Mellin Transform of Heaviside Step Function/Corollary

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

\(\ds \)

\(=\)

\(\ds \dfrac {c^s} s - 0\)

\(\ds \)

\(=\)

\(\ds \dfrac {c^s} s\)

$\blacksquare$