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.
Read more about this topic: Poisson Kernel
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