I consider the Laplace transform of the function $f(t) = e^{at}$, $t\geq 0$ and $a$ is some constant.
It has the form $$ \mathcal{L}\{f(t)\}(s) = \int_0^{\infty} e^{-s t} e^{a t} dt = \frac{1}{s-a} $$ and is well-defined for $Re(s)>a$.
I know from the general theory that the Laplace transform is well defined for $Re(s)>s_0$. In our case $s_0 = a$. Moreover, I know that in the region of the absolute convergence the Laplace transform is analytic.
In this particular example our Laplace transform is an analytic function not only for $Re(s)>a$, but also for $Re(s)<a$. It is only not defined when $s=a$.
I wonder if there exists a more general theorem which tells us under which conditions on $f(t)$ we can extend the Laplace transform on a negative half space and in this region it is also analytic.