DLMF:12.14.E26 (Q4266)

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DLMF:12.14.E26
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    Statements

    Defining formula
    W ⁡ ( 1 2 ⁢ μ 2 , - μ ⁢ t ⁢ 2 ) ∼ 2 1 2 ⁢ e 1 4 ⁢ π ⁢ μ 2 ⁢ l ⁢ ( μ ) ( t 2 - 1 ) 1 4 ⁢ ( sin ⁡ σ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ 𝒜 2 ⁢ s ⁢ ( t ) μ 4 ⁢ s + cos ⁡ σ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ 𝒜 2 ⁢ s + 1 ⁢ ( t ) μ 4 ⁢ s + 2 ) , asymptotic-expansion parabolic-W 1 2 superscript 𝜇 2 𝜇 𝑡 2 superscript 2 1 2 superscript 𝑒 1 4 𝜋 superscript 𝜇 2 𝑙 𝜇 superscript superscript 𝑡 2 1 1 4 𝜎 superscript subscript 𝑠 0 superscript 1 𝑠 subscript 𝒜 2 𝑠 𝑡 superscript 𝜇 4 𝑠 𝜎 superscript subscript 𝑠 0 superscript 1 𝑠 subscript 𝒜 2 𝑠 1 𝑡 superscript 𝜇 4 𝑠 2 {\displaystyle{\displaystyle W\left(\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)% \sim\frac{2^{\frac{1}{2}}e^{\frac{1}{4}\pi\mu^{2}}l(\mu)}{(t^{2}-1)^{\frac{1}{% 4}}}\left(\sin\sigma\sum_{s=0}^{\infty}(-1)^{s}\frac{{\cal{A}}_{2s}(t)}{\mu^{4% s}}+\cos\sigma\sum_{s=0}^{\infty}(-1)^{s}\frac{{\cal{A}}_{2s+1}(t)}{\mu^{4s+2}% }\right),}}
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    DLMF id
    DLMF:12.14.E26
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    Symbols used
    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.m2agdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
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