Almost every treatment of the Laplace transform that I come across talks about "the poles" of the Laplace transform function $F(s)$, thereby seeming to implicitly assume that $F(s)$ is meromorphic on the complex plane.
Can a Laplace transform function contain essential singularities, branch cuts, or other, more complicated ways to fail to be analytic? If so, are there any simple examples of such Laplace transforms? Does the usual inverse Laplace transform still work in these cases?