DLMF:9.6.E3 (Q2783)

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DLMF:9.6.E3
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    Statements

    Defining formula
    Ai ′ ⁡ ( z ) = - π - 1 ⁢ ( z / 3 ) ⁢ K ± 2 / 3 ⁡ ( ζ ) = ( z / 3 ) ⁢ ( I 2 / 3 ⁡ ( ζ ) - I - 2 / 3 ⁡ ( ζ ) ) = 1 2 ⁢ ( z / 3 ) ⁢ e - π ⁢ i / 6 ⁢ H 2 / 3 ( 1 ) ⁡ ( ζ ⁢ e π ⁢ i / 2 ) = 1 2 ⁢ ( z / 3 ) ⁢ e - 5 ⁢ π ⁢ i / 6 ⁢ H - 2 / 3 ( 1 ) ⁡ ( ζ ⁢ e π ⁢ i / 2 ) = 1 2 ⁢ ( z / 3 ) ⁢ e π ⁢ i / 6 ⁢ H 2 / 3 ( 2 ) ⁡ ( ζ ⁢ e - π ⁢ i / 2 ) = 1 2 ⁢ ( z / 3 ) ⁢ e 5 ⁢ π ⁢ i / 6 ⁢ H - 2 / 3 ( 2 ) ⁡ ( ζ ⁢ e - π ⁢ i / 2 ) , diffop Airy-Ai 1 𝑧 superscript 𝜋 1 𝑧 3 modified-Bessel-second-kind plus-or-minus 2 3 𝜁 𝑧 3 modified-Bessel-first-kind 2 3 𝜁 modified-Bessel-first-kind 2 3 𝜁 1 2 𝑧 3 superscript 𝑒 𝜋 𝑖 6 Hankel-H-1-Bessel-third-kind 2 3 𝜁 superscript 𝑒 𝜋 𝑖 2 1 2 𝑧 3 superscript 𝑒 5 𝜋 𝑖 6 Hankel-H-1-Bessel-third-kind 2 3 𝜁 superscript 𝑒 𝜋 𝑖 2 1 2 𝑧 3 superscript 𝑒 𝜋 𝑖 6 Hankel-H-2-Bessel-third-kind 2 3 𝜁 superscript 𝑒 𝜋 𝑖 2 1 2 𝑧 3 superscript 𝑒 5 𝜋 𝑖 6 Hankel-H-2-Bessel-third-kind 2 3 𝜁 superscript 𝑒 𝜋 𝑖 2 {\displaystyle{\displaystyle\mathrm{Ai}'\left(z\right)=-\pi^{-1}(z/\sqrt{3})K_% {\pm 2/3}\left(\zeta\right)=(z/3)\left(I_{2/3}\left(\zeta\right)-I_{-2/3}\left% (\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})e^{-\pi i/6}{H^{(1)}_{2/3}}\left(% \zeta e^{\pi i/2}\right)=\tfrac{1}{2}(z/\sqrt{3})e^{-5\pi i/6}{H^{(1)}_{-2/3}}% \left(\zeta e^{\pi i/2}\right)=\tfrac{1}{2}(z/\sqrt{3})e^{\pi i/6}{H^{(2)}_{2/% 3}}\left(\zeta e^{-\pi i/2}\right)=\tfrac{1}{2}(z/\sqrt{3})e^{5\pi i/6}{H^{(2)% }_{-2/3}}\left(\zeta e^{-\pi i/2}\right),}}
    0 references
    DLMF id
    DLMF:9.6.E3
    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.m2aadec
    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.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
     
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