Suppose that we have a real value function $F(x) : \mathbb{R}^+ \to \mathbb{R}^+$. We can compute the inverse Laplace transform (denoted by $f(y)$) by the Bromwich integral (Mellin's integral) such that $$ f(y) = \dfrac{1}{2\pi i} \int_{a-i \infty}^{a+i \infty} e^{yz}F(z) dz $$ here $a$ is larger than the singularities of $F(z)$. Now we assume that $F(z)$ is nice enough and the above integral converges.
The function $F(x)$ is real but $F(z)$ is complex in the computation of the inverse Laplace transform. To tackle it, can I define a new complex-valued function $G(z)$ such that $F(x) = G(x)$ for $x \in \mathbb{R}$? And can we still have the result $$ f(y) = \dfrac{1}{2\pi i} \int_{a-i \infty}^{a+i \infty} e^{yz}G(z) dz $$ for $y \in \mathbb{R}^+$?
Here $F(z)$ and $G(z)$ can be distinct if $z$ is not real and $F(z)$ exists. Since I found that the natural extension of $F(x)$ in my study, that is $F(z)$, may not be regular enough.