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