Completely monotone function

Summary

A completely monotone function is an infinitely differentiable function on $(0, \infty)$ whose derivatives alternate in sign: $f \ge 0$, $f' \le 0$, $f'' \ge 0$, and so on. It is the mirror image of an absolutely monotonic function under $x \mapsto -x$. By Bernstein's theorem, the completely monotone functions are exactly the Laplace transforms of positive measures on $[0, \infty)$. This makes them central in probability and potential theory.