In mathematics, the inverse Laplace transform of a function F(s) is the piecewisecontinuous and exponentiallyrestricted 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 dynamic systems.
Contents

Mellin's inverse formula 1

Software tools 2

See also 3

References 4

External links 5
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_{\gammaiT}^{\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 halfplane, 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 above 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
References

^ Cohen, A. M. (2007). "Inversion Formulae and Practical Results". Numerical Methods for Laplace Transform Inversion. Numerical Methods and Algorithms 5. p. 23.

^

^ Abate, J.; Valkó, P. P. (2004). "Multiprecision Laplace transform inversion". International Journal for Numerical Methods in Engineering 60 (5): 979.

Davies, B. J. (2002), Integral transforms and their applications (3rd ed.), Berlin, New York:

Manzhirov, A. V.; Polyanin, Andrei D. (1998), Handbook of integral equations, London:

Boas, Mary (1983), Mathematical Methods in the physical sciences, (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.
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