Poisson Kernel

The integral kernel in the Poisson integral, given by

(1)

for the open unit disk . Writing and taking gives

			(2)
			(3)
			(4)
			(5)
			(6)

(Krantz 1999, p. 93).

In three dimensions,

(7)

where and

(8)

The Poisson kernel for the -ball is

(9)

where is the outward normal derivative at point on a unit -sphere and

(10)

Let be harmonic on a neighborhood of the closed unit disk , then the reproducing property of the Poisson kernel states that for ,

(11)

(Krantz 1999, p. 94).