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Poisson Integral

There are at least two integrals called the Poisson integral. The first is also known as Bessel's second integral,

J_n(z)=((1/2z)^n)/(Gamma(n+1/2)Gamma(1/2))int_0^picos(zcostheta)sin^(2n)thetadtheta,

(1)

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),

u(z_0)=1/(2pi)int_0^(2pi)u(e^(ipsi))(1-|z_0|^2)/(|z_0-e^(ipsi)|^2)dpsi.

(2)

In polar coordinates on D^_(0,R),

u(z_0)=1/(2pi)int_0^(2pi)K(r,theta)phi(z_0+re^(itheta))dtheta,

(3)

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,

u(x,y,z)=1/(4pia)intint_(S)u(a^2-R^2)/((a^2+R^2-2aRcostheta)^(3/2))dS,

(5)

where

costheta=x·xi.

(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.

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Poisson Integral

Cite this as:

Weisstein, Eric W. "Poisson Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PoissonIntegral.html

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