= Solution
Reflect the source across the plane, defining $\mathbf r'^*=(x',y',-z')$ for an interior point $z'>0$. Superpose the direct <Green function> from part (b) and an image of opposite sign:
$$
\boxed{\Psi(\mathbf r,\mathbf r')=-\frac1{4\pi}\left[\frac{e^{-\kappa|\mathbf r-\mathbf r'|}}{|\mathbf r-\mathbf r'|}-\frac{e^{-\kappa|\mathbf r-\mathbf r'^*|}}{|\mathbf r-\mathbf r'^*|}\right],\qquad\kappa=|k|.}
$$
The direct term supplies the delta source in the upper half-space; the image singularity lies outside that domain and contributes no source there. On $z=0$ the two distances are equal, so the terms cancel, giving homogeneous <Dirichlet boundary conditions>. Both decay at infinity, which is the additional condition selecting this half-space solution. This is the <method of images> realization of the <Dirichlet image kernel for the three-dimensional modified Helmholtz equation>. It includes the unscreened case $k=0$.
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