Laplace transform operator

Definition of Laplace Transform

Let   $f(t)$   be a given function which is defined for   $t \ge 0$. If there exists a function   $F(s)$  so that
 

then   $F(s)$   is called the Laplace Transform of   $f(t)$, and will be denoted by   $\mathcal{L} \left\{f(t)\right\}$.   Notice the integrator   $e^{-st} \, dt$   where   $s$   is a parameter which may be real or complex.