Laplace Transform of Real Power

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Theorem

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.


Proof

\(\ds \laptrans {t^n}\) \(=\) \(\ds \int_0^\infty e^{-s t} t^n \rd t\) Definition of Laplace Transform
\(\ds \) \(=\) \(\ds \int_0^\infty e^{-u} \paren {\dfrac u s}^n \rd \paren {\dfrac u s}\) Integration by Substitution: $u := s t$ where $s > 0$ is assumed
\(\ds \) \(=\) \(\ds \dfrac 1 {s^{n + 1} } \int_0^\infty u^n e^{-u} \rd u\)
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma {n + 1} } {s^{n + 1} }\) Definition of Gamma Function

$\blacksquare$


Sources

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