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.