# Mellin Transform of Heaviside Step Function/Lemma

## Theorem

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 | $\quad$ | |||||||||

\(\displaystyle \therefore \ \ \) | \(\displaystyle \lim_{t \to + \infty} t^s\) | \(=\) | \(\displaystyle 0\) | $\quad$ | $\quad$ |

$\Box$