Laplace's equation for $\phi$ in cylindrical coordinates $(r, \theta, z)$, is

$$\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}+\frac{\partial^{2} \phi}{\partial z^{2}}=0$$

Use separation of variables to find an expression for the general solution to Laplace's equation in cylindrical coordinates that is $2 \pi$-periodic in $\theta$.

Find the bounded solution $\phi(r, \theta, z)$ that satisfies

$$\begin{aligned} \nabla^{2} \phi &=0 \quad z \geqslant 0, \quad 0 \leqslant r \leqslant 1 \\ \phi(1, \theta, z) &=e^{-4 z}(\cos \theta+\sin 2 \theta)+2 e^{-z} \sin 2 \theta \end{aligned}$$