Laplace equation in polar coordinates
= Laplace equation in polar coordinates
{c}
In plane <polar coordinates>, <Laplace's equation> is
$$
\frac1r\frac{\partial}{\partial r}
\left(r\frac{\partial\phi}{\partial r}\right)
+\frac1{r^2}\frac{\partial^2\phi}{\partial\theta^2}=0.
$$
<Separation of variables> gives radial powers $r^n,r^{-n}$ for positive angular modes and $1,\log r$ for the zero mode.