DLMF:13.16.E9 (Q4560)

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DLMF:13.16.E9
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    Statements

    Defining formula
    W κ , μ ⁡ ( z ) = e - 1 2 ⁢ z ⁢ z κ + c ⁢ ∫ 0 ∞ e - z ⁢ t ⁢ t c - 1 ⁢ 𝐅 1 2 ⁡ ( 1 2 + μ - κ , 1 2 - μ - κ c ; - t ) ⁢ d t , Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜅 𝑐 superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝑐 1 hypergeometric-bold-pFq 2 1 1 2 𝜇 𝜅 1 2 𝜇 𝜅 𝑐 𝑡 𝑡 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=e^{-\frac{1}{2}z}z^{% \kappa+c}\*\int_{0}^{\infty}e^{-zt}t^{c-1}{{}_{2}{\mathbf{F}}_{1}}\left({% \tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}-\mu-\kappa\atop c};-t\right)\mathrm{d}t,}}
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    DLMF id
    DLMF:13.16.E9
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    constraint
    | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\frac{1}{2}\pi}}
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    Symbols used
    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
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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.m2acdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ahdec
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    scaled (or Olver’s) generalized hypergeometric function
    Defining formula
    𝐅 q p ⁡ ( 𝐚 ; 𝐛 ; ) hypergeometric-bold-pFq 𝑝 𝑞 𝐚 𝐛 {\displaystyle{\displaystyle{{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{% \mathbf{a}};\NVar{\mathbf{b}};\right)\)\@add@PDF@RDFa@triples\end{document}}}
    xml-id
    C16.S2.E5.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
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