Explain poisson kernel and harmonic function

Reference no: EM13124360

Poisson Kernel and Harmonic Function

Let P_r(t)=R((1+z)/(1-z)), z=re^it be the Poisson kernel for the unit disc |z|<1.

Let U(theta) be a continous function of the interval [0,pi] with U(0)=U(pi)=0. Show that the function u(re^itheta)=1/2pi(integral from 0 to pi of {P_r(t-theta)-P_r(t+theta)}U(t)dt is harmonic in the half-disc {re^itheta,0<=r<1, 0<=theta<=pi} and has the following limiting behavior on the boundary: limz->e^itheta_0(u(z))=U(theta_0), where 0<theta_0<pi and u(x)=0, -1<x<1

Reference no: EM13124360

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