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) |
