Poisson kernel
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In potential theory, the Poisson kernel is the derivative of the Green's function for the two-dimensional Laplace equation, under circular symmetry, using Dirichlet boundary conditions. It is used for solving the two-dimensional Dirichlet problem.
In practice, there are many different forms of the Poisson kernel in use. For example, in complex analysis, the Poisson kernel for a disc is often used as is the Poisson kernel for the upper half-plane, and both of these can be extended into n-dimensional space.
In the complex plane, the Poisson kernel for the unit disc is given by
This can be thought of in two ways: either as a function of r and θ, or as a family of functions of θ indexed by r.
One of the main reasons for the importance of the Poisson kernel in complex analysis is that the Poisson integral of the Poisson kernel gives a solution of the Dirichlet problem for the disc. The Dirichlet problem asks for a solution to Laplace's equation on the unit disk, subject to the Dirichlet boundary condition. If D = {z: | z | < 1} is the unit disc in C, and if f is a continuous function from 
into R, then the function u given by
is harmonic in D and agrees with f on the boundary of the disc.
For the unit ball B in Rn, the Poisson kernel takes the form
where 
and 
(the surface of B).
Then, if u(x) is a continuous function defined on S, the corresponding result is that the function P[u](x) defined by
-
P[u](x) = ∫ u(ζ)P(x,ζ)dσ(ζ) S
is harmonic on the ball B.
[edit] References
- John B. Conway (1978). Functions of One Complex Variable I. Springer-Verlag. ISBN 0-387-90328-3.
- S. Axler, P. Bourdon, W. Ramey (1992). Harmonic Function Theory. Springer-Verlag. ISBN 0-387-95218-7.
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