Mellin transform

The Mellin transform is an integral transform defined as follows:

$$F(s)={\int }_0^{\mathrm{\infty }}f(t)t^{s-1}𝑑t$$

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

$$F(s)=-{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}g(r)e^{-rs}𝑑r,$$

which is a bilateral Laplace transform.

(more to come)

Title	Mellin transform
Canonical name	MellinTransform
Date of creation	2015-02-17 15:10:57
Last modified on	2015-02-17 15:10:57
Owner	rspuzio (6075)
Last modified by	pahio (2872)
Numerical id	7
Author	rspuzio (2872)
Entry type	Definition
Classification	msc 44A15