Is there any function whose Laplace Transform does not have a pole or singularity?

Question

The Laplace Transforms of all functions I know have poles.

1. Can there be a function $F$ whose Laplace transform does not have a pole (or any other singularity)?

2. If so, is it possible that the function $F$ itself does not have poles or zeros?

Answer 1

Any function which decays sufficiently fast has a Laplace transform with no poles. Examples include $e^{-x^2/2}$ and $1_{[0,1]}$ (the function that is $1$ on $[0,1]$ and $0$ elsewhere) (there may appear to be a singularity in the latter's Laplace transform, $(1-e^{-s})/s$ at $s=0$, but there actually isn't since $\int_0^1 \, dx=1$ is finite).