DLMF:32.11.E19 (Q9454)

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DLMF:32.11.E19
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    Statements

    Defining formula
    w ⁢ ( x ) = σ ⁢ 1 2 ⁢ x + σ ⁢ ρ ⁢ ( 2 ⁢ x ) - 1 / 4 ⁢ cos ⁡ ( ψ ⁢ ( x ) + θ ) + O ⁡ ( x - 1 ) , 𝑤 𝑥 𝜎 1 2 𝑥 𝜎 𝜌 superscript 2 𝑥 1 4 𝜓 𝑥 𝜃 Big-O superscript 𝑥 1 {\displaystyle{\displaystyle w(x)=\sigma\sqrt{\tfrac{1}{2}x}+\sigma\rho(2x)^{-% 1/4}\cos\left(\psi(x)+\theta\right)+O\left(x^{-1}\right),}}
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    DLMF id
    DLMF:32.11.E19
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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.m2aadec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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    Q12533
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C32.S1.XMD3.m1mdec
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    Q12641
    Defining formula
    σ 𝜎 {\displaystyle{\displaystyle\sigma}}
    xml-id
    C32.S11.XMD11.m1dec
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    Q12642
    Defining formula
    ρ 𝜌 {\displaystyle{\displaystyle\rho}}
    xml-id
    C32.S11.XMD12.m1dec
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    Q12643
    Defining formula
    θ 𝜃 {\displaystyle{\displaystyle\theta}}
    xml-id
    C32.S11.XMD13.m1dec
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    Q12644
    Defining formula
    ψ ⁢ ( z ) 𝜓 𝑧 {\displaystyle{\displaystyle\psi(z)}}
    xml-id
    C32.S11.XMD14.m1dec
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