If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:

$f(x) = \int ds \hat{f}(s) \exp(-sx)$

what conditions should I impose over $f(x)$?

In other words, what are the conditions for the Fourier-Mellin-Bromwich integral

$\hat{f}(s) = \frac{1}{2\pi i} \int_{\gamma - i\infty}^{\gamma + i\infty} f(x) \exp(sx) dx$

to exist?