DLMF:9.6.E21 (Q2801)

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DLMF:9.6.E21
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    Statements

    Defining formula
    Ai ⁡ ( z ) = 1 2 ⁢ π - 1 / 2 ⁢ z - 1 / 4 ⁢ W 0 , 1 / 3 ⁡ ( 2 ⁢ ζ ) = 3 - 1 / 6 ⁢ π - 1 / 2 ⁢ ζ 2 / 3 ⁢ e - ζ ⁢ U ⁡ ( 5 6 , 5 3 , 2 ⁢ ζ ) , Airy-Ai 𝑧 1 2 superscript 𝜋 1 2 superscript 𝑧 1 4 Whittaker-confluent-hypergeometric-W 0 1 3 2 𝜁 superscript 3 1 6 superscript 𝜋 1 2 superscript 𝜁 2 3 superscript 𝑒 𝜁 Kummer-confluent-hypergeometric-U 5 6 5 3 2 𝜁 {\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\tfrac{1}{2}\pi^{-1/2}z^% {-1/4}W_{0,1/3}\left(2\zeta\right)=3^{-1/6}\pi^{-1/2}\zeta^{2/3}e^{-\zeta}U% \left(\tfrac{5}{6},\tfrac{5}{3},2\zeta\right),}}
    0 references
    DLMF id
    DLMF:9.6.E21
    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.m1andec
    0 references
    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2adec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2andec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aldec
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    Q11381
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1tdec
    0 references
    change of variable (locally)
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C9.S6.XMD1.m1idec
    0 references
     
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