Mellin Transform of Heaviside Step Function/Corollary
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Theorem
Let $c$ be a constant real number.
Let $\map {u_c} t$ be the Heaviside step function.
Let $\MM$ be the Mellin transform.
Then:
- $\map {\MM \set {\map u {c - t} } } s = \dfrac {c^s} s$
for $c > 0, \map \Re s > 0$
Proof
\(\ds \map {\MM \set {\map u {c - t} } } s\) | \(=\) | \(\ds \int_0^{\to +\infty} t^{s - 1} \map u {c - t} \rd t\) | Definition of Mellin Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^c t^{s - 1} \rd t\) | Definition of Heaviside Step Function: integrand is elsewhere zero | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigintlimits {\dfrac {t^s} s} {t \mathop = 0} {t \mathop = c}\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {c^s} s - 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {c^s} s\) |
$\blacksquare$