Dirichlet image kernel for the three-dimensional modified Helmholtz equation ($G_D(r,r^{\prime })=-\{e^{-\kappa |r-r^{\prime }|}/|r-r^{\prime }|-e^{-\kappa |r-r^{\prime \ast }|}/|r-r^{\prime \ast }|\}/(4\pi )$)

For $\kappa \ge 0$ in the upper half-space, let $r^{\prime \ast }$ be the reflection of the interior source $r^{\prime }$ across its boundary plane. The displayed decaying Green function solves $(\Delta -{\kappa }^2)G_D=\delta (r-r^{\prime })$ with zero Dirichlet boundary condition. The direct term supplies the source; the image source lies outside the domain and its opposite sign makes the two terms cancel on the plane. The minus sign relative to the positive static Yukawa Green function comes from the operator convention. The case $\kappa =0$ is the Coulomb limit. An infinity condition is needed for uniqueness on an unbounded domain.