Mellin Transform of Heaviside Step Function/Lemma

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

$\Box$