The base $r < c$, $\theta = \pi/2$ of a solid hemisphere $r \le c$, $0 \le \theta \le \pi/2$ is insulated. The temperature distribution on the hemispherical surface is $u = F(\theta)$. Derive the expression
$\infty$
$u(r, \theta) = \sum_{n=0} A_{2n} \left(\frac{r}{c}\right)^{2n} P_{2n}(\cos\theta)$,
where
$A_{2n} = (4n + 1) \int_0^{\pi/2} F(\theta) P_{2n}(\cos\theta) \sin\theta \,d\theta$
$(n = 0, 1, 2, \dots)$,
for the steady temperatures in the solid. Also, show that $u(r, \theta) = 1$ when $F(\theta) = 1$.
