DLMF:10.21.E38 (Q3338)

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DLMF:10.21.E38
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    Statements

    Defining formula
    J ν ⁡ ( j ν , m ′ ) ∼ ( - 1 ) m - 1 ⁢ ( 2 / ν ) 1 3 π ⁢ N ⁡ ( a m ′ ) ⁢ ∑ k = 0 ∞ β k ′ ν 2 ⁢ k / 3 , asymptotic-expansion Bessel-J 𝜈 diffop zero-Bessel-J 𝜈 𝑚 1 superscript 1 𝑚 1 superscript 2 𝜈 1 3 𝜋 modulus-Airy-N zero-derivative-Airy-Ai 𝑚 superscript subscript 𝑘 0 subscript superscript 𝛽 ′ 𝑘 superscript 𝜈 2 𝑘 3 {\displaystyle{\displaystyle J_{\nu}\left(j_{\nu,m}'\right)\sim(-1)^{m-1}\frac% {(2/\nu)^{\frac{1}{3}}}{\pi N\left(a^{\prime}_{m}\right)}\sum_{k=0}^{\infty}% \frac{\beta^{\prime}_{k}}{\nu^{2k/3}},}}
    0 references
    DLMF id
    DLMF:10.21.E38
    0 references
    Symbols used
    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.m2addec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11amdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ajdec
    0 references
    Airy modulus function
    Defining formula
    N ⁡ ( z ) modulus-Airy-N 𝑧 {\displaystyle{\displaystyle N\left(\NVar{z}\right)}}
    xml-id
    C9.S8.E7.m2aadec
    0 references
    Q11456
    Defining formula
    j ν , m zero-Bessel-J 𝜈 𝑚 {\displaystyle{\displaystyle j_{\NVar{\nu},\NVar{m}}}}
    xml-id
    C10.S21.SS1.p2.m11akdec
    0 references
    Q11395
    Defining formula
    a k ′ zero-derivative-Airy-Ai 𝑘 {\displaystyle{\displaystyle a^{\prime}_{\NVar{k}}}}
    xml-id
    C9.S9.SS1.p1.m11adec
    0 references
    Q11438
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C10.S1.XMD1.m1ndec
    0 references
    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1ldec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1addec
    0 references
    Q11463
    Defining formula
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C10.S21.XMD5.m1fdec
    0 references
     
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