Search results
Results From The WOW.Com Content Network
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.
In mathematics, the Poisson boundary is a probability space associated to a random walk. It is an object designed to encode the asymptotic behaviour of the random walk, i.e. how trajectories diverge when the number of steps goes to infinity.
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. [1] The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the ...
Siméon Denis Poisson. Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics.For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate the corresponding electrostatic or gravitational (force) field.
In mathematics, a Green's function (sometimes improperly termed a Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if is a linear differential operator, then
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 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.
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