DLMF:9.6.E8 (Q2788)

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DLMF:9.6.E8
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    Bi ( - z ) = z / 3 ( J - 1 / 3 ( ζ ) - J 1 / 3 ( ζ ) ) = 1 2 z / 3 ( e 2 π i / 3 H 1 / 3 ( 1 ) ( ζ ) + e - 2 π i / 3 H 1 / 3 ( 2 ) ( ζ ) ) = 1 2 z / 3 ( e π i / 3 H - 1 / 3 ( 1 ) ( ζ ) + e - π i / 3 H - 1 / 3 ( 2 ) ( ζ ) ) , Airy-Bi 𝑧 𝑧 3 Bessel-J 1 3 𝜁 Bessel-J 1 3 𝜁 1 2 𝑧 3 superscript 𝑒 2 𝜋 𝑖 3 Hankel-H-1-Bessel-third-kind 1 3 𝜁 superscript 𝑒 2 𝜋 𝑖 3 Hankel-H-2-Bessel-third-kind 1 3 𝜁 1 2 𝑧 3 superscript 𝑒 𝜋 𝑖 3 Hankel-H-1-Bessel-third-kind 1 3 𝜁 superscript 𝑒 𝜋 𝑖 3 Hankel-H-2-Bessel-third-kind 1 3 𝜁 {\displaystyle{\displaystyle\mathrm{Bi}\left(-z\right)=\sqrt{z/3}\left(J_{-1/3% }\left(\zeta\right)-J_{1/3}\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}% \left(e^{2\pi i/3}{H^{(1)}_{1/3}}\left(\zeta\right)+e^{-2\pi i/3}{H^{(2)}_{1/3% }}\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/3}{H^{(1)}_{-% 1/3}}\left(\zeta\right)+e^{-\pi i/3}{H^{(2)}_{-1/3}}\left(\zeta\right)\right),}}
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    DLMF:9.6.E8
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