DLMF:13.21.E14 (Q4622): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Whittaker confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
Defining formula:

W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}

\WhittakerconfhyperW{\NVar{\kappa}}{\NVar{\mu}}@{\NVar{z}}
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
xml-id: C13.S14.E3.m2addec

Revision as of 16:19, 2 January 2020

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DLMF:13.21.E14
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    Statements

    Defining formula
    W κ , μ ⁡ ( x ) = e - μ ⁢ π ⁢ i π ⁢ Γ ⁡ ( κ + μ + 1 2 ) ⁢ Γ ⁡ ( κ - μ + 1 2 ) ⁢ c ⁢ ( κ , μ ) ⁢ Ψ ⁢ ( κ , μ , x ) ⁢ ( sin ⁡ ( κ ⁢ π - μ ⁢ π ) ⁢ J 2 ⁢ μ ⁡ ( ζ ⁢ κ ) - cos ⁡ ( κ ⁢ π - μ ⁢ π ) ⁢ Y 2 ⁢ μ ⁡ ( ζ ⁢ κ ) + env ⁢ Y 2 ⁢ μ ⁡ ( ζ ⁢ κ ) ⁢ O ⁡ ( κ - 1 ) ) , Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑥 superscript 𝑒 𝜇 𝜋 imaginary-unit 𝜋 Euler-Gamma 𝜅 𝜇 1 2 Euler-Gamma 𝜅 𝜇 1 2 𝑐 𝜅 𝜇 Ψ 𝜅 𝜇 𝑥 𝜅 𝜋 𝜇 𝜋 Bessel-J 2 𝜇 𝜁 𝜅 𝜅 𝜋 𝜇 𝜋 Bessel-Y-Weber 2 𝜇 𝜁 𝜅 envelope-Bessel-Y 2 𝜇 𝜁 𝜅 Big-O superscript 𝜅 1 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(x\right)=\frac{e^{-\mu\pi% \mathrm{i}}}{\pi}\Gamma\left(\kappa+\mu+\tfrac{1}{2}\right)\*\Gamma\left(% \kappa-\mu+\tfrac{1}{2}\right)\*c(\kappa,\mu)\Psi(\kappa,\mu,x)\*\left(\sin% \left(\kappa\pi-\mu\pi\right)J_{2\mu}\left(\sqrt{\zeta\kappa}\right)-\cos\left% (\kappa\pi-\mu\pi\right)Y_{2\mu}\left(\sqrt{\zeta\kappa}\right)+\mathrm{env}% \mskip-2.0mu Y_{2\mu}\left(\sqrt{\zeta\kappa}\right)O\left(\kappa^{-1}\right)% \right),}}
    0 references
    DLMF id
    DLMF:13.21.E14
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2acdec
    0 references
    Bessel function of the second kind
    Defining formula
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2aadec
    0 references
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2agdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
    0 references
    Whittaker confluent hypergeometric function
    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.m2addec
    0 references
     
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