Formula 10.4.17
$Ai'(-z)=-\frac{1}{3}z[J_{-2/3}(\zeta)-J_{2/3}(\zeta)]=\frac{1}{2}(z/\sqrt{3})[e^{-\pi i/6}H^{(1)}_{2/3}(\zeta)+e^{\pi i/6}H^{(2)}_{2/3}(\zeta)]$