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:
“The world was a huge ball then, the universe a might harmony of ellipses, everything moved mysteriously, incalculable distances through the ether.
We used to feel the awe of the distant stars upon us. All that led to was the eighty-eight naval guns, ersatz, and the night air-raids over cities. A magnificent spectacle.
After the collapse of the socialist dream, I came to America.”
—John Dos Passos (1896–1970)