Mellin’s inverse formula

It may be proven, that if a function F(s) has the inverse Laplace transform f(t), i.e. a piecewise continuous and exponentially real function f satisfying the condition


then f(t) is uniquely determined when not regarded as different such functions which differ from each other only in a point set having Lebesgue measure zero.

The inverse Laplace transform is directly given by Mellin’s inverse formula


by the Finn R. H. Mellin (1854—1933).  Here it must be integrated along a straight line parallelMathworldPlanetmathPlanetmath to the imaginary axisMathworldPlanetmath and intersecting the real axis in the point γ which must be chosen so that it is greater than the real parts of all singularities of F(s).

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

Title Mellin’s inverse formula
Canonical name MellinsInverseFormula
Date of creation 2013-03-22 14:23:02
Last modified on 2013-03-22 14:23:02
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Result
Classification msc 44A10
Synonym inverse Laplace transformation
Synonym Bromwich integral
Synonym Fourier-Mellin integral
Related topic InverseLaplaceTransformOfDerivatives
Related topic HjalmarMellin
Related topic TelegraphEquation