In mathematics, the inverse Laplace transform of a function is the piecewise-continuous and exponentially-restricted[clarification needed] real function which has the property:
where denotes the Laplace transform.
It can be proven that, if a function has the inverse Laplace transform , then 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.
^Cohen, A. M. (2007). "Inversion Formulae and Practical Results". Numerical Methods for Laplace Transform Inversion. Numerical Methods and Algorithms. Vol. 5. pp. 23–44. doi:10.1007/978-0-387-68855-8_2. ISBN 978-0-387-28261-9.
^Lerch, M. (1903). "Sur un point de la théorie des fonctions génératrices d'Abel". Acta Mathematica. 27: 339–351. doi:10.1007/BF02421315. hdl:10338.dmlcz/501554.
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