DLMF:13.16.E7 (Q4558)

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DLMF:13.16.E7
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    Statements

    test
    W κ , μ ⁡ ( z ) = ( - 1 ) n ⁢ e - 1 2 ⁢ z ⁢ z 1 2 - μ - n Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ Γ ⁡ ( 1 2 - μ - κ ) ⁢ ∫ 0 ∞ M - κ , μ ⁡ ( t ) ⁢ e - 1 2 ⁢ t ⁢ t n + μ - 1 2 t + z ⁢ d t , Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 1 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 1 2 𝜇 𝑛 Euler-Gamma 1 2 𝜇 Euler-Gamma 1 2 𝜇 𝜅 superscript subscript 0 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑡 superscript 𝑒 1 2 𝑡 superscript 𝑡 𝑛 𝜇 1 2 𝑡 𝑧 𝑡 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=\frac{(-1)^{n}e^{-% \frac{1}{2}z}z^{\frac{1}{2}-\mu-n}}{\Gamma\left(1+2\mu\right)\Gamma\left(\frac% {1}{2}-\mu-\kappa\right)}\*\int_{0}^{\infty}\frac{M_{-\kappa,\mu}\left(t\right% )e^{-\frac{1}{2}t}t^{n+\mu-\frac{1}{2}}}{t+z}\mathrm{d}t,}}
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    DLMF id
    DLMF:13.16.E7
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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 ) < n < | ℜ ⁡ μ | + ℜ ⁡ κ < 1 2 1 2 𝜇 𝑛 𝜇 𝜅 1 2 {\displaystyle{\displaystyle-\Re(1+2\mu)<n<\left|\Re\mu\right|+\Re\kappa<% \tfrac{1}{2}}}
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    | ph ⁡ z | < π ph 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}
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    n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle n=0,1,2,\dots}}
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    - ℜ ⁡ ( 1 + 2 ⁢ μ ) < n < | ℜ ⁡ μ | + ℜ ⁡ κ < 1 2 1 2 𝜇 𝑛 𝜇 𝜅 1 2 {\displaystyle{\displaystyle-\Re(1+2\mu)<n<\left|\Re\mu\right|+\Re\kappa<% \tfrac{1}{2}}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
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    Whittaker confluent hypergeometric function
    test
    M κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E2.m2addec
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    Whittaker confluent hypergeometric function
    test
    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.m2abdec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
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    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1abdec
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