Poisson Integral
There are at least two integrals called the Poisson integral. The first is also known as Bessel's second integral,
where J_n(z) is a Bessel function of the first kind and Gamma(x) is a gamma function. It can be derived from Sonine's integral. With n=0, the integral becomes Parseval's integral.
In complex analysis, let u:U->R be a harmonic function on a neighborhood of the closed disk D^_(0,1), then for any point z_0 in the open disk D(0,1),
In polar coordinates on D^_(0,R),
where R=|z_0| and K(r,theta) is the Poisson kernel. For a circle,
| u(x,y)=1/(2pi)int_0^(2pi)u(acosphi,asinphi)(a^2-R^2)/(a^2+R^2-2aRcos(theta-phi))dphi. |
(4)
|
For a sphere,
where
| costheta=x·xi. |
(6)
|
See also
Bessel Function of the First Kind, Circle, Harmonic Function, Parseval's Integral, Poisson Kernel, Sonine's Integral, SphereExplore with Wolfram|Alpha
WolframAlpha
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 IntegralCite this as:
Weisstein, Eric W. "Poisson Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PoissonIntegral.html