Mellin Transform of Power Times Function

Theorem

Let $t^n: \R \to \R$ be $t$ to the $n$th power for some $n \in \N_{\ge 0}$.

Let $\MM$ be the Mellin transform.

Then:

$\map {\MM \set {t^n \map f t} } s = \map {\MM \set {\map f t} } {s + n}$

given that both transforms exist.

$\ds \map {\MM \set {t^n \map f t} } s$

$=$

$\ds \int_0^{\to +\infty} t^{s - 1} t^n \map f t \rd t$

$\ds $

$=$

$\ds \int_0^{\to +\infty} t^{\paren {s + n} - 1} \map f t \rd t$

$\ds $

$=$

$\ds \map {\MM \set {\map f t} } {s + n}$

$\blacksquare$