Complex Analysis: Theory and Applications · Problem 10.2.10

Applications of Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · dirichlet-problem-harmonic-functions

Problem

Find the function

\phi(x,y)

that is harmonic in the portion of the upper half-plane

\operatorname{Im}(z) > 0

that lies outside the circle

|z| = 1

and has the boundary values

\phi(x,y) = 1 \text{ for } x + iy = z = e^{i\theta},\; 0 < \theta < \pi;

\phi(x,0) = 0 \text{ for } |x| > 1.

Hint: Use the mapping

w = -\frac{1}{z}

and the result of Example 10.9.