# Laplace Transform of Reciprocal of Square Root

## 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

$\blacksquare$