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DLMF:13.20.E16 - MaRDI portal

DLMF:13.20.E16 (Q5302)

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DLMF:13.20.E16
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    Statements

    DLMF defining formula
    W κ , μ ⁡ ( x ) = ( 1 2 ⁢ μ ) - 1 4 ⁢ ( κ + μ e ) 1 2 ⁢ ( κ + μ ) ⁢ Φ ⁢ ( κ , μ , x ) ⁢ ( U ⁡ ( μ - κ , ζ ⁢ 2 ⁢ μ ) + env ⁢ U ⁡ ( μ - κ , ζ ⁢ 2 ⁢ μ ) ⁢ O ⁡ ( μ - 2 3 ) ) , Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑥 superscript 1 2 𝜇 1 4 superscript 𝜅 𝜇 𝑒 1 2 𝜅 𝜇 Φ 𝜅 𝜇 𝑥 parabolic-U 𝜇 𝜅 𝜁 2 𝜇 envelope-Parabolic-U 𝜇 𝜅 𝜁 2 𝜇 Big-O superscript 𝜇 2 3 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(x\right)=\left(\tfrac{1}{2}% \mu\right)^{-\frac{1}{4}}\*\left(\frac{\kappa+\mu}{e}\right)^{\frac{1}{2}(% \kappa+\mu)}\*\Phi(\kappa,\mu,x)\*\left(U\left(\mu-\kappa,\zeta\sqrt{2\mu}% \right)+\mathrm{env}\mskip-1.0mu U\left(\mu-\kappa,\zeta\sqrt{2\mu}\right)O% \left(\mu^{-\frac{2}{3}}\right)\right),}}
    0 references
    Symbols used
    order not exceeding
    DLMF defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2afdec
    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.m2acdec
    0 references
    Q11555
    DLMF defining formula
    env ⁢ U ⁡ ( c , x ) envelope-Parabolic-U 𝑐 𝑥 {\displaystyle{\displaystyle\mathrm{env}\mskip-1.0mu U\left(\NVar{c},\NVar{x}% \right)}}
    xml-id
    C14.S15.SS5.p1.m7adec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    Q11291
    DLMF defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1abdec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1mdec
    0 references
    function (locally)
    DLMF defining formula
    Φ ⁢ ( κ , μ , x ) Φ 𝜅 𝜇 𝑥 {\displaystyle{\displaystyle\Phi(\kappa,\mu,x)}}
    xml-id
    C13.S20.XMD4.m1cdec
    0 references

    Identifiers

     
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