DLMF:10.20.E6 (Q3255)

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DLMF:10.20.E6
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    Statements

    Defining formula
    H ν ( 1 ) ⁡ ( ν ⁢ z ) H ν ( 2 ) ⁡ ( ν ⁢ z ) } ∼ 2 ⁢ e ∓ π ⁢ i / 3 ⁢ ( 4 ⁢ ζ 1 - z 2 ) 1 4 ⁢ ( Ai ⁡ ( e ± 2 ⁢ π ⁢ i / 3 ⁢ ν 2 3 ⁢ ζ ) ν 1 3 ⁢ ∑ k = 0 ∞ A k ⁢ ( ζ ) ν 2 ⁢ k + e ± 2 ⁢ π ⁢ i / 3 ⁢ Ai ′ ⁡ ( e ± 2 ⁢ π ⁢ i / 3 ⁢ ν 2 3 ⁢ ζ ) ν 5 3 ⁢ ∑ k = 0 ∞ B k ⁢ ( ζ ) ν 2 ⁢ k ) , asymptotic-expansion selection Hankel-H-1-Bessel-third-kind 𝜈 𝜈 𝑧 Hankel-H-2-Bessel-third-kind 𝜈 𝜈 𝑧 2 superscript 𝑒 minus-or-plus 𝜋 𝑖 3 superscript 4 𝜁 1 superscript 𝑧 2 1 4 Airy-Ai superscript 𝑒 plus-or-minus 2 𝜋 𝑖 3 superscript 𝜈 2 3 𝜁 superscript 𝜈 1 3 superscript subscript 𝑘 0 subscript 𝐴 𝑘 𝜁 superscript 𝜈 2 𝑘 superscript 𝑒 plus-or-minus 2 𝜋 𝑖 3 diffop Airy-Ai 1 superscript 𝑒 plus-or-minus 2 𝜋 𝑖 3 superscript 𝜈 2 3 𝜁 superscript 𝜈 5 3 superscript subscript 𝑘 0 subscript 𝐵 𝑘 𝜁 superscript 𝜈 2 𝑘 {\displaystyle{\displaystyle\rselection{{H^{(1)}_{\nu}}\left(\nu z\right)\\ {H^{(2)}_{\nu}}\left(\nu z\right)}\sim 2e^{\mp\pi i/3}\left(\frac{4\zeta}{1-z^% {2}}\right)^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left(e^{\pm 2\pi i/3}\nu^{% \frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}\sum_{k=0}^{\infty}\frac{A_{k}(% \zeta)}{\nu^{2k}}+\frac{e^{\pm 2\pi i/3}\mathrm{Ai}'\left(e^{\pm 2\pi i/3}\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.E6
    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
    Bessel function of the third kind (or Hankel function)
    Defining formula
    H ν ( 1 ) ⁡ ( z ) Hankel-H-1-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E5.m2adec
    0 references
    Bessel function of the third kind (or Hankel function)
    Defining formula
    H ν ( 2 ) ⁡ ( z ) Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E6.m2adec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1bdec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1edec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1bdec
    0 references
    Q11451
    Defining formula
    ζ ⁢ ( z ) 𝜁 𝑧 {\displaystyle{\displaystyle\zeta(z)}}
    xml-id
    C10.S20.XMD1.m1edec
    0 references
    coefficients (locally)
    Defining formula
    A k ⁢ ( ζ ) subscript 𝐴 𝑘 𝜁 {\displaystyle{\displaystyle A_{k}(\zeta)}}
    xml-id
    C10.S20.XMD4.m1bdec
    0 references
     
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