Poisson Kernel - On The Ball

On The Ball

For the ball of radius r, in Rn, the Poisson kernel takes the form

where, (the surface of ), and is the surface area of the unit sphere.

Then, if u(x) is a continuous function defined on S, the corresponding Poisson integral is the function P(x) defined by

It can be shown that P(x) is harmonic on the ball and that P(x) extends to a continuous function on the closed ball of radius r, and the boundary function coincides with the original function u.