As I read in Wikipedia, there are two types of Laplace transforms
One-sided Laplace transform: $F(s) = \int_{0}^\infty e^{-st} f(t) dt$
Two-sided Laplace transform: $F(s) = \int_{-\infty}^\infty e^{-st} f(t) dt$
But they give only one formula for Inverse Laplace transform:
$\hspace{3.0cm} f(t) = \frac{1}{2\pi i} \lim_{T \to \infty} \int_{\gamma - i T}^{\gamma + i T} e^{st} F(s) ds$
My question is that, does the type of Laplace transform I use affects the Inverse formula ?
p.s:
I've proved the Inverse Laplace transform above corresponding to Two-sided Laplace transform using Fourier transform. But I've not come up with any idea of proving the correctness of the Inverse Laplace transform corresponding to One-sided Laplace transform.
According to my proof, the Inverse transform above is correct for One-sided transform if $f$ satisfies $f(t) = 0$ $\forall t < 0$. In other words,
$\hspace{3.0cm} f(t) = \frac{1}{2\pi i} \lim_{T \to \infty} \int_{\gamma - i T}^{\gamma + i T} e^{st} F(s) ds$, $\forall t \geq 0$