DLMF:10.20.E4 (Q3253)

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DLMF:10.20.E4
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    Statements

    Defining formula
    J ν ⁡ ( ν ⁢ z ) ∼ ( 4 ⁢ ζ 1 - z 2 ) 1 4 ⁢ ( Ai ⁡ ( ν 2 3 ⁢ ζ ) ν 1 3 ⁢ ∑ k = 0 ∞ A k ⁢ ( ζ ) ν 2 ⁢ k + Ai ′ ⁡ ( ν 2 3 ⁢ ζ ) ν 5 3 ⁢ ∑ k = 0 ∞ B k ⁢ ( ζ ) ν 2 ⁢ k ) , asymptotic-expansion Bessel-J 𝜈 𝜈 𝑧 superscript 4 𝜁 1 superscript 𝑧 2 1 4 Airy-Ai superscript 𝜈 2 3 𝜁 superscript 𝜈 1 3 superscript subscript 𝑘 0 subscript 𝐴 𝑘 𝜁 superscript 𝜈 2 𝑘 diffop Airy-Ai 1 superscript 𝜈 2 3 𝜁 superscript 𝜈 5 3 superscript subscript 𝑘 0 subscript 𝐵 𝑘 𝜁 superscript 𝜈 2 𝑘 {\displaystyle{\displaystyle J_{\nu}\left(\nu z\right)\sim\left(\frac{4\zeta}{% 1-z^{2}}\right)^{\frac{1}{4}}\*\left(\frac{\mathrm{Ai}\left(\nu^{\frac{2}{3}}% \zeta\right)}{\nu^{\frac{1}{3}}}\sum_{k=0}^{\infty}\frac{A_{k}(\zeta)}{\nu^{2k% }}+\frac{\mathrm{Ai}'\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}% \sum_{k=0}^{\infty}\frac{B_{k}(\zeta)}{\nu^{2k}}\right),}}
    0 references
    DLMF id
    DLMF:10.20.E4
    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.m1adec
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    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2adec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
    0 references
    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1dec
    0 references
     
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