Complex Analysis: Theory and Applications · Problem 3.3.12

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · polar-harmonic-functions

Problem

Use the polar form of Laplace's equation given in Exercise 11 to show that the following functions are harmonic. (a)

u(r,\theta) = \left(r + \dfrac{1}{r}\right)\cos\theta

and

v(r,\theta) = \left(r - \dfrac{1}{r}\right)\sin\theta

. (b)

u(r,\theta) = r^n \cos n\theta

and

v(r,\theta) = r^n \sin n\theta