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
Famous quotes containing the word ball:
“Any balance we achieve between adult and parental identities, between children’s and our own needs, works only for a time—because, as one father says, “It’s a new ball game just about every week.” So we are always in the process of learning to be parents.”
—Joan Sheingold Ditzion, Dennie, and Palmer Wolf. Ourselves and Our Children, by Boston Women’s Health Book Collective, ch. 2 (1978)