DLMF:9.6.E5 (Q2785)

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DLMF:9.6.E5
No description defined

    Statements

    Defining formula
    Bi ′ ⁡ ( z ) = ( z / 3 ) ⁢ ( I 2 / 3 ⁡ ( ζ ) + I - 2 / 3 ⁡ ( ζ ) ) = 1 2 ⁢ ( z / 3 ) ⁢ ( e π ⁢ i / 3 ⁢ H 2 / 3 ( 1 ) ⁡ ( ζ ⁢ e - π ⁢ i / 2 ) + e - π ⁢ i / 3 ⁢ H 2 / 3 ( 2 ) ⁡ ( ζ ⁢ e π ⁢ i / 2 ) ) = 1 2 ⁢ ( z / 3 ) ⁢ ( e - π ⁢ i / 3 ⁢ H - 2 / 3 ( 1 ) ⁡ ( ζ ⁢ e - π ⁢ i / 2 ) + e π ⁢ i / 3 ⁢ H - 2 / 3 ( 2 ) ⁡ ( ζ ⁢ e π ⁢ i / 2 ) ) , diffop Airy-Bi 1 𝑧 𝑧 3 modified-Bessel-first-kind 2 3 𝜁 modified-Bessel-first-kind 2 3 𝜁 1 2 𝑧 3 superscript 𝑒 𝜋 𝑖 3 Hankel-H-1-Bessel-third-kind 2 3 𝜁 superscript 𝑒 𝜋 𝑖 2 superscript 𝑒 𝜋 𝑖 3 Hankel-H-2-Bessel-third-kind 2 3 𝜁 superscript 𝑒 𝜋 𝑖 2 1 2 𝑧 3 superscript 𝑒 𝜋 𝑖 3 Hankel-H-1-Bessel-third-kind 2 3 𝜁 superscript 𝑒 𝜋 𝑖 2 superscript 𝑒 𝜋 𝑖 3 Hankel-H-2-Bessel-third-kind 2 3 𝜁 superscript 𝑒 𝜋 𝑖 2 {\displaystyle{\displaystyle\mathrm{Bi}'\left(z\right)=(z/\sqrt{3})\left(I_{2/% 3}\left(\zeta\right)+I_{-2/3}\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3}% )\left(e^{\pi i/3}{H^{(1)}_{2/3}}\left(\zeta e^{-\pi i/2}\right)+e^{-\pi i/3}{% H^{(2)}_{2/3}}\left(\zeta e^{\pi i/2}\right)\right)=\tfrac{1}{2}(z/\sqrt{3})% \left(e^{-\pi i/3}{H^{(1)}_{-2/3}}\left(\zeta e^{-\pi i/2}\right)+e^{\pi i/3}{% H^{(2)}_{-2/3}}\left(\zeta e^{\pi i/2}\right)\right),}}
    0 references
    DLMF id
    DLMF:9.6.E5
    0 references
    Symbols used
    Q11081
    Defining formula
    Bi ⁡ ( z ) Airy-Bi 𝑧 {\displaystyle{\displaystyle\mathrm{Bi}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m2aadec
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    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.m2acdec
    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.m2acdec
    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
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2acdec
    0 references
    Q11381
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1ddec
    0 references
    change of variable (locally)
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C9.S6.XMD1.m1ddec
    0 references
     
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