DLMF:32.11.E13 (Q9448)

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DLMF:32.11.E13
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    Statements

    Defining formula
    w ⁢ ( x ) = d ⁢ | x | - 1 / 4 ⁢ sin ⁡ ( ϕ ⁢ ( x ) - χ ) + O ⁡ ( | x | - 5 / 4 ⁢ ln ⁡ | x | ) , 𝑤 𝑥 𝑑 superscript 𝑥 1 4 italic-ϕ 𝑥 𝜒 Big-O superscript 𝑥 5 4 𝑥 {\displaystyle{\displaystyle w(x)=d|x|^{-1/4}\sin\left(\phi(x)-\chi\right)+O% \left(|x|^{-5/4}\ln|x|\right),}}
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    DLMF id
    DLMF:32.11.E13
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    constraint
    x → - ∞ → 𝑥 {\displaystyle{\displaystyle x\to-\infty}}
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    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2adec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2addec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
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    Q12533
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C32.S1.XMD3.m1jdec
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    Q12638
    Defining formula
    χ 𝜒 {\displaystyle{\displaystyle\chi}}
    xml-id
    C32.S11.XMD10.m1dec
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    Q12639
    Defining formula
    ϕ ⁢ ( z ) italic-ϕ 𝑧 {\displaystyle{\displaystyle\phi(z)}}
    xml-id
    C32.S11.XMD8.m1dec
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    Q12640
    Defining formula
    d 𝑑 {\displaystyle{\displaystyle d}}
    xml-id
    C32.S11.XMD9.m1dec
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