Poisson Integral

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Poisson Integral

 

(1) An integral of the form

where r and Φ are polar coordinates and θ is a parameter that varies over the closed interval [0, 2π]. Poisson’s integral expresses the values of a function u(r, Φ) that is harmonic within a circle of radius R in terms of the function’s values f(θ) on the boundary of this circle. The function u(r, Φ) is the solution of the Dirichlet problem for the circle. The Poisson integral was first examined by S. D. Poisson in 1823. A rigorous theory of the Poisson integral was constructed by H. Schwarz in 1869.

(2) The integral

which is encountered in probability theory and in certain problems in mathematical physics. Poisson suggested an extremely simple method of calculating this integral. Since the integral was first calculated in 1729 by L. Euler, it is sometimes called the Euler-Poisson integral.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
We denote the Smirnov class by [N.sub.*](U), which consists of all holomorphic functions f on U such that log(1 + [absolute value of (f(z))]) [less than or equal to] Q[[phi]](z) (z [member of] U) for some [phi] [member of] [L.sup.1](T), [phi] [greater than or equal to] 0, where the right side denotes the Poisson integral of [phi] on U.
and [N.sub.*](D) the set of all holomorphic functions f on D satisfying log(1 + [absolute value of (f(z))]) [less than or equal to] P[[phi]](z) (z [member of] D) for some [phi] [member of] [L.sup.1](R), [phi] [greater than or equal to] 0, where the right side denotes the Poisson integral of f on D.
Our aim is to find the solution of the Dirichlet boundary value problem for the Poisson equation through the Poisson integral formula.
Note that X(x) is the Poisson integral of u(y')[[chi].sub.B(t)] W), where [[chi].sub.B(t)] is the characteristic function of B(t).
In [12], Neumann's function for the sphere in [R.sup.3] is constructed using the classical method of images and expressed in terms of eigenvalues associated with the surface, leading to an analogue of the Poisson integral as a solution to the Neumann problem for the sphere.
Topuria (Georgian Technical U., Georgia) investigates the boundary properties of the differentiated Poisson integral for various domains such as circle, ball, half-plane, half-space, and bicylinder.
While intended for relative novices in the field, the author notes that readers should be acquainted with basic and complex analysis and the theory of the Poisson integral in the unit disk.