 Solution

Set $\kappa =|k|$. The differential operator in the modified Helmholtz equation depends only on $k^2={\kappa }^2$. Let $G$ be its decaying Green function. Acting on the proposed integral with the operator, in the sense of distributions, givesThe last step is the sifting property of the Dirac delta. ThusFor a regular rapidly decaying source, the $1/|\mathbf{r}-{\mathbf{r}}^{\prime }|$ singularity of the kernel found in part (ii) is locally integrable in three dimensions, and its convolution with the source tends to zero at infinity. This verifies both the equation and the decay condition. The sign of the kernel below corresponds to ${\nabla }^2-k^2$; reversing the sign of the entire operator reverses the kernel.