DLMF:12.10.E36 (Q4201)

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DLMF:12.10.E36
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    Statements

    Defining formula
    U ′ ⁡ ( - 1 2 ⁢ μ 2 , μ ⁢ t ⁢ 2 ) ∼ ( 2 ⁢ π ) 1 2 ⁢ μ 2 3 ⁢ g ⁢ ( μ ) ϕ ⁢ ( ζ ) ⁢ ( Ai ⁡ ( μ 4 3 ⁢ ζ ) μ 4 3 ⁢ ∑ s = 0 ∞ C s ⁢ ( ζ ) μ 4 ⁢ s + Ai ′ ⁡ ( μ 4 3 ⁢ ζ ) ⁢ ∑ s = 0 ∞ D s ⁢ ( ζ ) μ 4 ⁢ s ) , asymptotic-expansion diffop parabolic-U 1 1 2 superscript 𝜇 2 𝜇 𝑡 2 superscript 2 𝜋 1 2 superscript 𝜇 2 3 𝑔 𝜇 italic-ϕ 𝜁 Airy-Ai superscript 𝜇 4 3 𝜁 superscript 𝜇 4 3 superscript subscript 𝑠 0 subscript 𝐶 𝑠 𝜁 superscript 𝜇 4 𝑠 diffop Airy-Ai 1 superscript 𝜇 4 3 𝜁 superscript subscript 𝑠 0 subscript 𝐷 𝑠 𝜁 superscript 𝜇 4 𝑠 {\displaystyle{\displaystyle U'\left(-\frac{1}{2}\mu^{2},\mu t\sqrt{2}\right)% \sim\frac{(2\pi)^{\frac{1}{2}}\mu^{\frac{2}{3}}g(\mu)}{\phi(\zeta)}\*\left(% \frac{\mathrm{Ai}\left(\mu^{\frac{4}{3}}\zeta\right)}{\mu^{\frac{4}{3}}}\sum_{% s=0}^{\infty}\frac{C_{s}(\zeta)}{\mu^{4s}}+\mathrm{Ai}'\left(\mu^{\frac{4}{3}}% \zeta\right)\sum_{s=0}^{\infty}\frac{D_{s}(\zeta)}{\mu^{4s}}\right),}}
    0 references
    DLMF id
    DLMF:12.10.E36
    0 references
    Symbols used
    Q10919
    Defining formula
    Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m1aadec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11aodec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    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.m1ahdec
    0 references
    Q11526
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C12.S1.XMD5.m1ydec
    0 references
    Q11534
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C12.S10.XMD21.m1adec
    0 references
    function (locally)
    Defining formula
    ϕ ⁢ ( ζ ) italic-ϕ 𝜁 {\displaystyle{\displaystyle\phi(\zeta)}}
    xml-id
    C12.S10.XMD22.m1adec
    0 references
    Q11537
    Defining formula
    C s ⁢ ( ζ ) subscript 𝐶 𝑠 𝜁 {\displaystyle{\displaystyle C_{s}(\zeta)}}
    xml-id
    C12.S10.XMD27.m1dec
    0 references
    Q11538
    Defining formula
    D s ⁢ ( ζ ) subscript 𝐷 𝑠 𝜁 {\displaystyle{\displaystyle D_{s}(\zeta)}}
    xml-id
    C12.S10.XMD28.m1dec
    0 references
     
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