There are at least two integrals called the Poisson integral. The first is also known as Bessel's second integral,

(1)

where  is a Bessel function of the first kind and  is a gamma function. It can be derived from Sonine's integral. With , the integral becomes Parseval's integral.

In complex analysis, let  be a harmonic function on a neighborhood of the closed disk , then for any point  in the open disk ,

(2)

In polar coordinates on ,

(3)

where  and  is the Poisson kernel. For a circle,

(4)

For a sphere,

(5)

where

(6)

REFERENCES:

Krantz, S. G. "The Poisson Integral." §7.3.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 92-93, 1999.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 373-374, 1953.