TITLE: Integral transform and $\frac{1}{n!}$.
QUESTION [6 upvotes]: Probably this is a trivial question, but I am unable to find an answer: is there a function $v(x)$ such that 
$$
\int_{0}^\infty x^n e^{v(x)} dx =\frac{1}{n!}
$$
for all positiv integer n?

REPLY [4 votes]: Suppose you are not wedded to the interval $(0,\+\infty)$.  
If $C$ is the unit circle in the complex plane, oriented in the counter-clockwise direction as usual, then
$$
\frac{1}{2\pi i}\oint_C z^n \frac{e^{1/z}}{z}\;dz = \frac{1}{n!}
$$
for $n=0,1,2,\dots$