TITLE: Is there an integral representation for $\frac{1}{n!}$?
QUESTION [8 upvotes]: I know that $n!$ has various integral representations, for instance the $\Gamma$ function. I was wondering if $\frac{1}{n!}$ has an integral representation.

REPLY [4 votes]: Take Cauchy's differentiation formula:
$$
f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^{n+1}}\, dz
$$
and plug a holomorphic $f$ such that $f^{(n)}(a)=1$.
For example, $f(z)=\exp(z)$, $a=0$, and $\gamma$ the unit circle:
$$
\frac{1}{n!} = \frac{1}{2\pi i} \oint_\gamma \frac{e^z}{z^{n+1}}\, dz
$$
Does that count?