TITLE: Does sinc function have any special inverse function defined?
QUESTION [9 upvotes]: 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.

REPLY [2 votes]: What does monotonicity have to do with it? The sin(x) function is not monotonic, even not 1:1, and yet there is an inverse function arcsin(x). I think there was just no need enough to give a name to the inverse of the sinc(x) function. It doesn't really matter, you can find numerically what the function is easily. If you want to give it a name, say, arcsinc(x), feel free to do that.