DLMF:13.21.E25 (Q5330)

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DLMF:13.21.E25
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    Statements

    DLMF defining formula
    W - κ , μ ⁡ ( x ⁢ e π ⁢ i ) = e - ( κ - 1 6 ) ⁢ π ⁢ i ⁢ c ^ ⁢ ( κ , μ ) ⁢ Ψ ^ ⁢ ( κ , μ , x ) ⁢ ( Ai ⁡ ( κ 2 3 ⁢ ζ ^ ⁢ e 2 3 ⁢ π ⁢ i ) + envBi ⁡ ( κ 2 3 ⁢ ζ ^ ) ⁢ O ⁡ ( κ - 1 ) ) , Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑥 superscript 𝑒 𝜋 imaginary-unit superscript 𝑒 𝜅 1 6 𝜋 imaginary-unit ^ 𝑐 𝜅 𝜇 ^ Ψ 𝜅 𝜇 𝑥 Airy-Ai superscript 𝜅 2 3 ^ 𝜁 superscript 𝑒 2 3 𝜋 imaginary-unit envelope-Airy-Bi superscript 𝜅 2 3 ^ 𝜁 Big-O superscript 𝜅 1 {\displaystyle{\displaystyle W_{-\kappa,\mu}\left(xe^{\pi\mathrm{i}}\right)=e^% {-(\kappa-\frac{1}{6})\pi\mathrm{i}}\widehat{c}(\kappa,\mu)\widehat{\Psi}(% \kappa,\mu,x)\left(\mathrm{Ai}\left(\kappa^{\frac{2}{3}}\widehat{\zeta}e^{% \frac{2}{3}\pi\mathrm{i}}\right)+\mathrm{envBi}\left(\kappa^{\frac{2}{3}}% \widehat{\zeta}\right)O\left(\kappa^{-1}\right)\right),}}
    0 references
    Symbols used
    Q10893
    DLMF defining formula
    Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m1acdec
    0 references
    order not exceeding
    DLMF defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2amdec
    0 references
    Whittaker confluent hypergeometric function
    DLMF defining formula
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2aidec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2amdec
    0 references
    Q11057
    DLMF defining formula
    envBi ⁡ ( x ) envelope-Airy-Bi 𝑥 {\displaystyle{\displaystyle\mathrm{envBi}\left(\NVar{x}\right)}}
    xml-id
    C2.S8.SS3.p4.m6abdec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2akdec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2agdec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1vdec
    0 references
    function (locally)
    DLMF defining formula
    c ^ ⁢ ( κ , μ ) ^ 𝑐 𝜅 𝜇 {\displaystyle{\displaystyle\widehat{c}(\kappa,\mu)}}
    xml-id
    C13.S21.XMD4.m1cdec
    0 references
    function (locally)
    DLMF defining formula
    Ψ ^ ⁢ ( κ , μ , x ) ^ Ψ 𝜅 𝜇 𝑥 {\displaystyle{\displaystyle\widehat{\Psi}(\kappa,\mu,x)}}
    xml-id
    C13.S21.XMD6.m1ddec
    0 references

    Identifiers

     
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