Inverse of $\frac{e^{iw}(-1+e^{imw})}{(-1+e^{iw})}+\frac{e^{-imw}(-1+e^{imw})}{(-1+e^{iw})}$

Question

I would like to compute the inverse of the function

$$f(w)=\frac{e^{iw}(-1+e^{imw})}{(-1+e^{iw})}+\frac{e^{-imw}(-1+e^{imw})}{(-1+e^{iw})}.$$

What could be a possible approach to find an expression of $w$?

Answer 1

$\dfrac{e^{iw}(-1+e^{imw})}{(-1+e^{iw})}+\dfrac{e^{-imw}(-1+e^{imw})}{(-1+e^{iw})} = \dfrac{-e^{iw}+e^{i(m+1)w}-e^{-imw}+1}{(-1+e^{iw})}= -1+\dfrac{e^\frac{iw}{2}(2\cos((\frac{1}{2}+m)\omega))}{e^\frac{iw}{2}(2\cos(\omega))}=-1+\dfrac{(\cos((\frac{1}{2}+m)\omega))}{(\cos(\omega))}$.

This function (in signal processing) is known as discrete Sinc function (because DFT of discrete rect function a.k.a. moving average results in discrete Sinc in digital frequency domain) and is not an injective function hence there is no inverse. However by restricting its domain you can define an inverse for it.