(i) Consider the Poisson equation $\nabla^{2} \psi(\mathbf{r})=f(\mathbf{r})$ with forcing term $f$ on the infinite domain $\mathbb{R}^{3}$ with $\lim _{|\mathbf{r}| \rightarrow \infty} \psi=0$. Derive the Green's function $G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=-1 /\left(4 \pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)$ for this equation using the divergence theorem. [You may assume without proof that the divergence theorem is valid for the Green's function.]

(ii) Consider the Helmholtz equation

$$\tag{†} \nabla^{2} \psi(\mathbf{r})+k^{2} \psi(\mathbf{r})=f(\mathbf{r})$$

where $k$ is a real constant. A Green's function $g\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ for this equation can be constructed from $G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ of (i) by assuming $g\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=U(r) G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ where $r=\left|\mathbf{r}-\mathbf{r}^{\prime}\right|$ and $U(r)$ is a regular function. Show that $\lim _{r \rightarrow 0} U(r)=1$ and that $U$ satisfies the equation

$$\tag{‡} \frac{d^{2} U}{d r^{2}}+k^{2} U(r)=0$$

(iii) Take the Green's function with the specific solution $U(r)=e^{i k r}$ to Eq. ($‡$) and consider the Helmholtz equation $(†)$ on the semi-infinite domain $z>0, x, y \in \mathbb{R}$. Use the method of images to construct a Green's function for this problem that satisfies the boundary conditions

$$\frac{\partial g}{\partial z^{\prime}}=0 \text { on } z^{\prime}=0 \quad \text { and } \quad \lim _{|\mathbf{r}| \rightarrow \infty} g\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=0$$

(iv) A solution to the Helmholtz equation on a bounded domain can be constructed in complete analogy to that of the Poisson equation using the Green's function in Green's 3rd identity

$$\psi(\mathbf{r})=\int_{\partial V}\left[\psi\left(\mathbf{r}^{\prime}\right) \frac{\partial g\left(\mathbf{r}, \mathbf{r}^{\prime}\right)}{\partial n^{\prime}}-g\left(\mathbf{r}, \mathbf{r}^{\prime}\right) \frac{\partial \psi\left(\mathbf{r}^{\prime}\right)}{\partial n^{\prime}}\right] d S^{\prime}+\int_{V} f\left(\mathbf{r}^{\prime}\right) g\left(\mathbf{r}, \mathbf{r}^{\prime}\right) d V^{\prime},$$

where $V$ denotes the volume of the domain, $\partial V$ its boundary and $\partial / \partial n^{\prime}$ the outgoing normal derivative on the boundary. Now consider the homogeneous Helmholtz equation $\nabla^{2} \psi(\mathbf{r})+k^{2} \psi(\mathbf{r})=0$ on the domain $z>0, x, y \in \mathbb{R}$ with boundary conditions $\psi(\mathbf{r})=0$ at $|\mathbf{r}| \rightarrow \infty$ and

$$\left.\frac{\partial \psi}{\partial z}\right|_{z=0}= \begin{cases}0 & \text { for } \rho>a \\ A & \text { for } \rho \leqslant a\end{cases}$$

where $\rho=\sqrt{x^{2}+y^{2}}$ and $A$ and $a$ are real constants. Construct a solution in integral form to this equation using cylindrical coordinates $(z, \rho, \varphi)$ with $x=\rho \cos \varphi, y=\rho \sin \varphi$.