Complex Analysis: Theory and Applications · Problem 5.2.7

Elementary Functions — Howell & Matthews

Mathematics — Complex Analysis · harmonic-functions

Problem

Show that the following are harmonic functions in the right half-plane

\{z : \mathrm{Re}\, z > 0\}

. ((a))

u(x,y) = \ln(x^2 + y^2)

. ((b))

v(x,y) = \arctan\left(\frac{y}{x}\right)