Harmonic Function
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Date
2014-02-02
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Publisher
Addis Ababa University
Abstract
Harmonic functions are closely connected to analytic functions. Since the
real and imaginary parts of analytic functions are harmonic functions.
The theories of harmonic functions have many interesting features. Among
these interesting features are mean value property; maximum principle and
these properties with Poisson integral enables to solve Dirichlet problem.
The Poisson integral formula shows that if u(z) is harmonic in a disk and
continuous on the closed disk, then its value at any interior point is com-
pletely determined by its value on the boundary circle.
These facts suggest the following two questions:
(a) Given a real valued bounded piecewise continuous function u(ei_) on the
unit circle, do we obtain a harmonic function v(z) through the Poisson inte-
gral;
v(z) =
1
2_
Z 2_
0
u(ei_)Pr(_ t)d_; z = reit?
(b) If so, do the boundary values of v(z) agree with u(ei_)?
The answer is a_rmative and the unique solution is given by;
v(z) =
(
pu; ifz 2 D(0; 1)
u(z); ifz 2 @D(0; 1)
The above two questions leads to the problem of _nding a function that is
harmonic in a region and has pre assigned values on the boundary which is
known as the Dirichlet problem.
Description
Keywords
Harmonic, Function