Conformal Mappings and the Riemann Mapping Theorem
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Date
2010-06-06
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Addis Ababa University
Abstract
It is easy to point out that any function of a complex-variable can be
considered as a mapping from one complex plane into another. A great deal
of attention is devoted to the study of holomorphic functions. The reason for
this is that many problems from the theory of holomorphic functions can be
solved according to the following procedures;
1. Solve the problem for the simplest possible type of domain;
2. Express the desired solution in terms of the one already found with
the aid of a mapping.
A non-constant holomorphic function maps a domain of
the z −planeonto another domain f (z) of the w−plane .At points where
'
f z 0 such a map has the remarkable property that it is conformal. This
means that any two smooth curves intersecting in map into curves which
intersect at the same angle in f .By means of conformal mapping,
problems of fluid flow, electrostatics and other fields can be mapped into
simpler problems of the same general sort in f .Solution of the problem
in f then solves the original problem in .Conformal mapping also
gives geometrical insight into analytic(holomorphic) questions. This seminar
paper includes definitions and basic properties of holomorphic functions
,conformal mappings(the bilinear transformations, The Schwarz-Christoffel
transformation),Normal Families, the Riemann Mapping Theorem, which
characterizes those domains that can be mapped conformally onto the unit
open disk and concludes with one of the consequences of the Riemann
mapping theorem.
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Keywords
Conformal, Mappings, Riemann Mapping, Theorem