The solution to the Dirichlet problem on the half-space $D=\{\mathbf{x}=(x, y, z): z>0\}$ :

$$\nabla^{2} u(\mathbf{x})=0, \quad \mathbf{x} \in D, \quad u(\mathbf{x}) \rightarrow 0 \quad \text { as } \quad|\mathbf{x}| \rightarrow \infty, \quad u(x, y, 0)=h(x, y)$$

is given by the formula

$$u\left(\mathbf{x}_{0}\right)=u\left(x_{0}, y_{0}, z_{0}\right)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} h(x, y) \frac{\partial}{\partial n} G\left(\mathbf{x}, \mathbf{x}_{0}\right) d x d y$$

where $n$ is the outward normal to $\partial D$.

State the boundary conditions on $G$ and explain how $G$ is related to $G_{3}$, where

$$G_{3}\left(\mathbf{x}, \mathbf{x}_{0}\right)=-\frac{1}{4 \pi} \frac{1}{\left|\mathbf{x}-\mathbf{x}_{0}\right|}$$

is the fundamental solution to the Laplace equation in three dimensions.

Using the method of images find an explicit expression for the function $\frac{\partial}{\partial n} G\left(\mathbf{x}, \mathbf{x}_{0}\right)$ in the formula.