Mellin Transform of Heaviside Step Function/Lemma
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Theorem
It has been suggested that this page be renamed. In particular: 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. To discuss this page in more detail, feel free to use the talk page. |
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 | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{t \mathop \to +\infty} t^s\) | \(=\) | \(\ds 0\) |
$\blacksquare$