DLMF:12.10.E25 (Q4189)

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DLMF:12.10.E25
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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 ∞ u ¯ s ⁢ ( t ) ( t 2 + 1 ) 3 2 ⁢ s ⁢ 1 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 ¯ 𝑢 𝑠 𝑡 superscript superscript 𝑡 2 1 3 2 𝑠 1 superscript 𝜇 2 𝑠 {\displaystyle{\displaystyle U\left(\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)% \sim\frac{\overline{g}(\mu)e^{-\mu^{2}{\overline{\xi}}}}{(t^{2}+1)^{\frac{1}{4% }}}\sum_{s=0}^{\infty}\frac{\overline{u}_{s}(t)}{(t^{2}+1)^{\frac{3}{2}s}}% \frac{1}{\mu^{2s}},}}
    0 references
    DLMF id
    DLMF:12.10.E25
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    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11ajdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
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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.m1addec
    0 references
    Q11526
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C12.S1.XMD5.m1qdec
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    DLMF:12.10.E26
    Defining formula
    ξ ¯ ¯ 𝜉 {\displaystyle{\displaystyle\overline{\xi}}}
    xml-id
    C12.S10.XMD15.m1dec
    0 references
    solution (locally)
    Defining formula
    u ¯ s ⁢ ( t ) subscript ¯ 𝑢 𝑠 𝑡 {\displaystyle{\displaystyle\overline{u}_{s}(t)}}
    xml-id
    C12.S10.XMD16.m1dec
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