DLMF:12.10.E3 (Q4157)

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DLMF:12.10.E3
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    Statements

    Defining formula
    U ⁡ ( - 1 2 ⁢ μ 2 , μ ⁢ t ⁢ 2 ) ∼ g ⁢ ( μ ) ⁢ e - μ 2 ⁢ ξ ( t 2 - 1 ) 1 4 ⁢ ∑ s = 0 ∞ 𝒜 s ⁢ ( t ) μ 2 ⁢ s , asymptotic-expansion parabolic-U 1 2 superscript 𝜇 2 𝜇 𝑡 2 𝑔 𝜇 superscript 𝑒 superscript 𝜇 2 𝜉 superscript superscript 𝑡 2 1 1 4 superscript subscript 𝑠 0 subscript 𝒜 𝑠 𝑡 superscript 𝜇 2 𝑠 {\displaystyle{\displaystyle U\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)% \sim\frac{g(\mu)e^{-\mu^{2}\xi}}{(t^{2}-1)^{\frac{1}{4}}}\sum_{s=0}^{\infty}% \frac{{\cal A}_{s}(t)}{\mu^{2s}},}}
    0 references
    DLMF id
    DLMF:12.10.E3
    0 references
    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    Q11320
    Defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1adec
    0 references
    Q11526
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C12.S1.XMD5.m1dec
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    DLMF:12.10.E7
    Defining formula
    ξ 𝜉 {\displaystyle{\displaystyle\xi}}
    xml-id
    C12.S10.XMD1.m1dec
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    coefficients (locally)
    Defining formula
    𝒜 s ⁢ ( t ) subscript 𝒜 𝑠 𝑡 {\displaystyle{\displaystyle\mathcal{A}_{s}(t)}}
    xml-id
    C12.S10.XMD2.m1dec
    0 references
    expansion (locally)
    Defining formula
    g ⁢ ( μ ) 𝑔 𝜇 {\displaystyle{\displaystyle g(\mu)}}
    xml-id
    C12.S10.XMD7.m1dec
    0 references
     
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