Laplace Transform of Power

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Theorem

Laplace Transform of Positive Integer Power

Let $\laptrans f$ denote the Laplace transform of a function $f$.

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


Then:

$\laptrans {t^n} = \dfrac {n!} { s^{n + 1} }$

for $\map \Re s > 0$.


Laplace Transform of Real Power

Let $n$ be a constant real number such that $n > -1$

Let $f: \R \to \R$ be the real function defined as:

$\map f t = t^n$


Then $f$ has a Laplace transform given by:

\(\ds \laptrans {\map f t}\) \(=\) \(\ds \int_0^\infty e^{-s t} t^n \rd t\)
\(\ds \) \(=\) \(\ds \frac {\map \Gamma {n + 1} } {s^{n + 1} }\)

where $\Gamma$ denotes the gamma function.


Laplace Transform of Complex Power

Let $q$ be a constant complex number with $\map \Re q > -1$

Let $t^q: \R_{>0} \to \C$ be a branch of the complex power multifunction chosen such that $f$ is continuous on the half-plane $\map \Re s > 0$.


Then $f$ has a Laplace transform given by:

$\laptrans {t^q} = \dfrac {\map \Gamma {q + 1} } {s^{q + 1} }$

where $\Gamma$ denotes the gamma function.