# 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.

(more to come)

Title Mellin transform MellinTransform 2015-02-17 15:10:57 2015-02-17 15:10:57 rspuzio (6075) pahio (2872) 7 rspuzio (2872) Definition msc 44A15