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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.

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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.

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    $\begingroup$ $xe^x$ is not monotonic. It is if you restrict to subsets of the domain where it is monotonic you can define branches of the inverse, just as one could for $\mathrm{sinc}$ to define local inverses, and as one does for $\sin$ to define $\arcsin$. $\endgroup$ – Jonas Meyer Oct 28 '13 at 3:20
  • $\begingroup$ @JonasMeyer -- yes you are right it has two branches. What peterwhy said is good for the other functions. $\endgroup$ – Betty Mock Oct 28 '13 at 18:14
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My intuition for the lack of inverses for your two functions is that they have infinite branches, and the infinite branches are not in a simple relation. Contrast with Lambert W function which has two branches, and $\sin^{-1}$ which the branches are related in simple enough periodic manner.

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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.

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