# DLMF:9.6.E2 (Q2782)

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DLMF:9.6.E2
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## Statements

${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\pi^{-1}\sqrt{z/3}K_{\pm 1% /3}\left(\zeta\right)=\tfrac{1}{3}\sqrt{z}\left(I_{-1/3}\left(\zeta\right)-I_{% 1/3}\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}e^{2\pi i/3}{H^{(1)}_{1/3}% }\left(\zeta e^{\pi i/2}\right)=\tfrac{1}{2}\sqrt{z/3}e^{\pi i/3}{H^{(1)}_{-1/% 3}}\left(\zeta e^{\pi i/2}\right)=\tfrac{1}{2}\sqrt{z/3}e^{-2\pi i/3}{H^{(2)}_% {1/3}}\left(\zeta e^{-\pi i/2}\right)=\tfrac{1}{2}\sqrt{z/3}e^{-\pi i/3}{H^{(2% )}_{-1/3}}\left(\zeta e^{-\pi i/2}\right),}}$
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DLMF:9.6.E2
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${\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}$
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${\displaystyle{\displaystyle{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}$
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${\displaystyle{\displaystyle{H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}$
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${\displaystyle{\displaystyle\pi}}$
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${\displaystyle{\displaystyle\mathrm{e}}}$
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${\displaystyle{\displaystyle\mathrm{i}}}$
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${\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}$
${\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}$
${\displaystyle{\displaystyle z}}$
${\displaystyle{\displaystyle\zeta}}$