DLMF:9.6.E7 (Q2787)

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DLMF:9.6.E7
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    Statements

    Defining formula
    Ai ′ ⁡ ( - z ) = ( z / 3 ) ⁢ ( J 2 / 3 ⁡ ( ζ ) - J - 2 / 3 ⁡ ( ζ ) ) = 1 2 ⁢ ( z / 3 ) ⁢ ( e - π ⁢ i / 6 ⁢ H 2 / 3 ( 1 ) ⁡ ( ζ ) + e π ⁢ i / 6 ⁢ H 2 / 3 ( 2 ) ⁡ ( ζ ) ) = 1 2 ⁢ ( z / 3 ) ⁢ ( e - 5 ⁢ π ⁢ i / 6 ⁢ H - 2 / 3 ( 1 ) ⁡ ( ζ ) + e 5 ⁢ π ⁢ i / 6 ⁢ H - 2 / 3 ( 2 ) ⁡ ( ζ ) ) , diffop Airy-Ai 1 𝑧 𝑧 3 Bessel-J 2 3 𝜁 Bessel-J 2 3 𝜁 1 2 𝑧 3 superscript 𝑒 𝜋 𝑖 6 Hankel-H-1-Bessel-third-kind 2 3 𝜁 superscript 𝑒 𝜋 𝑖 6 Hankel-H-2-Bessel-third-kind 2 3 𝜁 1 2 𝑧 3 superscript 𝑒 5 𝜋 𝑖 6 Hankel-H-1-Bessel-third-kind 2 3 𝜁 superscript 𝑒 5 𝜋 𝑖 6 Hankel-H-2-Bessel-third-kind 2 3 𝜁 {\displaystyle{\displaystyle\mathrm{Ai}'\left(-z\right)=(z/3)\left(J_{2/3}% \left(\zeta\right)-J_{-2/3}\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})% \left(e^{-\pi i/6}{H^{(1)}_{2/3}}\left(\zeta\right)+e^{\pi i/6}{H^{(2)}_{2/3}}% \left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{-5\pi i/6}{H^{(1)}_% {-2/3}}\left(\zeta\right)+e^{5\pi i/6}{H^{(2)}_{-2/3}}\left(\zeta\right)\right% ),}}
    0 references
    DLMF id
    DLMF:9.6.E7
    0 references
    Symbols used
    Q10919
    Defining formula
    Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m1acdec
    0 references
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aadec
    0 references
    Bessel function of the third kind (or Hankel function)
    Defining formula
    H ν ( 1 ) ⁡ ( z ) Hankel-H-1-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E5.m2aedec
    0 references
    Bessel function of the third kind (or Hankel function)
    Defining formula
    H ν ( 2 ) ⁡ ( z ) Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E6.m2aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    Q11381
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1fdec
    0 references
    change of variable (locally)
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C9.S6.XMD1.m1fdec
    0 references
     
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