Gamma integral
= Gamma integral
{c}
For $a,s>0$, the change of variable $u=at$ in the defining integral for the <Gamma function> gives
$$
\int_0^\infty t^{s-1}e^{-at}\,dt=a^{-s}\Gamma(s).
$$
In particular, $\Gamma(1/2)=\sqrt\pi$ implies
$$
u^{-1/2}=\frac1{\sqrt\pi}\int_0^\infty t^{-1/2}e^{-ut}\,dt.
$$