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.

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