calculus - Frullani integral for $f(x)=e^{x}$ in a complex context

I should prove the Frullani integral equality

$$
\int_{0}^{\infty} (1-e^{zx}) \frac{\beta}{x} e^{-\gamma x}dx = \beta \log \left(1- \frac{z}{\gamma}\right)
$$
for $z \in \mathbb{C}$ with non-positive real part.

I should first consider $z \leq 0$ and use
$$

\frac{e^{-\gamma x} - e^{-(\gamma - z)x}}{x}=\int_{\gamma}^{\gamma - z}e^{-y x}dy$$
and then change the order of integration. These steps are clear (see also Frullani integral for $f(x)=e^{-\lambda x }$).

But then I should use analytic extension to show that the formula is valid for $z \in \mathbb{C}$ with non-positive real part. I need help for this step.

Thank you in advance!