The Riemann Mapping Theorem
نویسنده
چکیده
We will develop some of the basic concepts of complex function theory and prove a number of useful results concerning holomorphic functions. We will focus on derivatives, zeros, and sequences of holomorphic functions. This will lead to a brief discussion of the significance of biholomorphic mappings and allow us to prove the Riemann mapping theorem.
منابع مشابه
The existence results for a coupled system of nonlinear fractional differential equations with multi-point boundary conditions
In this paper, we study a coupled system of nonlinear fractional differential equations with multi-point boundary condi- tions. The differential operator is taken in the Riemann-Liouville sense. Applying the Schauder fixed-point theorem and the contrac- tion mapping principle, two existence results are obtained for the following system D^{alpha}_{0+}x(t)=fleft(t,y(t),D^{p}_{0+}y(t)right), t in (0,...
متن کاملA Constructive Method for Numerically Computing Conformal Mappings for Gearlike Domains
The Riemann mapping theorem asserts that the open unit disk D = {z| |z| < 1} is conformally equivalent to each simply connected domain G in the complex plane, whose boundary consists of at least two points, i.e., there exists a function f , analytic and univalent function on D , such that f maps D onto G . More precisely, if do is an arbitrary point in D and go is an arbitrary point in G, then ...
متن کاملMath 506: Complex Variables
Part I: Single Variables: Review and extensions. CR equations and analyticity; Cauchy-Goursat theorem and Cauchy integral formula, Louiville’s theorem; Morera’s theorem; maximum modulus principle; Laurent series and singularities; Riemann extension theorem; residues; Schwartz’s lemma; open mapping theorem; analytic continuation and the dilogarithm; linear fractional transformations; spaces of a...
متن کاملThe Riemann Mapping Theorem for semianalytic domains and o-minimality
We consider the Riemann Mapping Theorem in the case of a bounded simply connected and semianalytic domain. We show that the germ at 0 of the Riemann map (i.e. bihilomorphic map) from the upper half plane to such a domain can be realized in a certain quasianalytic class used by Ilyashenko in his work on Hilbert’s 16 problem if the angle of the domain at the boundary point to which 0 is mapped, i...
متن کاملA note on spectral mapping theorem
This paper aims to present the well-known spectral mapping theorem for multi-variable functions.
متن کامل