DLMF:9.10.E22 (Q2907)

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DLMF:9.10.E22
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    Statements

    Defining formula
    ∫ 0 ∞ ∫ t ∞ ⋯ ⁢ ∫ t ∞ Ai ⁡ ( - t ) ⁢ ( d t ) n = 2 ⁢ cos ⁡ ( 1 3 ⁢ ( n - 1 ) ⁢ π ) 3 ( n + 2 ) / 3 ⁢ Γ ⁡ ( 1 3 ⁢ n + 2 3 ) , superscript subscript 0 superscript subscript 𝑡 ⋯ superscript subscript 𝑡 Airy-Ai 𝑡 superscript 𝑡 𝑛 2 1 3 𝑛 1 𝜋 superscript 3 𝑛 2 3 Euler-Gamma 1 3 𝑛 2 3 {\displaystyle{\displaystyle\int_{0}^{\infty}\!\!\int_{t}^{\infty}\!\!\!\!% \cdots\int_{t}^{\infty}\mathrm{Ai}\left({-}t\right)(\mathrm{d}t)^{n}=\frac{2% \cos\left(\tfrac{1}{3}(n-1)\pi\right)}{3^{(n+2)/3}\Gamma\left(\tfrac{1}{3}n+% \tfrac{2}{3}\right)},}}
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    DLMF id
    DLMF:9.10.E22
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    constraint
    n = 1 , 2 , … . 𝑛 1 2 … {\displaystyle{\displaystyle n=1,2,\ldots.}}
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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.m1aldec
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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.m2addec
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