Suppose I can show that
\begin{align}\label{eq:1} \int_0^{\infty}e^{-zx}f(x)\text{d} x = F(z) \end{align}
holds for all $z$ such that $\Re(z) > b$. Suppose further that $F(z)$ is analytic for all $\Re(z) > a$ and that $b > a$. Can I conclude that the above equality holds also on the larger region $\Re(z) > a$?