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:

$z_{0}=re^{i\theta }\mapsto P_{r}(\theta )={\frac {1}{2\pi }}{\frac {1-r^{2}}{1-2r\cos \theta +r^{2}}}$

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

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

As the real part of a holomorphic function

We can also define the Poisson kernel as the real part of the following holomorphic function on the open unit disk:

${\frac {1}{2\pi }}{\frac {1+z}{1-z}}$

here $z=re^{i\theta }$

This also tells us that the harmonic conjugate of the Poisson kernel is given by the function:

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