Suppose I have a function $F(s)$ which is the (two sided) laplace transform of some function $f(t)$. Are there any statements I can make about $f(t)$ when I raise $F(s)$ to a positive power $\alpha$?
In perhaps nicer notation $$ F(s) = \mathcal{B}\{f(t)\} $$ $$ [F(s)]^{\alpha} = \mathcal{B}\{?\},\quad \alpha \in \mathbb{R}^{+} $$
I've tried a bunch of simple things already, but I just can't figure out a way of bringing the exponent $\alpha$ inside of the integral of the transform whenever $\alpha$ isn't an integer.