Complex Analysis: Theory and Applications · Problem 10.2.4

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 upper half-plane

\operatorname{Im}(z) > 0

and has the boundary values

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