= Dirichlet image kernel for the three-dimensional Helmholtz equation
{c}
{title2=$G_D=-[e^{ik|r-r'|}/|r-r'|-e^{ik|r-r'^*|}/|r-r'^*|]/(4\pi)$}
For the upper half-space and the sign convention $(\Delta+k^2)G_D=\delta_{r'}$, reflect the source $r'=(x',y',z')$ to $r'^*=(x',y',-z')$ and use the displayed difference. This is the negative of the conventional outgoing kernel, whose source sign is $-\delta$. The <method of images> makes the <Dirichlet boundary condition> zero on the plane. With $R=|r'-r|$ and $\cos\theta=z'/R$, its outward normal derivative there is
$$
\boxed{\partial_nG_D=-\frac{ik}{2\pi}\frac{e^{ikR}}R\left(1+\frac{i}{kR}\right)\cos\theta.}
$$
The outward <unit normal> is $-e_z$. Differentiating $e^{ikR}/R$ gives $e^{ikR}(ik/R-1/R^2)$; the real and reflected source contributions add in the normal derivative. <Green's second identity> therefore expresses a solution of the <Helmholtz equation> as $\phi(r')=\int_{z=0}\phi(r)\partial_nG_D(r,r')\,dS$ when the boundary term at infinity vanishes. Suitable decay or compatible outgoing <Sommerfeld radiation conditions> are needed; vanishing at infinity without further control is not a general uniqueness condition for a real-frequency Helmholtz problem.
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