Editing Inverse Laplace transform

{{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.

==Mellin's inverse formula==
An integral formula for the inverse [[Laplace transform]], called the ''Mellin's inverse formula'', the ''[[Thomas John I'Anson Bromwich|Bromwich]] integral'', or the ''[[Joseph Fourier|Fourier]]–[[Hjalmar Mellin|Mellin]] integral'', is given by the [[line integral]]:
:<math>f(t) = \mathcal{L}^{-1} \{F(s)\}(t) =  \frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}F(s)\,ds</math>
where the integration is done along the vertical line <math>\textrm{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 <math>F(s)</math> and <math>F(s)</math> is bounded on the line, for example if the contour path is in the [[region of convergence]]. If all singularities are in the left half-plane, or <math>F(s)</math> is an [[entire function]], then <math>\gamma</math> can be set to zero and the above inverse integral formula becomes identical to the [[inverse Fourier transform]].

In practice, computing the complex integral can be done by using the [[Cauchy residue theorem]].

==Post's inversion formula==
'''Post's inversion formula''' for [[Laplace transform]]s, named after [[Emil Leon Post|Emil Post]],<ref name="Post1930">{{cite journal|last1=Post|first1=Emil L.|title=Generalized differentiation|journal=Transactions of the American Mathematical Society|volume=32|issue=4|year=1930|pages=723–781|issn=0002-9947|doi=10.1090/S0002-9947-1930-1501560-X|doi-access=free}}</ref> is a simple-looking but usually impractical formula for evaluating an inverse Laplace transform.

The statement of the formula is as follows: Let <math>f(t)</math> be a continuous function on the interval <math>[0,\infty)</math> of exponential order, i.e.

: <math>\sup_{t>0} \frac{f(t)}{e^{bt}} < \infty</math>

for some real number <math>b</math>. Then for all <math>s > b</math>, the Laplace transform for <math>f(t)</math> exists and is infinitely differentiable with respect to <math>s</math>. Furthermore, if <math>F(s)</math> is the Laplace transform of <math>f(t)</math>, then the inverse Laplace transform of <math>F(s)</math> is given by

: <math>f(t) = \mathcal{L}^{-1} \{F\}(t)
 = \lim_{k \to \infty} \frac{(-1)^k}{k!} \left( \frac{k}{t} \right) ^{k+1} F^{(k)} \left( \frac{k}{t} \right)</math>

for <math>t > 0</math>, where <math>F^{(k)}</math> is the <math>k</math>-th derivative of <math>F</math> with respect to <math>s</math>.

As can be seen from the formula, the need to evaluate derivatives of arbitrarily high orders renders this formula impractical for most purposes.
   
With the advent of powerful personal computers, the main efforts to use this formula have come from dealing with approximations or asymptotic analysis of the Inverse Laplace transform, using the [[Grunwald–Letnikov differintegral]] to evaluate the derivatives.   
    
Post's inversion has attracted interest due to the improvement in computational science and the fact that it is not necessary to know where the [[Pole (complex analysis)|poles]] of <math>F(s)</math> lie, which make it possible to calculate the asymptotic behaviour for big <math>x</math> using inverse [[Mellin transform]]s for several arithmetical functions related to the [[Riemann hypothesis]].

==Software tools==
* [http://reference.wolfram.com/mathematica/ref/InverseLaplaceTransform.html InverseLaplaceTransform] performs symbolic inverse transforms in [[Mathematica]]
* [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 journal | last1 = Abate | first1 = J. | last2 = Valkó | first2 = P. P. | doi = 10.1002/nme.995 | title = Multi-precision Laplace transform inversion | journal = International Journal for Numerical Methods in Engineering | volume = 60 | issue = 5 | pages = 979 | year = 2004 | bibcode = 2004IJNME..60..979A | s2cid = 119889438 }}</ref>
* [http://www.mathworks.co.uk/help/symbolic/ilaplace.html ilaplace] {{Webarchive|url=https://web.archive.org/web/20140903152047/http://www.mathworks.co.uk/help/symbolic/ilaplace.html |date=2014-09-03 }} performs symbolic inverse transforms in [[MATLAB]]
* [http://www.mathworks.co.uk/matlabcentral/fileexchange/32824-numerical-inversion-of-laplace-transforms-in-matlab Numerical Inversion of Laplace Transforms in Matlab]
* [https://www.mathworks.com/matlabcentral/fileexchange/71511-a-cme-based-numerical-inverse-laplace-transformation-method Numerical Inversion of Laplace Transforms based on concentrated matrix-exponential functions] in Matlab

==See also==
* [[Inverse Fourier transform]]
* [[Poisson summation formula]]

==References==
{{Reflist}}

==Further reading==
* {{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}}
* {{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}}
* {{Citation | last1=Boas | first1=Mary | year=1983 | title=Mathematical Methods in the physical sciences | publisher=[[John Wiley & Sons]] | isbn=0-471-04409-1 | page=[https://archive.org/details/mathematicalmeth00boas/page/662 662] | url-access=registration | url=https://archive.org/details/mathematicalmeth00boas/page/662 }} (p.&nbsp;662 or search Index for "Bromwich Integral", a nice explanation showing the connection to the Fourier transform)
* {{Citation | last1=Widder | first1=D. V. | title=The Laplace Transform | publisher=[[Princeton University Press]] | year=1946}}
* [http://www.rose-hulman.edu/~bryan/invlap.pdf Elementary inversion of the Laplace transform]. Bryan, Kurt. Accessed June 14, 2006.

==External links==
* [http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.

{{PlanetMath attribution|id=5877|title=Mellin's inverse formula}}

[[Category:Transforms]]
[[Category:Complex analysis]]
[[Category:Integral transforms]]
[[Category:Laplace transforms]]