TITLE: Laplace Transform
QUESTION [5 upvotes]: How can one show that 1/$e^s$  is not the laplace transform of any function?
Note that function here does not include distributions like dirac delta function.

REPLY [5 votes]: Suppose that $e^{-s}$ is the Laplace transform of some function $f(t)$. Then, $$f(t) = \frac{1}{2\pi i} \lim_{T \to \infty} \int_{\gamma -iT}^{\gamma+iT} e^{st}e^{-s}\ ds = \delta(t-1).$$
By uniqueness of the inverse Laplace transform, this is the only such "function."