\[\operatorname{Ai}\left(z\right)=\pi^{-1}\sqrt{z/3}K_{\pm 1/3}\left(\zeta\right%
)=\tfrac{1}{3}\sqrt{z}\left(I_{-1/3}\left(\zeta\right)-I_{1/3}\left(\zeta%
\right)\right)=\tfrac{1}{2}\sqrt{z/3}e^{2\pi i/3}{H^{(1)}_{1/3}}\left(\zeta e^%
{\pi i/2}\right)=\tfrac{1}{2}\sqrt{z/3}e^{\pi i/3}{H^{(1)}_{-1/3}}\left(\zeta e%
^{\pi i/2}\right)=\tfrac{1}{2}\sqrt{z/3}e^{-2\pi i/3}{H^{(2)}_{1/3}}\left(%
\zeta e^{-\pi i/2}\right)=\tfrac{1}{2}\sqrt{z/3}e^{-\pi i/3}{H^{(2)}_{-1/3}}%
\left(\zeta e^{-\pi i/2}\right),\]