DLMF:9.5.E8 (Q2780)

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DLMF:9.5.E8
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    Statements

    Defining formula
    Ai ⁡ ( z ) = e - ζ ⁢ ζ - 1 / 6 π ⁢ ( 48 ) 1 / 6 ⁢ Γ ⁡ ( 5 6 ) ⁢ ∫ 0 ∞ e - t ⁢ t - 1 / 6 ⁢ ( 2 + t ζ ) - 1 / 6 ⁢ d t , Airy-Ai 𝑧 superscript 𝑒 𝜁 superscript 𝜁 1 6 𝜋 superscript 48 1 6 Euler-Gamma 5 6 superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 1 6 superscript 2 𝑡 𝜁 1 6 𝑡 {\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\frac{e^{-\zeta}\zeta^{% \ifrac{-1}{6}}}{\sqrt{\pi}(48)^{\ifrac{1}{6}}\Gamma\left(\frac{5}{6}\right)}% \int_{0}^{\infty}e^{-t}t^{-\ifrac{1}{6}}\left(2+\frac{t}{\zeta}\right)^{-% \ifrac{1}{6}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:9.5.E8
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    constraint
    | ph ⁡ z | < 2 3 ⁢ π phase 𝑧 2 3 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\frac{2}{3}\pi}}
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    Symbols used
    Q10919
    Defining formula
    Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m1aedec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
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