What is the name of $\frac{e^z-1}{z}$ and how to invert it? 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.
 A: As for the name, according to wikipedia the Todd genus is given by:
$$\mathrm{Td}(z)=\frac{z}{1-e^{-z}}.$$
So, $f(z)=1/\mathrm{Td}(-z)$.
A: the coefficients of $f^{-1}(z)$ is just bernoulli numbers 
$$\frac{z}{e^z-1}=\sum_{n=0}^{\infty}B_n\frac{z^n}{n!}$$
and we have Bernoulli numbers of higher order $\alpha$
$$\big(\frac{z}{e^z-1}\big)^\alpha=\sum_{n=0}^{\infty}B_n^{(\alpha)}\frac{z^n}{n!}$$
where $\alpha=-1$ in the case $f(z)$ and Todd genus can be computed in terms of Bernoulli numbers
A: 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).
A: A related thread (about relations between two real branches of lambert function):
Equation between the two branches of the lambert w function
A (perhaps more complete?) solution involving both the real branches could be written as:
$z=W_0(-\frac{e^\frac{-1}{f}}{f})-W_{-1}(-\frac{e^\frac{-1}{f}}{f})=-\frac{1}{f}-W_{-1}(-\frac{e^\frac{-1}{f}}{f})$ for  $f >1 $
$z=W_{-1}(-\frac{e^\frac{-1}{f}}{f})-W_{0}(-\frac{e^\frac{-1}{f}}{f})=-\frac{1}{f}-W_{0}(-\frac{e^\frac{-1}{f}}{f})$ for  $f < 1 $
