Poisson kernel

The weighting function that reconstructs a harmonic function on a disk from its boundary values.

The Poisson kernel is the weighting function used to recover a harmonic function's values inside a disk from its prescribed values on the boundary circle, by integrating the boundary data against the kernel around the circle. It is built directly from the disk's geometry -- it weights boundary values near a given interior point more heavily than distant ones -- and the resulting integral, the Poisson integral formula, is the disk's solution to the Dirichlet problem, which conformal mapping then transplants to any region that can be mapped to a disk.

Examples

For the unit disk, the Poisson kernel weights the boundary value at angle $\phi$ more heavily the closer $\phi$ is to the angle of the interior point in question.
Solving the Dirichlet problem on a square by first mapping it to the unit disk still ends with an integral against the disk's Poisson kernel.