Poisson integral

In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson.

Poisson kernels commonly find applications in control theory and two-dimensional problems in electrostatics. In practice, the definition of Poisson kernels are often extended to n-dimensional problems.

In the complex plane, the Poisson kernel for the unit disc is given by

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.

If ${\displaystyle D=\{z:|z|<1\}}$ is the open unit disc in C, T is the boundary of the disc, and f a function on T that lies in L¹(T), then the function u given by

is harmonic in D and has a radial limit that agrees with f almost everywhere on the boundary T of the disc.

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