In mathematics, the inverse Laplace transform of a function
denotes the Laplace transform.
has the inverse Laplace transform
is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same).
[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.
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: where the integration is done along the vertical line
is greater than the real part of all singularities of
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
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 for Laplace transforms, named after Emil Post,[3] is a simple-looking but usually impractical formula for evaluating an inverse Laplace transform.
be a continuous function on the interval
of exponential order, i.e. for some real number
exists and is infinitely differentiable with respect to
, then the inverse Laplace transform of
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 poles of
lie, which make it possible to calculate the asymptotic behaviour for big
using inverse Mellin transforms for several arithmetical functions related to the Riemann hypothesis.
This article incorporates material from Mellin's inverse formula on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.