= Solution
Let $\mathbf x_0^-=(x_0,y_0,-z_0)$ be the reflection of the source across the plane. The <method of images> gives the <Dirichlet Green function>
$$
\boxed{G_H(\mathbf x;\mathbf x_0)=-\frac1{4\pi}\left(\frac1{|\mathbf x-\mathbf x_0|}-\frac1{|\mathbf x-\mathbf x_0^-|}\right).}
$$
The two source distances agree on $z=0$, so the boundary value vanishes. The reflected source lies outside the upper half-space, leaving only the required <Dirac delta distribution> inside. Both terms vanish at infinity.
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