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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.
In mathematics, the Poisson formula, named after Siméon Denis Poisson, may refer to: Poisson distribution in probability; Poisson summation formula in Fourier analysis; Poisson kernel in complex or harmonic analysis; Poisson–Jensen formula in complex analysis
The next steps in the study of the Dirichlet's problem were taken by Karl Friedrich Gauss, William Thomson (Lord Kelvin) and Peter Gustav Lejeune Dirichlet, after whom the problem was named, and the solution to the problem (at least for the ball) using the Poisson kernel was known to Dirichlet (judging by his 1850 paper submitted to the ...
The proof utilizes the symmetry of the Poisson kernel using the Hardy–Littlewood maximal function for the circle.; The analogous theorem is frequently defined for the Hardy space over the upper-half plane and is proved in much the same way.
Let P r denote the Poisson kernel on the unit circle T. For a distribution f on the unit circle, set () = < < | () |, where the star indicates convolution between the distribution f and the function e iθ → P r (θ) on the circle.
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.
Let be the unit disk in the complex plane with the Poincaré metric = | | (| |). Then, for | | < and | | =, the Busemann function is given by [2] = (| | | |),where the term in brackets on the right hand side is the Poisson kernel for the unit disk and corresponds to the radial geodesic from the origin towards , = (/).
In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane. It is one of the few stable distributions with a probability density function that can be expressed analytically, the others being the normal distribution and the Lévy distribution.