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.
The Poisson kernel is important in complex analysis because its integral against a function defined on the unit circle — the Poisson integral — gives the extension of a function defined on the unit circle to a harmonic function on the unit disk. By definition, harmonic functions are solutions to Laplace's equation, and, in two dimensions, harmonic functions are equivalent to meromorphic functions. Thus, the two-dimensional Dirichlet problem is essentially the same problem as that of finding a meromorphic extension of a function defined on a boundary.
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.
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