DLMF:10.67.E5 (Q3850)

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DLMF:10.67.E5
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    Statements

    Defining formula
    ker ν ′ ⁡ x ∼ - e - x / 2 ⁢ ( π 2 ⁢ x ) 1 2 ⁢ ∑ k = 0 ∞ b k ⁢ ( ν ) x k ⁢ cos ⁡ ( x 2 + ( ν 2 + k 4 - 1 8 ) ⁢ π ) , asymptotic-expansion diffop Kelvin-ker 𝜈 1 𝑥 superscript 𝑒 𝑥 2 superscript 𝜋 2 𝑥 1 2 superscript subscript 𝑘 0 subscript 𝑏 𝑘 𝜈 superscript 𝑥 𝑘 𝑥 2 𝜈 2 𝑘 4 1 8 𝜋 {\displaystyle{\displaystyle\operatorname{ker}_{\nu}'x\sim-e^{-x/\sqrt{2}}% \left(\frac{\pi}{2x}\right)^{\frac{1}{2}}\sum_{k=0}^{\infty}\frac{b_{k}(\nu)}{% x^{k}}\*\cos\left(\frac{x}{\sqrt{2}}+\left(\frac{\nu}{2}+\frac{k}{4}-\frac{1}{% 8}\right)\pi\right),}}
    0 references
    DLMF id
    DLMF:10.67.E5
    0 references
    Symbols used
    Kelvin function
    Defining formula
    ker ν ⁡ ( x ) Kelvin-ker 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C10.S61.E2.m2acdec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11addec
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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.m2addec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1ddec
    0 references
    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1ddec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1ddec
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    expansion (locally)
    Defining formula
    b k ⁢ ( ν ) subscript 𝑏 𝑘 𝜈 {\displaystyle{\displaystyle b_{k}(\nu)}}
    xml-id
    C10.S17.XMD3.m1dec
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