Let $\Omega$ be a bounded region in the plane, with smooth boundary $\partial \Omega$. Green's second identity states that for any smooth functions $u, v$ on $\Omega$

$$\int_{\Omega}\left(u \nabla^{2} v-v \nabla^{2} u\right) \mathrm{d} x \mathrm{~d} y=\oint_{\partial \Omega} u(\mathbf{n} \cdot \nabla v)-v(\mathbf{n} \cdot \nabla u) \mathrm{d} s$$

where $\mathbf{n}$ is the outward pointing normal to $\partial \Omega$. Using this identity with $v$ replaced by

$$G_{0}\left(\mathbf{x} ; \mathbf{x}_{0}\right)=\frac{1}{2 \pi} \ln \left(\left\|\mathbf{x}-\mathbf{x}_{0}\right\|\right)=\frac{1}{4 \pi} \ln \left(\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}\right)$$

and taking care of the singular point $(x, y)=\left(x_{0}, y_{0}\right)$, show that if $u$ solves the Poisson equation $\nabla^{2} u=-\rho$ then

$$\begin{aligned} u(\mathbf{x})=-\int_{\Omega} G_{0}\left(\mathbf{x} ; \mathbf{x}_{0}\right) \rho\left(\mathbf{x}_{0}\right) \mathrm{d} x_{0} \mathrm{~d} y_{0} \\ &+\oint_{\partial \Omega}\left(u\left(\mathbf{x}_{0}\right) \mathbf{n} \cdot \nabla G_{0}\left(\mathbf{x} ; \mathbf{x}_{0}\right)-G_{0}\left(\mathbf{x} ; \mathbf{x}_{0}\right) \mathbf{n} \cdot \nabla u\left(\mathbf{x}_{0}\right)\right) \mathrm{d} s \end{aligned}$$

at any $\mathbf{x}=(x, y) \in \Omega$, where all derivatives are taken with respect to $\mathbf{x}_{0}=\left(x_{0}, y_{0}\right)$.

In the case that $\Omega$ is the unit disc $\|\mathbf{x}\| \leqslant 1$, use the method of images to show that the solution to Laplace's equation $\nabla^{2} u=0$ inside $\Omega$, subject to the boundary condition

$$u(1, \theta)=\delta(\theta-\alpha),$$

is

$$u(r, \theta)=\frac{1}{2 \pi} \frac{1-r^{2}}{1+r^{2}-2 r \cos (\theta-\alpha)}$$

where $(r, \theta)$ are polar coordinates in the disc and $\alpha$ is a constant.

[Hint: The image of a point $\mathbf{x}_{0} \in \Omega$ is the point $\mathbf{y}_{0}=\mathbf{x}_{0} /\left\|\mathbf{x}_{0}\right\|^{2}$, and then

$$\left\|\mathbf{x}-\mathbf{x}_{0}\right\|=\left\|\mathbf{x}_{0}\right\|\left\|\mathbf{x}-\mathbf{y}_{0}\right\|$$

for all $\mathbf{x} \in \partial \Omega .]$