44A10-MellinsInverseFormula.tex

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\pmtitle{Mellin's inverse formula}
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\pmsynonym{inverse Laplace transformation}{MellinsInverseFormula}
\pmsynonym{Bromwich integral}{MellinsInverseFormula}
\pmsynonym{Fourier-Mellin integral}{MellinsInverseFormula}
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It may be proven, that if a function $F(s)$ has the {\em inverse Laplace transform} $f(t)$, i.e. a piecewise continuous and exponentially \PMlinkescapetext{restricted} 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 {\em 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.
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