The Laplace equation in plane polar coordinates has the form

$$\nabla^{2} \phi=\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}\right] \phi(r, \theta)=0 .$$

Using separation of variables, derive the general solution to the equation that is singlevalued in the domain $1<r<2$.

For

$$f(\theta)=\sum_{n=1}^{\infty} A_{n} \sin n \theta$$

solve the Laplace equation in the annulus with the boundary conditions:

$$\nabla^{2} \phi=0, \quad 1<r<2, \quad \phi(r, \theta)= \begin{cases}f(\theta), & r=1 \\ f(\theta)+1, & r=2\end{cases}$$