Is $s/log(s)$ the Laplace transform of some functions f(x)?

Question

I am trying to prove that $ F(s) = s/log(s) $ is not the Laplace transform of some functions f(s).
Here is my resoaning:

• F(s) is discontinuos for s=1. For $s=1$ the funtion goes to $\infty$
• $ \lim_{s \to \infty} {s \over log(s)} = \infty $

However I cannot find a theorem that clearly states my results.

Answer 1

There is a theorem which states that given any function $f$ of exponential order, $f$'s Laplace Transform must have its limit at infinity be $0$, you can find it in - for example - Joel L. Schiff's "The Laplace Transform: Theory and Applications" as Theorem 1.20