Definitions

# Poisson kernel

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

$P_r\left(theta\right) = sum_\left\{n=-infty\right\}^infty r^$
>e^{intheta} = frac{1-r^2}{1-2rcostheta +r^2} = Releft(frac{1+re^{itheta}}{1-re^{itheta}}right).

This can be thought of in two ways: either as a function of $r$ and $theta$, or as a family of functions of $theta$ 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 $D = \left\{z:|z|<1\right\}$ is the unit disc in C, and if f is a continuous function from $partial D$ into R, then the function u given by

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

is harmonic in D and agrees with f on the boundary of the disc.

For the ball of radius r, $B_\left\{r\right\}$, in Rn, the Poisson kernel takes the form

$P\left(x,zeta\right) = frac\left\{r^2-|x|^2\right\}\left\{romega _\left\{n\right\}|x-zeta|^n\right\}$

where $xin B_\left\{r\right\}$, $zetain S$ (the surface of $B_\left\{r\right\}$), and $omega _\left\{n\right\}$ is the surface area of the unit ball.

Then, if u(x) is a continuous function defined on S, the corresponding result is that the function P[u](x) defined by

$P\left[u\right]\left(x\right) = int_S u\left(zeta\right)P\left(x,zeta\right)dsigma\left(zeta\right)$

is harmonic on the ball $B_\left\{r\right\}$.