Definition:Mellin Transform

Definition

Let $\map f t: \R_{\ge 0} \to \C$ be a function of a real variable $t$.

The Mellin transform of $f$ is defined and denoted as:

$\ds \map {\MM \set {\map f t} } s := \int_0^{\to +\infty} t^{s - 1} \map f t \rd t$

wherever this improper integral exists.

Here $\map \MM f$ is a complex function of the variable $s$.

Also denoted as

The Mellin transform can also be denoted $\phi$, hence:

$\map \phi s := \map {\MM \set {\map f t} } s$

Also see

• Results about Mellin transforms can be found here.

Source of Name

This entry was named for Robert Hjalmar Mellin.

Sources

• Weisstein, Eric W. "Mellin Transform." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MellinTransform.html