DLMF:9.10.E6 (Q2888)

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DLMF:9.10.E6
No description defined

    Statements

    test
    ∫ - ∞ x Ai ⁡ ( t ) ⁢ d t = π - 1 / 2 ⁢ ( - x ) - 3 / 4 ⁢ cos ⁡ ( 2 3 ⁢ ( - x ) 3 / 2 + 1 4 ⁢ π ) + O ⁡ ( | x | - 9 / 4 ) , superscript subscript 𝑥 Airy-Ai 𝑡 𝑡 superscript 𝜋 1 2 superscript 𝑥 3 4 2 3 superscript 𝑥 3 2 1 4 𝜋 Big-O superscript 𝑥 9 4 {\displaystyle{\displaystyle\int_{-\infty}^{x}\mathrm{Ai}\left(t\right)\mathrm% {d}t=\pi^{-1/2}(-x)^{-3/4}\*\cos\left(\tfrac{2}{3}(-x)^{3/2}+\tfrac{1}{4}\pi% \right)+O\left(|x|^{-9/4}\right),}}
    0 references
    DLMF id
    DLMF:9.10.E6
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    constraint
    x → - ∞ → 𝑥 {\displaystyle{\displaystyle x\rightarrow-\infty}}
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    Symbols used
    Q10919
    test
    Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m1acdec
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    order not exceeding
    test
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2adec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
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    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
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    Q11382
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C9.S1.XMD2.m1bdec
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