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# Mellin transform

The *Mellin transform* is an integral transform defined as follows:

$F(s)=\int_{0}^{\infty}f(t)t^{{s-1}}\,dt$ |

Intuitively, it may be viewed as a continuous analogue of a power series — instead of synthetizing a function by summing multiples of integer powers, we integrate over all real powers. This transform is closely related to the Laplace transform — if we make a change of variables $t=e^{{-r}}$ and define $g$ by $f(e^{{-r}})=g(r)$, then the above integral becomes

$F(s)=-\int_{{-\infty}}^{{+\infty}}g(r)e^{{-rs}}\,dr,$ |

which is a bilateral Laplace transform.

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