DLMF:8.6.E9 (Q2523)

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DLMF:8.6.E9
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    Statements

    test
    Γ ⁡ ( - a , z ⁢ e ± π ⁢ i ) = e z ⁢ e ∓ π ⁢ i ⁢ a Γ ⁡ ( 1 + a ) ⁢ ∫ 0 ∞ t a ⁢ e - z ⁢ t t - 1 ⁢ d t , incomplete-Gamma 𝑎 𝑧 superscript 𝑒 plus-or-minus 𝜋 𝑖 superscript 𝑒 𝑧 superscript 𝑒 minus-or-plus 𝜋 imaginary-unit 𝑎 Euler-Gamma 1 𝑎 superscript subscript 0 superscript 𝑡 𝑎 superscript 𝑒 𝑧 𝑡 𝑡 1 𝑡 {\displaystyle{\displaystyle\Gamma\left(-a,ze^{\pm\pi i}\right)=\frac{e^{z}e^{% \mp\pi\mathrm{i}a}}{\Gamma\left(1+a\right)}\int_{0}^{\infty}\frac{t^{a}e^{-zt}% }{t-1}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:8.6.E9
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    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
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    ℜ ⁡ a > - 1 𝑎 1 {\displaystyle{\displaystyle\Re a>-1}}
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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.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.m2acdec
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