at the real software program (programs written in simple) in order that readers could be in a position to use this know-how once they have accomplished.

Acta Numerica 1995: Volume 4 (v. 4)

Acta Numerica has verified itself because the top discussion board for the presentation of definitive experiences of numerical research themes. Highlights of this year's factor comprise articles on sequential quadratic programming, mesh adaption, loose boundary difficulties, and particle tools in continuum computations.

Extra info for Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions (Universitext)

Sample text

Here n is a natural number. We want to formulate this result for Riemann surfaces. First, we notice that for every point a ∈ X of a Riemann surface there exists an analytic chart ϕ : U → E, a ∈ U , whose image is the unit disk. We choose an arbitrary analytic chart ψ : U → V , a ∈ U , and then replace U by the inverse image of a small disk around ψ(a). Now ϕ is obtained by restricting ψ to this inverse image and composing it with a conformal map from the small disk onto the unit disk. 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 .

An open subset U ⊂ X is called “small” if f maps U topologically onto an open set f (U ) and if there exists an analytic chart on Y , ∼ f (U ) −→ V (⊂ C open). The composition U −→ f (U ) −→ V is a chart on X. Obviously, these charts are analytically compatible. Hence they deﬁne a structure in the form Riemann surface on X. To prove uniqueness, we describe the elements ϕ : U → V of the maximal atlas. We can restrict ourselves to ϕ such that U is small in the sense that it is mapped biholomorphically onto the open set f (U ).

All that remains to be proved is that q can be extended. (For this, we can assume that P truly depends on z. ) Now we can interchange the roles of p and q. There exists a ﬁnite subset T ⊂ C such that the canonical projection q : X0 −→ C − T, X0 := {(a, b) ∈ C × (C − T ), P (a, b) = 0}, is locally topological and proper. One can choose T large enough such that X0 is a subset of X. The complement X − X0 is a ﬁnite set. 8, Corollary), it is suﬃcient to extend q to some compactiﬁcation of X0 . 7 (with q instead of p).