Let $k$ be a complex number such that $$im \;k>0$$
the function
$$u(x,y):=\frac{1}{4\pi} \frac{e^{ik|x-y|}}{|x-y|}, x,y\in \mathbb{R^3}, x\neq y$$
is a solution of the Helmholtz equation
what i tried is :
the equation $\nabla^2 u(\vec x)+k^2u(\vec x)=0$
reduced to $\nabla^2 u+k^2 u=\frac1{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial u}{\partial r}\right)+k^2u=0$ by using spherical symmetry
which can be $\frac{\partial^2}{\partial r^2}(r u)+k^2 (ru)=0 $
then $ru=C_1 \cos(kr)+C_2 \sin (kr)$ from here how to processed
thank you so much