I have the following equation $ $ \frac{2\kappa}{(k+\kappa)^2}=2i\ell e^{-2ik\ell} $ $ with $ \kappa, \ell \in \mathbb{R}$ and $ k\in \mathbb{C}$ which I want to solve for $ k$ .

Using

`Reduce[((2*κ)/(k+κ)^2)-2*x*I*Exp[-2*I*l*k]==0, k] `

Mathematica gives me the following solution:

$ (k\neq 0\land l=0\land \kappa =0)\lor \left(l\neq 0\land \kappa =-\frac{i}{l}\land k=0\right)\lor \left(c_1\in \mathbb{Z}\land \sqrt{-i \kappa l e^{-2 i \kappa l}}\neq 0\land l\neq 0\land \kappa +k\neq 0\land \left(k=\frac{-\kappa l+i \cdot\text{ProductLog}\left[c_1,-i \sqrt{-i \kappa l e^{-2 i \kappa l}}\right]}{l}\lor k=\frac{-\kappa l+i \cdot\text{ProductLog}\left[c_1,i \sqrt{-i \kappa l e^{-2 i \kappa l}}\right]}{l}\right)\right)$

I don’t really know how to deal with the ProductLog. I looked it up and know, that ProductLog$ [z]$ gives the principal solution for $ w$ in $ z=we^w$ , which is the Lambert $ W$ function. But what does this actually mean? Is there a way for me to compute without Mathematica, that this is a solution of my equation above?

Or are there maybe any better commands than "Reduce" to find solutions?

Thank you very much for your help!