= Solution
The domain is the three-dimensional upper <half-space>. Reflect the source point to $\mathbf r'^*=(x',y',-z')$. The <method of images> places an opposite source at this reflected point, giving the <Dirichlet image kernel for the three-dimensional modified Helmholtz equation>
$$
\boxed{G_D(\mathbf r,\mathbf r')=-\frac1{4\pi}\left[\frac{e^{-\kappa|\mathbf r-\mathbf r'|}}{|\mathbf r-\mathbf r'|}-\frac{e^{-\kappa|\mathbf r-\mathbf r'^*|}}{|\mathbf r-\mathbf r'^*|}\right],\qquad\kappa=|k|.}
$$
On $z=0$, the two <Euclidean distances> are equal, so the terms cancel exactly. The image lies outside the domain, so its contribution satisfies the homogeneous <modified Helmholtz equation> everywhere inside; only the original source contributes a <Dirac delta>. Both terms decay at infinity, including for $k=0$, and the homogeneous <Dirichlet boundary condition> follows.
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