Is there an integral representation for $\frac{1}{n!}$?

Question

I know that $n!$ has various integral representations, for instance the $\Gamma$ function. I was wondering if $\frac{1}{n!}$ has an integral representation.

Answer 1

$$ \dfrac{1}{2\pi} \int_0^{2\pi} e^{e^{i\theta}} e^{-in\theta}\; d\theta $$

Answer 2

Following @RobertIsrael, we have

$$\frac{1}{2\pi}\int_0^{2\pi}e^{e^{i\phi}}e^{-in\phi}d\phi=\oint_{|z|=1}e^{z}z^{-n}\frac{dz}{iz}\tag 1$$

We note that the integrand on the right-hand side of $(1)$ has a pole of order $n+1$ at $z=0$. The residue is given by

$$\text{Res}\left(-i\frac{e^z}{z^{n+1}},z=0\right)=\frac{1}{n!}\lim_{z\to 0}\frac{d^n}{dz^n}\left(z^{n+1}\frac{-ie^z}{z^{n+1}}\right)=\frac{-i}{n!}$$

Putting it all together reveals that

$$\frac{1}{2\pi}\int_0^{2\pi}e^{e^{i\phi}}e^{-in\phi}d\phi=2\pi i \frac{-i}{2\pi n!}=\frac{1}{n!}$$

as expected!

Answer 3

You can write the inverse Laplace transform of $1/s^{n+1}$, evaluated at $t=1$, as $1/n!$. The integral is $$ \int_{c-i\infty}^{c+i\infty}\frac{1}{2\pi is^{n+1}}e^s\,ds=\frac{1}{n!}, $$ for suitable real $c$.

Answer 4

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?