Chapter 13.3, Problem 6E

The steady-state temperature in a hemisphere of radius r = c is determined from

$\frac{{\partial }^{2}u}{\partial r^2}+\frac{2}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\theta }^2}+\frac{\cot \theta }{r^2}\frac{\partial u}{\partial \theta }=0,$

0 < r < c, 0 < θ < π/2

u(r, π/2) = 0, 0 < r < c

u(r, θ) = f(θ), 0 < θ < π/2

Solve for u(r, θ). [Hint: P_n(0) = 0 only if n is odd. Also see Problem 20 in Exercises 11.5.]