DLMF:9.6.E17 (Q2797)

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DLMF:9.6.E17
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    Statements

    Defining formula
    H 1 / 3 ( 1 ) ⁡ ( ζ ) = e - π ⁢ i / 3 ⁢ H - 1 / 3 ( 1 ) ⁡ ( ζ ) = e - π ⁢ i / 6 ⁢ 3 / z ⁢ ( Ai ⁡ ( - z ) - i ⁢ Bi ⁡ ( - z ) ) , Hankel-H-1-Bessel-third-kind 1 3 𝜁 superscript 𝑒 𝜋 𝑖 3 Hankel-H-1-Bessel-third-kind 1 3 𝜁 superscript 𝑒 𝜋 𝑖 6 3 𝑧 Airy-Ai 𝑧 𝑖 Airy-Bi 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{1/3}}\left(\zeta\right)=e^{-\pi i/3}{H^{% (1)}_{-1/3}}\left(\zeta\right)=e^{-\pi i/6}\sqrt{3/z}\left(\mathrm{Ai}\left(-z% \right)-i\mathrm{Bi}\left(-z\right)\right),}}
    0 references
    DLMF id
    DLMF:9.6.E17
    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.m1ajdec
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    Q11081
    Defining formula
    Bi ⁡ ( z ) Airy-Bi 𝑧 {\displaystyle{\displaystyle\mathrm{Bi}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m2ahdec
    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.m2ahdec
    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
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ahdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ahdec
    0 references
    Q11381
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1pdec
    0 references
     
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