Dirichlet problem in the circle and the Poisson kernel

Using series solve $\Delta u = 0$, $u(1,\theta) = 1+ \sum\limits_{n=1}^\infty \frac{1}{n^2}\sin(n\theta)$.

Using the series solution find the solution to $\Delta u = 0$, $u(1,\theta) = 1- \cos(\theta)$. Express the solution in Cartesian coordinates (that is, using $x$ and $y$).

1. Try and guess a solution to $\Delta u = -1$, $u(1,\theta) = 0$. Hint: Try a solution that only depends on $r$. Also, don't worry about the boundary condition at first.
2. Now solve $\Delta u = -1$, $u(1,\theta) = \sin(2\theta)$ using superposition.

Derive the Poisson kernel solution if the region is a circle of radius $\rho$ rather than 1. That is, solve $\Delta u = 0$, $u(\rho,\theta) = g(\theta)$.