Mellin's inverse formula

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In mathematics Mellin's inverse formula is used to invert the Laplace transform. It can be proven, that if a function F(s) has the inverse Laplace transform f(t), i.e. f is a piecewise continuous and exponentially restricted real function f satisfying the condition

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

then f(t) is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same).

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,

named after Hjalmar Mellin (Finland 1854 – 1933). The integration must be performed along a straight line parallel to the imaginary axis and intersecting the real axis at 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.

This article incorporates material from Mellin's inverse formula on PlanetMath, which is licensed under the GFDL.

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