Solving an equation with the Lambert W function

Question

I am trying to solve the following equation \begin{equation} -e^{-i2k\ell}=\frac{k-1}{k+1} \end{equation} for $k$. I thought it might be an idea to use the Lambert W function to do so, but my calculations are not really working out. Does anyone see if and maybe how this works? I would be really greatful! Thank you very much!

Answer 1

$$e^{-i2k\ell}=-\frac{k-1}{k+1}$$ Make $x=2i k$; this gives $$e^{-\ell x}=-\frac{x-2 i}{x+2 i}$$ and the solution is given in terms of the generalized Lambert function (have a look at equation $(4)$ in the paper).

This is nice from a formal point of view but not very practical. Let $k=a+ib$, cross multiply and separate the real and imaginary parts. We then need to solve $$F(a,b,\ell)=e^{2 b \ell} (b \sin (2 a \ell)+(a+1) \cos (2 a \ell))+a-1=0 \tag 1$$ $$G(a,b,\ell)=e^{2 b \ell} (b \cos (2 a \ell)-(a+1) \sin (2 a \ell))+b=0 \tag 2$$

To visualize where are (more or less precisely) the roots, make a contour plot of the function $$H(a,b,\ell)=F^2(a,b,\ell)+G^2(a,b,\ell)$$ for a given value of $\ell$.

For example, using $\ell=2.345$, there is a root close to $(1.0,-0.4)$ and Newton-Raphson method works like a charm and gives $a=0.978529$ and $b=-0.364096$.