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 integerpowers, we integrate over all real powers. This transform is closelyrelated 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 integralbecomes

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

which is a bilateral Laplace transform.

(more to come)