A First Course in Complex Analysis · Problem 6.3.10

Power Series — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · harmonic-functions

Problem

Suppose

u(r,\phi)

is a function

\mathbb{R}^2 \to \mathbb{R}

given in terms of polar coordinates. (a) Show that the Laplace equation for

u(r,\phi)

\frac{1}{r}u_r + u_{rr} + \frac{1}{r^2}u_{\phi\phi} = 0.

(b) Show that

u(r,\phi) = r^2\cos(2\phi)

is harmonic on

\mathbb{C}

. Generalize. (c) If

u(r,\phi)

depends only on

r

, when is

u

harmonic? (d) If

u(r,\phi)

depends only on

\phi

, when is

u

harmonic?