Solving the equation $-e^{-i2k\ell}=\frac{k-1}{k+1}$ for $k$

Question

I am trying to solve the following equation for $k$: \begin{equation*} -e^{-i2k\ell}=\frac{k-1}{k+1}\, , \end{equation*} where $\ell$ is a positive number (constant).

I would like to know how the solutions of $k$ look like. I expect it to be an infinite number of solutions. Does anyone have any ideas how to prove or show this?

Answer 1

This would be an interesting extension of the generalized Lambert function in the complex domain.

Let $c=2i \ell$ and write the equation as $$e^{-c k}=\frac{k-1}{k+1}$$

Now, have a look at equation $(4)$ in the linked paper.

Answer 2

Let's define :

$e^{-c k}=\frac{k-1}{k+1}$

we can write :

\begin{align} -ce^{-c k}=-\frac{k-1}{k+1}c & \iff -c(k+1)e^{-c(k+1)}=-(\sqrt k - 1)(\sqrt k + 1)ce^{-c} \\ \\ & \iff -c(k+1)e^{-c(k+1)}= -(\sqrt k - 1)(\sqrt k + 1)ce^{-c} \\ \\ & \iff \ln(-c(k+1)) - c(k+1)=- (\sqrt k - 1)(\sqrt k + 1)ce^c \\ \\ & \iff W(-(\sqrt k - 1)(\sqrt k + 1)ce^c) - c(k+1)= \ln(- (\sqrt k - 1)(\sqrt k + 1)ce^c) \\ \\ & \iff k = -\frac{c + \ln(- (\sqrt k - 1)(\sqrt k + 1)c) - W(-(\sqrt k - 1)(\sqrt k + 1)ce^c) + c }{c} \\ \\ & \iff k = -(2 + \ln(- (\sqrt k - 1)(\sqrt k + 1)c) - W(-(\sqrt k - 1)(\sqrt k + 1)ce^c)) \\ \\ & \iff k = -2 - \ln(- (\sqrt k - 1)(\sqrt k + 1)c) + W(-(\sqrt k - 1)(\sqrt k + 1)ce^c)) \\ \end{align}

This express the multiple solutions using the $W(-(\sqrt k - 1)(\sqrt k + 1)ce^c)$ Lambert function with $c=2i \ell$.