Mellin Transform of Heaviside Step Function

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 $c > 0, \map \Re s < 0$.

Corollary

$\map {\MM \set {\map u {c - t} } } s = \dfrac {c^s} s$

for $c > 0, \map \Re s > 0$

Proof

Lemma

Let $t \in \R$.

Let $s \in \C$ with $\map \Re s < 0$.

Then:

$\ds \lim_{t \mathop \to +\infty} t^s = 0$

\(\ds \map {\MM \set {\map {u_c} t} } s\)

\(=\)

\(\ds \int_0^{\to +\infty} t^{s - 1} \map {u_c} t \rd t\)

Definition of Mellin Transform

\(\ds \)

\(=\)

\(\ds \int_c^{\to +\infty} t^{s - 1} \rd t\)

Definition of Heaviside Step Function: integrand is elsewhere zero

\(\ds \)

\(=\)

\(\ds \intlimits {\dfrac {t^s} s} c {+\infty}\)

Primitive of Power

\(\ds \)

\(=\)

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

By Lemma

\(\ds \)

\(=\)

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

$\blacksquare$