DLMF:13.16.E6 (Q4557)

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DLMF:13.16.E6
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    Statements

    Defining formula
    W κ , μ ⁡ ( z ) = e - 1 2 ⁢ z ⁢ z κ + 1 Γ ⁡ ( 1 2 + μ - κ ) ⁢ Γ ⁡ ( 1 2 - μ - κ ) ⁢ ∫ 0 ∞ W - κ , μ ⁡ ( t ) ⁢ e - 1 2 ⁢ t ⁢ t - κ - 1 t + z ⁢ d t , Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜅 1 Euler-Gamma 1 2 𝜇 𝜅 Euler-Gamma 1 2 𝜇 𝜅 superscript subscript 0 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑡 superscript 𝑒 1 2 𝑡 superscript 𝑡 𝜅 1 𝑡 𝑧 𝑡 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=\frac{e^{-\frac{1}{2% }z}z^{\kappa+1}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}% {2}-\mu-\kappa\right)}\*\int_{0}^{\infty}\frac{W_{-\kappa,\mu}\left(t\right)e^% {-\frac{1}{2}t}t^{-\kappa-1}}{t+z}\mathrm{d}t,}}
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    DLMF id
    DLMF:13.16.E6
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    constraint
    | ph ⁡ z | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\pi}}
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    ℜ ⁡ ( 1 2 + μ - κ ) > max ⁡ ( 2 ⁢ ℜ ⁡ μ , 0 ) 1 2 𝜇 𝜅 2 𝜇 0 {\displaystyle{\displaystyle\Re(\frac{1}{2}+\mu-\kappa)>\max\left(2\Re\mu,0% \right)}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
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    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.m2aadec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
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