Show that Laplace's equation $\nabla^{2} \phi=0$ in polar coordinates $(r, \theta)$ has solutions proportional to $r^{\pm \alpha} \sin \alpha \theta, r^{\pm \alpha} \cos \alpha \theta$ for any constant $\alpha$.

Find the function $\phi$ satisfying Laplace's equation in the region $a<r<b, 0<\theta<\pi$, where $\phi(a, \theta)=\sin ^{3} \theta, \phi(b, \theta)=\phi(r, 0)=\phi(r, \pi)=0$.

[The Laplacian $\nabla^{2}$ in polar coordinates is

$$\left.\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}\right]$$