Inverse Laplace transform

Inmathematics,theinverse Laplace transformof afunction$F(s)$is arealfunction$f(t)$that is piecewise-continuous,exponentially-restricted (that is,$|f(t)|\leq Me^{\alpha t}$$\forall t\geq 0$for some constants$M>0$and$\alpha \in \mathbb {R}$) and has the property:

${\mathcal {L}}\{f\}(s)={\mathcal {L}}\{f(t)\}(s)=F(s),$

where${\mathcal {L}}$denotes theLaplace transform.

It can be proven that, if a function$F(s)$has the inverse Laplace transform$f(t)$,then$f(t)$is uniquely determined (considering functions which differ from each other only on a point set havingLebesgue measurezero as the same). This result was first proven byMathias Lerchin 1903 and is known as Lerch's theorem.

TheLaplace transformand the inverse Laplace transform together have a number of properties that make them useful for analysinglinear dynamical systems.

Mellin's inverse formula

An integral formula for the inverseLaplace transform,called theMellin's inverse formula,theBromwichintegral,or theFourier–Mellinintegral,is given by theline integral:

$f(t)={\mathcal {L}}^{-1}\{F(s)\}(t)={\frac {1}{2\pi i}}\lim _{T\to \infty }\int _{\gamma -iT}^{\gamma +iT}e^{st}F(s)\,ds$

where the integration is done along the vertical line$Re(s)=\gamma$in thecomplex planesuch that$\gamma$is greater than the real part of allsingularitiesof$F(s)$and$F(s)$is bounded on the line, for example if the contour path is in theregion of convergence.If all singularities are in the left half-plane, or$F(s)$is anentire function,then$\gamma$can be set to zero and the above inverse integral formula becomes identical to theinverse Fourier transform.

In practice, computing the complex integral can be done by using theCauchy residue theorem.

Post's inversion formula

Post's inversion formulaforLaplace transforms,named afterEmil Post,is a simple-looking but usually impractical formula for evaluating an inverse Laplace transform.

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

$\sup _{t>0}{\frac {f(t)}{e^{bt}}}<\infty$

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

$f(t)={\mathcal {L}}^{-1}\{F\}(t)=\lim _{k\to \infty }{\frac {(-1)^{k}}{k!}}\left({\frac {k}{t}}\right)^{k+1}F^{(k)}\left({\frac {k}{t}}\right)$

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

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 theGrunwald–Letnikov differintegralto 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 thepolesof$F(s)$lie, which make it possible to calculate the asymptotic behaviour for big$x$using inverseMellin transformsfor several arithmetical functions related to theRiemann hypothesis.

Software tools

• InverseLaplaceTransformperforms symbolic inverse transforms inMathematica
• Numerical Inversion of Laplace Transform with Multiple Precision Using the Complex Domainin Mathematica gives numerical solutions
• ilaplaceArchived2014-09-03 at theWayback Machineperforms symbolic inverse transforms inMATLAB
• Numerical Inversion of Laplace Transforms in Matlab
• Numerical Inversion of Laplace Transforms based on concentrated matrix-exponential functionsin Matlab

See also

• Inverse Fourier transform
• Poisson summation formula

Further reading

• Davies, B. J. (2002),Integral transforms and their applications(3rd ed.), Berlin, New York:Springer-Verlag,ISBN978-0-387-95314-4
• Manzhirov, A. V.; Polyanin, Andrei D. (1998),Handbook of integral equations,London:CRC Press,ISBN978-0-8493-2876-3
• Boas, Mary (1983),Mathematical Methods in the physical sciences,John Wiley & Sons,p.662,ISBN0-471-04409-1(p. 662 or search Index for "Bromwich Integral", a nice explanation showing the connection to the Fourier transform)
• Widder, D. V. (1946),The Laplace Transform,Princeton University Press
• Elementary inversion of the Laplace transform.Bryan, Kurt. Accessed June 14, 2006.

External links

• Tables of Integral Transformsat EqWorld: The World of Mathematical Equations.

This article incorporates material from Mellin's inverse formula onPlanetMath,which is licensed under theCreative Commons Attribution/Share-Alike License.