I came across this complex function in my work $f(z)=\frac{e^z1}{z}$. 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.

Let $y=(e^z1)/z$ and $x=1/y$. Then $xe^x=(xz)e^{xz}$. Hence $$xz=W(xe^x).$$ Here W is an appropriately chosen branch of the Lambert function (ProductLog[1,.] in Mathematica). 


As for the name, according to wikipedia the Todd genus is given by: $$\mathrm{Td}(z)=\frac{z}{1e^{z}}.$$ So, $f(z)=1/\mathrm{Td}(z)$. 


the coefficients of $f^{1}(z)$ is just bernoulli numbers $$\frac{z}{e^z1}=\sum_{n=0}^{\infty}B_n\frac{z^n}{n!}$$ and we have Bernoulli numbers of higher order $\alpha$ $$\big(\frac{z}{e^z1}\big)^\alpha=\sum_{n=0}^{\infty}B_n^{(\alpha)}\frac{z^n}{n!}$$ where $\alpha=1$ in the case $f(z)$ and Todd genus can be computed in terms of Bernoulli numbers 


A 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 $ 

