Poisson Integral
المؤلف:
Krantz, S. G
المصدر:
"The Poisson Integral." §7.3.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser
الجزء والصفحة:
...
27-12-2018
1164
Poisson Integral
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.
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