Inverse Laplace transform

In mathematics, the inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t) which has the property:

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

where ${\mathcal {L}}$ denotes the Laplace 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 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.

Mellin's inverse formula

An integral formula for the inverse Laplace transform, called the Mellin's inverse formula, the Bromwich integral, or the Fourier–Mellin integral, is given by the line integral:

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

where the integration is done along the vertical line Re(s) = γ in the complex plane such that γ is greater than the real part of all singularities of F(s). This ensures that the contour path is in the region of convergence. If all singularities are in the left half-plane, or F(s) is a smooth function on −∞ < Re(s) < ∞ (i.e., no singularities), then γ 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.

Software tools

• InverseLaplaceTransform performs symbolic inverse transforms in Mathematica
• Numerical Inversion of Laplace Transform with Multiple Precision Using the Complex Domain in Mathematica gives numerical solutions^[3]
• 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

1. Cohen, A. M. (2007). "Inversion Formulae and Practical Results". Numerical Methods for Laplace Transform Inversion. Numerical Methods and Algorithms. 5. p. 23. doi:10.1007/978-0-387-68855-8_2. ISBN 978-0-387-28261-9.
2. Lerch, M. (1903). "Sur un point de la théorie des fonctions génératrices d'Abel". Acta Mathematica. 27: 339. doi:10.1007/BF02421315.
3. Abate, J.; Valkó, P. P. (2004). "Multi-precision Laplace transform inversion". International Journal for Numerical Methods in Engineering. 60 (5): 979. doi:10.1002/nme.995.

• 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.