 Solution

By translation and rotation invariance, use a radial Green function $G(R)$, where $R=|\mathbf{r}-{\mathbf{r}}^{\prime }|$ is the Euclidean distance. Away from the source, the radial modified Helmholtz equation isWriting $u=RG$ reduces this to $u^{\prime \prime }-{\kappa }^{2}u=0$, where $\kappa =|k|$. For $\kappa >0$, decay eliminates $e^{\kappa R}/R$, leaving $G=A{e}^{-\kappa R}/R$. For $\kappa =0$, the two radial branches are a constant and $1/R$; decay again leaves $A/R$.

To fix the singular normalization, integrate the defining equation over a ball of radius $\epsilon$ about ${\mathbf{r}}^{\prime }$. The divergence theorem givesThe volume term is $O({\epsilon }^2)$, whereasConsequentlySource qualification. The printed $e^{-kR}$ is the decaying branch only when $k\ge 0$, although the source declares $k$ real. For negative $k$ it grows, so it cannot be the requested decaying Green function. The absolute value above covers every real $k$ and reduces to the printed form for nonnegative $k$.