= Dirichlet image kernel for the three-dimensional modified Helmholtz equation
{c}
{title2=$G_D(r,r')=-\{e^{-\kappa|r-r'|}/|r-r'|-e^{-\kappa|r-r'^*|}/|r-r'^*|\}/(4\pi)$}
For $\kappa\geq0$ in the upper half-space, let $r'^*$ be the reflection of the interior source $r'$ across its boundary plane. The displayed decaying <Green function> solves $(\Delta-\kappa^2)G_D=\delta(r-r')$ 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.
Back to article page
