Mellin Transform of Heaviside Step Function/Lemma

From ProofWiki
Jump to navigation Jump to search

Theorem




Let $t \in \R$.

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

Then:

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


Proof

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

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

$\blacksquare$