# Mellin Transform of Heaviside Step Function/Lemma

## 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 | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \lim_{t \mathop \to +\infty} t^s\) | \(=\) | \(\displaystyle 0\) |

$\blacksquare$