I want to apply something like a stationary phase approximation to the following expression
$\int_V d^3x' \frac{B(x')}{|x-x'|}e^{ik|x-x'|}$
with $x\in \mathbb{R}^3$, $k\rightarrow \infty$ and $B$ is slowly varying function over the Volume $V$.
For better clearness one can assume the volume $V$ to be a finite box $V=[0,L_x]\times [0,L_y]\times [0,L_z]$ and one can for example assume $B=\sin(\pi x/L_x)^2\sin(\pi y/L_y)^2\sin(\pi z/L_z)^2$, where $L_{x,y,z}\gg \frac{2\pi}{k}$.
My problem is that I cannot apply the standard method for stationary phases, e.g. https://en.wikipedia.org/wiki/Stationary_phase_approximation because my function $g=\frac{B(x')}{|x-x'|}$ is not smooth at $x=x'$ and the phase function $f=|x-x'|$ is not differentiable at $x=x'$...
The answer to my question at lowest order should be
$\int_V d^3x' \frac{B(x')}{|x-x'|}e^{ik|x-x'|}=-\frac{4\pi}{k^2}B(x)$, but I don't know how to show that rigorously...
Many thanks in advance