DLMF:9.10.E7 (Q2889)

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DLMF:9.10.E7
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    Statements

    Defining formula
    ∫ - ∞ x Bi ⁡ ( t ) ⁢ d t = π - 1 / 2 ⁢ ( - x ) - 3 / 4 ⁢ sin ⁡ ( 2 3 ⁢ ( - x ) 3 / 2 + 1 4 ⁢ π ) + O ⁡ ( | x | - 9 / 4 ) , superscript subscript 𝑥 Airy-Bi 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{Bi}\left(t\right)\mathrm% {d}t=\pi^{-1/2}(-x)^{-3/4}\*\sin\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.E7
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    constraint
    x → - ∞ → 𝑥 {\displaystyle{\displaystyle x\rightarrow-\infty}}
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    Symbols used
    Q11081
    Defining formula
    Bi ⁡ ( z ) Airy-Bi 𝑧 {\displaystyle{\displaystyle\mathrm{Bi}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m2abdec
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    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2aadec
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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.m2afdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
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