Editing Inverse Laplace transform (section)

{{Short description|Mathematical function}}

In [[mathematics]], the '''inverse Laplace transform''' of a [[function (mathematics)|function]] <math>F(s)</math> is a [[real number|real]] function <math>f(t)</math> that is  piecewise-[[continuous function|continuous]], exponentially-restricted (that is, <math>|f(t)|\leq Me^{\alpha t}</math> <math>\forall t \geq 0</math> for some constants <math>M > 0</math> and <math>\alpha \in \mathbb{R}</math>) and  has the property:

:<math>\mathcal{L}\{f\}(s) = \mathcal{L}\{f(t)\}(s) = F(s),</math>
where <math>\mathcal{L}</math> denotes the [[Laplace transform]].

It can be proven that, if a function <math>F(s)</math> has the inverse Laplace transform <math>f(t)</math>, then <math>f(t)</math> 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.<ref>{{Cite book | doi = 10.1007/978-0-387-68855-8_2| chapter = Inversion Formulae and Practical Results| title = Numerical Methods for Laplace Transform Inversion| volume = 5| pages = 23–44| series = Numerical Methods and Algorithms| year = 2007| last1 = Cohen | first1 = A. M. | isbn = 978-0-387-28261-9}}</ref><ref>{{Cite journal | doi = 10.1007/BF02421315| title = Sur un point de la théorie des fonctions génératrices d'Abel| journal = Acta Mathematica| volume = 27| pages = 339–351| year = 1903| last1 = Lerch | first1 = M. | author-link1 = Mathias Lerch| doi-access = free| hdl = 10338.dmlcz/501554| hdl-access = free}}</ref>

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