DLMF:9.6.E2 (Q2782)

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DLMF:9.6.E2
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    Statements

    Defining formula
    Ai ⁡ ( z ) = π - 1 ⁢ z / 3 ⁢ K ± 1 / 3 ⁡ ( ζ ) = 1 3 ⁢ z ⁢ ( I - 1 / 3 ⁡ ( ζ ) - I 1 / 3 ⁡ ( ζ ) ) = 1 2 ⁢ z / 3 ⁢ e 2 ⁢ π ⁢ i / 3 ⁢ H 1 / 3 ( 1 ) ⁡ ( ζ ⁢ e π ⁢ i / 2 ) = 1 2 ⁢ z / 3 ⁢ e π ⁢ i / 3 ⁢ H - 1 / 3 ( 1 ) ⁡ ( ζ ⁢ e π ⁢ i / 2 ) = 1 2 ⁢ z / 3 ⁢ e - 2 ⁢ π ⁢ i / 3 ⁢ H 1 / 3 ( 2 ) ⁡ ( ζ ⁢ e - π ⁢ i / 2 ) = 1 2 ⁢ z / 3 ⁢ e - π ⁢ i / 3 ⁢ H - 1 / 3 ( 2 ) ⁡ ( ζ ⁢ e - π ⁢ i / 2 ) , Airy-Ai 𝑧 superscript 𝜋 1 𝑧 3 modified-Bessel-second-kind plus-or-minus 1 3 𝜁 1 3 𝑧 modified-Bessel-first-kind 1 3 𝜁 modified-Bessel-first-kind 1 3 𝜁 1 2 𝑧 3 superscript 𝑒 2 𝜋 𝑖 3 Hankel-H-1-Bessel-third-kind 1 3 𝜁 superscript 𝑒 𝜋 𝑖 2 1 2 𝑧 3 superscript 𝑒 𝜋 𝑖 3 Hankel-H-1-Bessel-third-kind 1 3 𝜁 superscript 𝑒 𝜋 𝑖 2 1 2 𝑧 3 superscript 𝑒 2 𝜋 𝑖 3 Hankel-H-2-Bessel-third-kind 1 3 𝜁 superscript 𝑒 𝜋 𝑖 2 1 2 𝑧 3 superscript 𝑒 𝜋 𝑖 3 Hankel-H-2-Bessel-third-kind 1 3 𝜁 superscript 𝑒 𝜋 𝑖 2 {\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\pi^{-1}\sqrt{z/3}K_{\pm 1% /3}\left(\zeta\right)=\tfrac{1}{3}\sqrt{z}\left(I_{-1/3}\left(\zeta\right)-I_{% 1/3}\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}e^{2\pi i/3}{H^{(1)}_{1/3}% }\left(\zeta e^{\pi i/2}\right)=\tfrac{1}{2}\sqrt{z/3}e^{\pi i/3}{H^{(1)}_{-1/% 3}}\left(\zeta e^{\pi i/2}\right)=\tfrac{1}{2}\sqrt{z/3}e^{-2\pi i/3}{H^{(2)}_% {1/3}}\left(\zeta e^{-\pi i/2}\right)=\tfrac{1}{2}\sqrt{z/3}e^{-\pi i/3}{H^{(2% )}_{-1/3}}\left(\zeta e^{-\pi i/2}\right),}}
    0 references
    DLMF id
    DLMF:9.6.E2
    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.m1adec
    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
    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
    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.m2adec
    0 references
    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2adec
    0 references
    Q11381
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1adec
    0 references
    change of variable (locally)
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C9.S6.XMD1.m1adec
    0 references
     
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