DLMF:12.14.E35 (Q4277)

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DLMF:12.14.E35
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    Statements

    Defining formula
    W ′ ⁡ ( - 1 2 ⁢ μ 2 , μ ⁢ t ⁢ 2 ) ∼ - μ 2 ⁢ l ⁢ ( μ ) ⁢ ( t 2 + 1 ) 1 4 ⁢ ( sin ⁡ σ ¯ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ v ¯ 2 ⁢ s ⁢ ( t ) ( t 2 + 1 ) 3 ⁢ s ⁢ μ 4 ⁢ s + cos ⁡ σ ¯ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ v ¯ 2 ⁢ s + 1 ⁢ ( t ) ( t 2 + 1 ) 3 ⁢ s + 3 2 ⁢ μ 4 ⁢ s + 2 ) , asymptotic-expansion diffop parabolic-W 1 1 2 superscript 𝜇 2 𝜇 𝑡 2 𝜇 2 𝑙 𝜇 superscript superscript 𝑡 2 1 1 4 ¯ 𝜎 superscript subscript 𝑠 0 superscript 1 𝑠 subscript ¯ 𝑣 2 𝑠 𝑡 superscript superscript 𝑡 2 1 3 𝑠 superscript 𝜇 4 𝑠 ¯ 𝜎 superscript subscript 𝑠 0 superscript 1 𝑠 subscript ¯ 𝑣 2 𝑠 1 𝑡 superscript superscript 𝑡 2 1 3 𝑠 3 2 superscript 𝜇 4 𝑠 2 {\displaystyle{\displaystyle W'\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)% \sim-\frac{\mu}{\sqrt{2}}l(\mu)(t^{2}+1)^{\frac{1}{4}}\left(\sin\overline{% \sigma}\sum_{s=0}^{\infty}\frac{(-1)^{s}\overline{v}_{2s}(t)}{(t^{2}+1)^{3s}% \mu^{4s}}+\cos\overline{\sigma}\sum_{s=0}^{\infty}\frac{(-1)^{s}\overline{v}_{% 2s+1}(t)}{(t^{2}+1)^{3s+\frac{3}{2}}\mu^{4s+2}}\right),}}
    0 references
    DLMF id
    DLMF:12.14.E35
    0 references
    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11ajdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aedec
    0 references
    Q11543
    Defining formula
    W ⁡ ( a , x ) parabolic-W 𝑎 𝑥 {\displaystyle{\displaystyle W\left(\NVar{a},\NVar{x}\right)}}
    xml-id
    C12.S14.SS1.p1.m1asdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aedec
    0 references
    Q11526
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C12.S1.XMD5.m1hdec
    0 references
    solution (locally)
    Defining formula
    v ¯ s ⁢ ( t ) subscript ¯ 𝑣 𝑠 𝑡 {\displaystyle{\displaystyle\overline{v}_{s}(t)}}
    xml-id
    C12.S10.XMD18.m1dec
    0 references
    DLMF:12.14.E27
    Defining formula
    σ 𝜎 {\displaystyle{\displaystyle\sigma}}
    xml-id
    C12.S14.XMD12.m1ddec
    0 references
    DLMF:12.14.E28
    Defining formula
    l ⁢ ( μ ) 𝑙 𝜇 {\displaystyle{\displaystyle l(\mu)}}
    xml-id
    C12.S14.XMD13.m1hdec
    0 references
    DLMF:12.14.E36
    Defining formula
    σ ¯ ¯ 𝜎 {\displaystyle{\displaystyle\overline{\sigma}}}
    xml-id
    C12.S14.XMD15.m1adec
    0 references
     
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