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,}}
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    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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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1agdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3agdec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1abdec
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    Q11381
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1ddec
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    Q11383
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C9.S5.XMD1.m1adec
    0 references
     
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