Inverse Laplace transform
It can be proven, that if a function has the inverse Laplace transform , i.e. is a piecewise-continuous and exponentially-restricted real function satisfying the condition
then is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same).
Mellin's inverse formula
where the integration is done along the vertical line in the complex plane such that 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 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.
If F(s) is the Laplace transform of the function f(t),then f(t) is called the inverse Laplace transform of F(s).
- InverseLaplaceTransform performs symbolic inverse transforms in Mathematica
- Numerical Inversion of Laplace Transform with Multiple Precision Using the Complex Domain in Mathematica gives numerical solutions
- ilaplace performs symbolic inverse transforms in MATLAB
- Numerical Inversion of Laplace Transforms in Matlab
- Inverse Fourier transform
- Post's inversion formula, an alternative formula for the inverse Laplace transform.
- 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)
- Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.