Inverse Laplace transform

In mathematics, the inverse Laplace transform of a function ${\displaystyle F(s)}$ is the piecewise-continuous and exponentially-restricted^{[clarification needed]} real function ${\displaystyle f(t)}$ which 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.