Inverse Laplace transform

In mathematics, the inverse Laplace transform of a function ${\displaystyle F(s)}$ is a real function ${\displaystyle f(t)}$ that is piecewise-continuous, exponentially-restricted (that is, ${\displaystyle |f(t)|\leq Me^{\alpha t}}$ ${\displaystyle \forall t\geq 0}$ for some constants ${\displaystyle M>0}$ and ${\displaystyle \alpha \in \mathbb {R} }$) and has the property:

${\displaystyle {\mathcal {L}}\{f\}(s)={\mathcal {L}}\{f(t)\}(s)=F(s),}$

where ${\displaystyle {\mathcal {L}}}$ denotes the Laplace transform.

It can be proven that, if a function ${\displaystyle F(s)}$ has the inverse Laplace transform ${\displaystyle f(t)}$, then ${\displaystyle f(t)}$ is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same). This result was first proven by Mathias Lerch in 1903 and is known as Lerch's theorem.^[1][2]

The Laplace transform and the inverse Laplace transform together have a number of properties that make them useful for analysing linear dynamical systems.