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

$P_r(\theta) = \sum_{n=-\infty}^\infty r^{|n|}e^{in\theta} = \frac{1-r^2}{1-2r\cos\theta +r^2} = Re\left(\frac{1+re^{i\theta}}{1-re^{i\theta}}\right).$

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 $\partial D$ into R, then the function u given by

$u(re^{i\theta}) = \frac{1}{2\pi}\int_{-\pi}^\pi P_r(\theta-t)f(e^{it})dt$

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

$P(x,\zeta) = \frac{1-|x|^2}{|x-\zeta|^n}$

where $x\in B$ and $\zeta\in S$ (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.