DLMF:9.6.E6 (Q2786)

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DLMF:9.6.E6
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    Statements

    Defining formula
    Ai ⁡ ( - z ) = ( z / 3 ) ⁢ ( J 1 / 3 ⁡ ( ζ ) + J - 1 / 3 ⁡ ( ζ ) ) = 1 2 ⁢ z / 3 ⁢ ( e π ⁢ i / 6 ⁢ H 1 / 3 ( 1 ) ⁡ ( ζ ) + e - π ⁢ i / 6 ⁢ H 1 / 3 ( 2 ) ⁡ ( ζ ) ) = 1 2 ⁢ z / 3 ⁢ ( e - π ⁢ i / 6 ⁢ H - 1 / 3 ( 1 ) ⁡ ( ζ ) + e π ⁢ i / 6 ⁢ H - 1 / 3 ( 2 ) ⁡ ( ζ ) ) , Airy-Ai 𝑧 𝑧 3 Bessel-J 1 3 𝜁 Bessel-J 1 3 𝜁 1 2 𝑧 3 superscript 𝑒 𝜋 𝑖 6 Hankel-H-1-Bessel-third-kind 1 3 𝜁 superscript 𝑒 𝜋 𝑖 6 Hankel-H-2-Bessel-third-kind 1 3 𝜁 1 2 𝑧 3 superscript 𝑒 𝜋 𝑖 6 Hankel-H-1-Bessel-third-kind 1 3 𝜁 superscript 𝑒 𝜋 𝑖 6 Hankel-H-2-Bessel-third-kind 1 3 𝜁 {\displaystyle{\displaystyle\mathrm{Ai}\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^{\pi i/6}{H^{(1)}_{1/3}}\left(\zeta\right)+e^{-\pi i/6}{H^{(2)}_{1/3}}% \left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{-\pi i/6}{H^{(1)}_{-1% /3}}\left(\zeta\right)+e^{\pi i/6}{H^{(2)}_{-1/3}}\left(\zeta\right)\right),}}
    0 references
    DLMF id
    DLMF:9.6.E6
    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.m1abdec
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    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.m2adec
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    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.m2addec
    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.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
    0 references
    Q11381
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1edec
    0 references
    change of variable (locally)
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C9.S6.XMD1.m1edec
    0 references
     
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