Complex Analysis: Theory and Applications · Problem 10.3.3

Applications of Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · poisson-integral-formula

Problem

Use Poisson's integral formula for the upper half-plane to conclude

\phi(x,y) = e^{-y}\cos x = \frac{y}{\pi}\int_{-\infty}^{\infty} \frac{\cos t}{(x-t)^2+y^2}\,dt.