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.
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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, $f(z)=1/\mathrm{Td}(z)$. 

