A First Course in Complex Analysis · Problem 2.5.19

Differentiation — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · harmonic-functions

Problem

Show that the following functions are harmonic and find a harmonic conjugate: ((a))

u(x,y) = x^2 - y^2

((b))

u(x,y) = \sinh(x)\sin(y)

((c))

u(x,y) = x^3 - 3xy^2

((d))

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

for

(x,y) \neq (0,0)