I came across this complex function in my work $f(z)=\frac{e^z-1}{z}$. Is there a reference to $f(z)$? What is its name in the literature? More importantly, is the function inversible? If so, what is $f^{-1}(z)$? Thanks.



What is the name of $\frac{e^z-1}{z}$ and how to invert it?
Let $y=(e^z-1)/z$ and $x=-1/y$. Then $xe^x=(x-z)e^{x-z}$. Hence
$$x-z=W(xe^x).$$
Here W is an appropriately chosen branch of the Lambert function (ProductLog[-1,.] in Mathematica).


