DLMF:12.10.E31 (Q4195)

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DLMF:12.10.E31
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    Statements

    Defining formula
    U ⁡ ( - 1 2 ⁢ μ 2 , μ ⁢ t ⁢ 2 ) ∼ h ⁢ ( μ ) ⁢ e - μ 2 ⁢ ξ ( t 2 - 1 ) 1 4 ⁢ ∑ s = 0 ∞ 𝖠 s ⁢ ( τ ) μ 2 ⁢ s , asymptotic-expansion parabolic-U 1 2 superscript 𝜇 2 𝜇 𝑡 2 ℎ 𝜇 superscript 𝑒 superscript 𝜇 2 𝜉 superscript superscript 𝑡 2 1 1 4 superscript subscript 𝑠 0 subscript 𝖠 𝑠 𝜏 superscript 𝜇 2 𝑠 {\displaystyle{\displaystyle U\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)% \sim\frac{h(\mu)e^{-\mu^{2}\xi}}{(t^{2}-1)^{\frac{1}{4}}}\sum_{s=0}^{\infty}% \frac{\mathsf{A}_{s}(\tau)}{\mu^{2s}},}}
    0 references
    DLMF id
    DLMF:12.10.E31
    0 references
    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11amdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ahdec
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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.m1afdec
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    Q11526
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C12.S1.XMD5.m1vdec
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    DLMF:12.10.E7
    Defining formula
    ξ 𝜉 {\displaystyle{\displaystyle\xi}}
    xml-id
    C12.S10.XMD1.m1edec
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    DLMF:12.10.E32
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C12.S10.XMD19.m1dec
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    coefficients (locally)
    Defining formula
    𝖠 s ⁢ ( τ ) subscript 𝖠 𝑠 𝜏 {\displaystyle{\displaystyle\mathsf{A}_{s}(\tau)}}
    xml-id
    C12.S10.XMD20.m1dec
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