Gamma function asymptotic at zero

The functional equation and $\Gamma (1)=1$ give $\Gamma (z)\sim 1/z$ as $z\to 0$ away from the negative real axis.

The beta--gamma identity gives $B(z,z)=\Gamma (z{)}^2/\Gamma (2z)\sim 2/z$ as $z\to 0$ through the right half-plane.