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.

Read more about this topic:  Poisson Kernel

Famous quotes containing the word ball:

    Don’t tell me what delusion he entertains regarding God, or what mountebank he follows in politics, or what he springs from, or what he submits to from his wife. Simply tell me how he makes his living. It is the safest and surest of all known tests. A man who gets his board and lodging on this ball in an ignominious way is inevitably an ignominious man.
    —H.L. (Henry Lewis)