Mellin Transform of Power Times Function
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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.
Proof
\(\ds \map {\MM \set {t^n \map f t} } s\) | \(=\) | \(\ds \int_0^{\to +\infty} t^{s - 1} t^n \map f t \rd t\) | Definition of Mellin Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^{\to +\infty} t^{\paren {s + n} - 1} \map f t \rd t\) | Exponent Combination Laws | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\MM \set {\map f t} } {s + n}\) | Definition of Mellin Transform |
$\blacksquare$