Solution
= Solution
Green's identity with $\partial G/\partial n=0$ leaves only the prescribed normal derivative. Since $\partial u/\partial n=-p$ and, on the plane,
$$
G=-\frac1{2\pi\sqrt{(x-x_0)^2+(y-y_0)^2+z_0^2}},
$$
the decaying solution is
$$
\boxed{
u(x_0,y_0,z_0)
=-\frac1{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
\frac{p(x,y)}{\sqrt{(x-x_0)^2+(y-y_0)^2+z_0^2}},dx,dy}.
$$
Thus
$$
\boxed{f(X,Y,z_0)=-\frac1{2\pi\sqrt{X^2+Y^2+z_0^2}}.}
$$