# 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

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

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

 $f(t)=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}e^{st}F(s)\,ds,$

by the Finn R. H. Mellin (1854—1933).  Here it must be integrated along a straight line parallel to the imaginary axis and intersecting the real axis in the point $\gamma$ 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