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).

REFERENCES:

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1090, 2000.

Krantz, S. G. "The Poisson Kernel." §7.3.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 93, 1999.