Question

Inverse Laplace transformation

Answer 1

Step-by-step explanation:

In mathematics, the inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t) which has the property:

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

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

It can be proven that, if a function F(s) has the inverse Laplace transform f(t), then 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.