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)
|
