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

$$ℒ\{f(t)\}=F(s),$$

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

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

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