A First Course in Complex Analysis · Problem 2.5.27

Differentiation — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · harmonic-functions

Problem

Suppose

u

is a harmonic function. Show that

v

defined (up to a constant) by

v(x,y) = \int_{(x_0,y_0)}^{(x,y)} \left( -u_y \, dx + u_x \, dy \right)

is a harmonic conjugate of

u