Find the Fourier Series of the function

$$f(\theta)= \begin{cases}1 & 0 \leq \theta<\pi \\ -1 & \pi \leq \theta<2 \pi\end{cases}$$

Find the solution $\phi(r, \theta)$ of the Poisson equation in two dimensions inside the unit disk $r \leq 1$

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

subject to the boundary condition $\phi(1, \theta)=0$.

[Hint: The general solution of $r^{2} R^{\prime \prime}+r R^{\prime}-n^{2} R=r^{2}$ is $R=a r^{n}+b r^{-n}-r^{2} /\left(n^{2}-4\right) .$ ]

From the solution, show that

$$\int_{r \leq 1} f \phi d A=-\frac{4}{\pi} \sum_{n \text { odd }} \frac{1}{n^{2}(n+2)^{2}}$$