Mellin Transform of Heaviside Step Function/Lemma

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Theorem


Let $t \in \R$.

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

Then:

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


Proof

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

\(\displaystyle \cmod {\lim_{t \mathop \to + \infty} t^s}\) \(=\) \(\displaystyle \lim_{t \mathop \to +\infty} \cmod {t^s}\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{t \mathop \to +\infty} \cmod {e^{s \ln t} }\) Definition of Power to Complex Number
\(\displaystyle \) \(=\) \(\displaystyle \lim_{t \mathop \to +\infty} \cmod {e^{\paren {a + b i} \ln t} }\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{t \mathop \to +\infty} \cmod {e^{a \ln t} e^{i b \ln t} }\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{t \mathop \to +\infty} \cmod {e^{a \ln t} } \cmod {e^{i b \ln t} }\) Modulus of Product
\(\displaystyle \) \(=\) \(\displaystyle \lim_{t \mathop \to +\infty} \cmod {e^{a \ln t} }\) Modulus of Exponential of Imaginary Number is One
\(\displaystyle \) \(=\) \(\displaystyle \lim_{t \mathop \to +\infty} \cmod {t^a}\) Definition of Power to Complex Number
\(\displaystyle \) \(=\) \(\displaystyle \lim_{t \mathop \to +\infty} \cmod {\dfrac 1 {t^{-a} } }\)
\(\displaystyle \) \(=\) \(\displaystyle 0\) Sequence of Powers of Reciprocals is Null Sequence
\(\displaystyle \leadsto \ \ \) \(\displaystyle \lim_{t \mathop \to +\infty} t^s\) \(=\) \(\displaystyle 0\)

$\blacksquare$