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)
|
See also
Bessel Function of the First Kind,
Circle,
Harmonic
Function,
Parseval's Integral,
Poisson
Kernel,
Sonine's Integral,
Sphere
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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.Referenced on Wolfram|Alpha
Poisson Integral
Cite this as:
Weisstein, Eric W. "Poisson Integral."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PoissonIntegral.html
Subject classifications
