Complex Analysis: Theory and Applications · Problem 10.3.4

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 ϕ(x,y)=e−ysin⁡x=yπ∫−∞∞sin⁡t(x−t)2+y2 dt.\phi(x,y) = e^{-y}\sin x = \frac{y}{\pi}\int_{-\infty}^{\infty} \frac{\sin t}{(x-t)^2+y^2}\,dt.

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