Laplace Transform of Reciprocal of Square Root
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Theorem
- $\laptrans {\dfrac 1 {\sqrt t} } = \sqrt {\dfrac \pi s}$
where $\laptrans f$ denotes the Laplace transform of the real function $f$.
Proof
| \(\ds \laptrans {\dfrac 1 {\sqrt t} }\) | \(=\) | \(\ds \laptrans {t^{-1 / 2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\map \Gamma {1 / 2} } {s^{1 / 2} }\) | Laplace Transform of Real Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\sqrt \pi} {\sqrt s}\) | Gamma Function of One Half | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {\dfrac \pi s}\) | simplifying |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: The Gamma Function: $32$