PlanetMath: Mellin transform

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Mellin transform

(Definition)

The Mellin transform is an integral transform defined as follows:

$\displaystyle F(s) = \int_0^\infty f(t) t^{s-1} \, dt$

Intuitively, it may be viewed as a continuous analogue of a power series -- instead of synthesizing 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 changes of variables $t = e^{-r}$ and define

by $f(e^{-r}) = g(r)$ , then the above integral becomes

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

which is a bilateral Laplace transform.

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"Mellin transform" is owned by rspuzio.

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Cross-references: integral, variables, Laplace transform, Transform, real, powers, integer, multiples, summing, function, power series, continuous, integral transform
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This is version 2 of Mellin transform, born on 2008-05-14, modified 2008-05-15.
Object id is 10589, canonical name is MellinTransform.
Accessed 294 times total.

Classification:

44A15 (Integral transforms, operational calculus :: Special transforms )

Pending Errata and Addenda

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