We know that $y=xe^x$ cannot be solved for $x$ using elementary functions.
The Lagrange inversion theorem can be used for finding a "new" function that would be the inverse function of the above equation. This special function is named "Lambert W Function"
So for $y=xe^x$, $x=W(y)$.
There are many equations that can be solved through Lambert's W Function. However it seems that some common equations in Optics or Control Theory, like $y=\dfrac{\sin x}{x}$ or $y=e^{-x}\cos x$ cannot be solved with Lambert W Function.
I wonder if there are already any specials, Lambert-W-like, functions for those cases, or their inverse functions still remain undefined.
Best Answer
For a function to be invertible it must be monotonic. y = $xe^x$ is monotonic. However, sinx/x and $e^{-x}$cosx are monotonic only in small intervals. So you certainly can't have a universal inverse for either of them.