Laplace Transform of Positive Integer Power

Theorem
Let $t^n: \R \to \R$ be $t$ to the $n$th power for some $n \in \N_{\ge 0}$.

Let $\mathcal L$ be the Laplace Transform.

Then:


 * $\displaystyle \mathcal L \left\{ {t^n} \right\} = \frac {n!} { s^{n+1} }$

for $\operatorname{Re}\left({s}\right) > 1$.

Proof
The proof proceeds by induction on $n$ for $t^n$.

Basis for the Induction
The case $t^1 = t$ is proved in Laplace Transform of Identity Mapping.

This is the basis for the induction.

Induction Hypothesis
Fix $n \in \N$ with $n \ge 1$.

Assume:


 * $\displaystyle \mathcal L \left\{ {t^n} \right\} = \frac {n!} { s^{n+1} }$

This is our induction hypothesis.

Induction Step
This is our induction step:

From Integration by Parts:


 * $\displaystyle \int fg' \, \mathrm dt = fg - \int f'g \, \mathrm dt$

Here:

So:

Evaluating at $t = 0$ and $t \to +\infty$:

The result follows by the Principle of Mathematical Induction.