Solution
= Solution
Reflect the source point $\mathbf r_0=(x_0,y_0,z_0)$ across the plane to $\mathbf r_0^*=(x_0,y_0,-z_0)$. Equal-sign source and image make the normal derivatives cancel on the plane. Thus the <method of images> gives the Neumann Green function
$$
\boxed{
G(\mathbf r;\mathbf r_0)
=-\frac1{4\pi}
\left(\frac1{|\mathbf r-\mathbf r_0|}
+\frac1{|\mathbf r-\mathbf r_0^*|}\right)},
\qquad
\left.\frac{\partial G}{\partial z}\right|_{z=0}=0.
$$