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Poisson Kernel

The integral kernel in the Poisson integral, given by

K(psi)=1/(2pi)(1-|z_0|^2)/(|z_0-e^(ipsi)|^2)
(1)

for the open unit disk D(0,1). Writing z_0=re^(itheta) and taking D(0,R) gives

K(r,theta)=1/(2pi)R[(R+re^(itheta))/(R-re^(itheta))]
(2)
=1/(2pi)R[((R+re^(itheta))(R-re^(-itheta)))/((R-re^(itheta))(R-re^(-itheta)))]
(3)
=1/(2pi)R[(R^2-rR(e^(itheta)-e^(-itheta))-r^2)/(R^2-rR(e^(itheta)+e^(-itheta))+r^2)]
(4)
=1/(2pi)R[(R^2+2irRsintheta-r^2)/(R^2-2Rrcostheta+r^2)]
(5)
=1/(2pi)(R^2-r^2)/(R^2-2Rrcostheta+r^2)
(6)

(Krantz 1999, p. 93).

In three dimensions,

u(y)=(R(R^2-a^2))/(4pi)int_0^(2pi)int_0^pi(f(theta,phi)sinthetadthetadphi)/((R^2+a^2-2aRcosgamma)^(3/2)),
(7)

where a=|y| and

cosgamma=y·[Rcosthetasinphi; Rsinthetasinphi; Rcosphi].
(8)

The Poisson kernel for the n-ball is

P(x,z)=1/(2-n)(D_(n)v)(z),
(9)

where D_(n) is the outward normal derivative at point z on a unit n-sphere and

v(z)=|z-x|^(2-n)-|x|^(2-n)|(x)/(|x|^2)|^(2-n).
(10)

Let u be harmonic on a neighborhood of the closed unit disk D^_(0,1), then the reproducing property of the Poisson kernel states that for z in D(0,1),

u(z)=1/(2pi)int_0^(2pi)u(e^(ipsi))(1-|z|^2)/(|z-e^(ipsi)|^2)dpsi
(11)

(Krantz 1999, p. 94).

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