Poisson kernel

Definition

The Poisson kernel is the kernel for the Poisson integral operator, the integral operator that solves the Dirichlet problem for a disk. This kernel is defined by the following expression for the open unit disk centered at the origin:

$P_{r}(\theta )={\frac {1}{2\pi }}{\frac {1-r^{2}}{1-2r\cos \theta +r^{2}}}$

If, instead of the unit disk, we're working with a disk of radius $R$, the Poisson kernel for that is given by:

${\frac {1}{2\pi }}{\frac {R^{2}-r^{2}}{R^{2}-2Rr\cos \theta +r^{2}}}$