Inverse Laplace transform: Difference between revisions
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Gareth Jones (talk | contribs) add references for original proof of uniqueness of Laplace transform |
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:<math>\mathcal{L}_t\{f(t)\}(s) = F(s),\ \forall s \in \mathbb R</math> |
:<math>\mathcal{L}_t\{f(t)\}(s) = F(s),\ \forall s \in \mathbb R</math> |
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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). |
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.<ref>{{cite doi|10.1007/978-0-387-68855-8_2}}</ref><ref>{{cite doi|10.1007/BF02421315}} |
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The [[Laplace transform]] and the inverse Laplace transform together have a number of properties that make them useful for analysing [[linear dynamic system]]s. |
The [[Laplace transform]] and the inverse Laplace transform together have a number of properties that make them useful for analysing [[linear dynamic system]]s. |
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Revision as of 16:42, 25 July 2014
In mathematics, the inverse Laplace transform of a function F(s) is the function f(t) which has the property , or alternatively , where denotes the Laplace transform.
It can be proven, that if a function has the inverse Laplace transform , i.e. is a piecewise-continuous and exponentially-restricted real function satisfying the condition
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.[1]Cite error: A <ref> tag is missing the closing </ref> (see the help page).
- ilaplace performs symbolic inverse transforms in MATLAB
- Numerical Inversion of Laplace Transforms in Matlab
See also
- Inverse Fourier transform
- Post's inversion formula, an alternative formula for the inverse Laplace transform.
References
- ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/978-0-387-68855-8_2, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with
|doi=10.1007/978-0-387-68855-8_2instead.
- Davies, B. J. (2002), Integral transforms and their applications (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95314-4
- Manzhirov, A. V.; Polyanin, Andrei D. (1998), Handbook of integral equations, London: CRC Press, ISBN 978-0-8493-2876-3
- Boas, Mary (1983), Mathematical Methods in the physical sciences, John Wiley & Sons, p. 662, ISBN 0-471-04409-1 (p. 662 or search Index for "Bromwich Integral", a nice explanation showing the connection to the fourier transform)
External links
- Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
This article incorporates material from Mellin's inverse formula on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.