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| In [[mathematics]], the '''inverse Laplace transform''' of a function ''F''(''s'') is the function ''f''(''t'') which has the property
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| <math>\mathcal{L}\left\{ f\right\}(s) = F(s)</math>,
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| or alternatively <math>\mathcal{L}_t\left\{ f(t)\right\}(s) = F(s)</math>,
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| where <math>\mathcal{L}</math> denotes the [[Laplace transform]].
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| It can be proven, that if a function <math>F(s)</math> has the inverse Laplace transform <math>f(t)</math>, i.e. <math>f</math> is a piecewise-continuous and exponentially-restricted real function <math>f</math> satisfying the condition
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| :<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).
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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.
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| ==Mellin's inverse formula==
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| An integral formula for the inverse [[Laplace transform]], called the '''Bromwich integral''', the '''Fourier–Mellin integral''', and '''Mellin's inverse formula''', is given by the [[line integral]]:
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| :<math>f(t) = \mathcal{L}^{-1} \{F\}(t) = \mathcal{L}^{-1}_s \{F(s)\}(t) = \frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}F(s)\,ds,</math>
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| where the integration is done along the vertical line <math>Re(s)=\gamma</math> in the [[complex plane]] such that <math>\gamma</math> is greater than the real part of all [[Mathematical singularity|singularities]] of ''F''(''s''). This ensures that the contour path is in the [[region of convergence]].
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| If all singularities are in the left half-plane, or ''F''(''s'') is a [[smooth function]] on - ∞ < ''Re''(''s'') < ∞ (i.e. no [[Mathematical singularity|singularities]]), then <math>\gamma</math> can be set to zero and the above inverse integral formula above becomes identical to the [[inverse Fourier transform]].
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| In practice, computing the complex integral can be done by using the [[Cauchy residue theorem]].
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| It is named after [[Hjalmar Mellin]], [[Joseph Fourier]] and [[Thomas John I'Anson Bromwich]].
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| If F(s) is the Laplace transform of the function f(t),then f(t) is called the inverse Laplace transform of F(s).
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| ==Software tools==
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| * [http://reference.wolfram.com/mathematica/ref/InverseLaplaceTransform.html InverseLaplaceTransform] performs symbolic inverse transforms in [[Mathematica]]
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| * [http://library.wolfram.com/infocenter/MathSource/5026/ Numerical Inversion of Laplace Transform with Multiple Precision Using the Complex Domain] in Mathematica gives numerical solutions<ref>{{cite doi|10.1002/nme.995}}</ref>
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| * [http://www.mathworks.co.uk/help/symbolic/ilaplace.html ilaplace] performs symbolic inverse transforms in [[MATLAB]]
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| * [http://www.mathworks.co.uk/matlabcentral/fileexchange/32824-numerical-inversion-of-laplace-transforms-in-matlab Numerical Inversion of Laplace Transforms in Matlab]
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| ==See also==
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| * [[Inverse Fourier transform]]
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| * [[Post's inversion formula]], an alternative formula for the inverse Laplace transform.
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| == References ==
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| {{Reflist}}
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| * {{Citation | last1=Davies | first1=B. J. | title=Integral transforms and their applications | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | isbn=978-0-387-95314-4 | year=2002}}
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| * {{Citation | last1=Manzhirov | first1=A. V. | last2=Polyanin | first2=Andrei D. | title=Handbook of integral equations | publisher=[[CRC Press]] | location=London | isbn=978-0-8493-2876-3 | year=1998}}
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| * {{Citation | last1=Boas | first1=Mary | year=1983 | title=Mathematical Methods in the physical sciences | publisher=[[John Wiley & Sons]] | isbn=0-471-04409-1 | page=662 }} (p. 662 or search Index for "Bromwich Integral", a nice explanation showing the connection to the fourier transform)
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| ==External links==
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| * [http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.
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| {{PlanetMath attribution|id=5877|title=Mellin's inverse formula}}
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| [[Category:Transforms]]
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| [[Category:Complex analysis]]
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| [[Category:Integral transforms]]
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