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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 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
where P is the Poisson kernel. Any function f on the disc determines a function on the group of Möbius transformations of the disc by setting F(g) = f(g(0)). Then the Poisson formula has the form = | | = ^ () where m is the Haar measure on the boundary. This function is then harmonic in the sense that it satisfies the mean-value property with ...
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 ...
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 .
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.
When I say 'K' of 'G' above, I mean "used for kernels in general", not the Poisson kernel in particular. The only book I have that actually talks about the Poisson kernel is a slim volume on linear operators, that, while good, wouldn't be authoritative. (although it does use K). linas 02:53, 5 November 2008 (UTC) I understood.
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 , = (/).