Let $x = (x_1, x_2, x_3)$ be a point in $\mathbb{R}^3$. I am trying to calculate
$$\Delta \frac{e^{ik|x|}}{|x|}$$
Doing this 'manually' be calculating second derivatives is really long and tedious due to the Euclidean distance function $|\cdot|$. Is there some identities that can be used to make it more efficient, and clearer?
Use spherical polar coordinates. Your function becomes $\frac{e^{ikr}}{r}$ with $r \in \mathbb{R}.$ And it is well known that $$\Delta f =\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial f}{\partial r}) + \frac{1}{r^2 \sin \varphi} \frac{\partial}{\partial \varphi}(\sin \varphi \frac{\partial f}{\partial \varphi}) + \frac{1}{r^2 \sin^2 \varphi} \frac{\partial^2 f}{\partial \theta^2}.$$ You will only have to compute the first term.