Suppose I have a function $f(t)$, whose domain is $(0,\infty)$.
I take Laplace transform of the function: $F(s)=\int_0^{\infty}e^{-st}f(t)dt$.
It exists only for $Re(s)>0$. If $s$ is real and $s\le0$, then the integral diverges, if $s$ is complex and $Re(s)<0$, then the integral doesn't exist.
So, I have $F(s)$. Now, I try to approximate it with a rational function $R(s)$.
How do I find Inverse Laplace of $R(s)$ to approximate $f(t)$? The problem is that $R(s)$ is defined for all $s$, not only positive, like $F(s)$.
What I want to do is to find Laplace transform, approximate it with a rational function and then take inverse Laplace transform to see how this final function approximates original function.