Complex Analysis: Theory and Applications · Problem 10.3.2

Applications of Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · poisson-integral-formula

Problem

Use Poisson's integral formula to find the harmonic function

\phi(x,y)

in the upper half-plane that takes on the boundary values

\phi(t,0) = U(t) = 0, \text{ for } t < 0;

\phi(t,0) = U(t) = t, \text{ for } 0 < t < 1;

\phi(t,0) = U(t) = 1, \text{ for } 1 < t.