Mellin Transform of Heaviside Step Function/Lemma

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

$\blacksquare$