# Mellin Transform of Heaviside Step Function/Lemma

## Theorem

 It has been suggested that this article or section be renamed: This looks like too useful a result to bury as a lemma to something specialised. Recommend we extract it into a result page in its own right -- if it does not already so exist, which it might. One may discuss this suggestion on the talk page.

Let $t \in \R$.

Let $s \in \C$ with $\Re \left({s}\right) < 0$.

Then:

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

## Proof

Let $s = a + b i$, where $a, b \in \R$, $a<0$.

 $\displaystyle \left\vert \lim_{t \to + \infty} t^s \right\vert$ $=$ $\displaystyle \lim_{t \to + \infty} \left\vert t^s \right\vert$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \lim_{t \to + \infty} \left\vert e^{s \ln t} \right\vert$ $\quad$ Definition of Power to Complex Number $\quad$ $\displaystyle$ $=$ $\displaystyle \lim_{t \to + \infty} \left\vert e^{(a + b i) \ln t} \right\vert$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \lim_{t \to + \infty} \left\vert e^{a \ln t} e^{i b \ln t} \right\vert$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \lim_{t \to + \infty} \left\vert e^{a \ln t} \right\vert \left\vert e^{i b \ln t} \right\vert$ $\quad$ Modulus of Product $\quad$ $\displaystyle$ $=$ $\displaystyle \lim_{t \to + \infty} \left\vert e^{a \ln t} \right\vert$ $\quad$ Modulus of Exponential of Imaginary Number is One $\quad$ $\displaystyle$ $=$ $\displaystyle \lim_{t \to + \infty} \left\vert t^a \right\vert$ $\quad$ Definition of Power to Complex Number $\quad$ $\displaystyle$ $=$ $\displaystyle \lim_{t \to + \infty} \left\vert \dfrac{1}{t^{-a} } \right\vert$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle 0$ $\quad$ Power of Reciprocal $\quad$ $\displaystyle \therefore \ \$ $\displaystyle \lim_{t \to + \infty} t^s$ $=$ $\displaystyle 0$ $\quad$ $\quad$

$\Box$