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In mathematics, and specifically 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 disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson.
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
where f(z) denotes the extension of f by Poisson integral. F is holomorphic in the unit disk with |F(z)| ≤ 1. The restriction of F to a countable family of concentric circles gives a sequence of functions in L ∞ (T) which has a weak g limit in L ∞ (T) with Poisson integral F. By the L 2 results, g is the radial limit for almost all angles ...
For example, the solution to the Dirichlet problem for the unit disk in R 2 is given by the Poisson integral formula. If f {\displaystyle f} is a continuous function on the boundary ∂ D {\displaystyle \partial D} of the open unit disk D {\displaystyle D} , then the solution to the Dirichlet problem is u ( z ) {\displaystyle u(z)} given by
A singular integral of non-convolution type T associated to a Calderón–Zygmund kernel K is called a Calderón–Zygmund operator when it is bounded on L 2, that is, there is a C > 0 such that ‖ ‖ ‖ ‖, for all smooth compactly supported ƒ.
the Poisson kernel is the real part of the integrand above; the real part of a holomorphic function is harmonic and determines the holomorphic function up to addition of a scalar; the above formula defines a holomorphic function, the real part of which is given by the previous theorem
This page was last edited on 11 June 2009, at 02:03 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may ...
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.