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The domain is the three-dimensional upper half-space. Reflect the source point to ${\mathbf{r}}^{\prime \ast }=(x^{\prime },y^{\prime },-z^{\prime })$. The method of images places an opposite source at this reflected point, giving the Dirichlet image kernel for the three-dimensional modified Helmholtz equationOn $z=0$, 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.

Reflect the source across the plane, defining ${\mathbf{r}}^{\prime \ast }=(x^{\prime },y^{\prime },-z^{\prime })$ for an interior point $z^{\prime }>0$. Superpose the direct Green function from part (b) and an image of opposite sign: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 $k=0$.