Write down the general form of the solution in polar coordinates $(r, \theta)$ to Laplace's equation in two dimensions.

Solve Laplace's equation for $\phi(r, \theta)$ in $0<r<1$ and in $1<r<\infty$, subject to the conditions

$$\begin{gathered} \phi \rightarrow 0 \quad \text { as } \quad r \rightarrow 0 \text { and } r \rightarrow \infty \\ \left.\phi\right|_{r=1+}=\left.\phi\right|_{r=1-} \quad \text { and }\left.\quad \frac{\partial \phi}{\partial r}\right|_{r=1+}-\left.\frac{\partial \phi}{\partial r}\right|_{r=1-}=\cos 2 \theta+\cos 4 \theta . \end{gathered}$$