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'Mellin's inverse formula'
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| Title of object: |
Mellin's inverse formula |
| Canonical Name: |
MellinsInverseFormula |
| Type: |
Result |
| Created on: |
2004-05-31 15:35:28 |
| Modified on: |
2004-10-27 14:59:54 |
| Classification: |
msc:44A10 |
| Synonyms: |
Mellin's inverse formula=inverse Laplace transformation
Mellin's inverse formula=Bromwich integral
Mellin's inverse formula=Fourier-Mellin integral |
Preamble:
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Content:
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 the {\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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