DLMF:5.11.E10 (Q2132)

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DLMF:5.11.E10
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    Statements

    test
    Γ ⁡ ( z ) = e - z ⁢ z z ⁢ ( 2 ⁢ π z ) 1 / 2 ⁢ ( ∑ k = 0 K - 1 g k z k + R K ⁢ ( z ) ) , Euler-Gamma 𝑧 superscript 𝑒 𝑧 superscript 𝑧 𝑧 superscript 2 𝜋 𝑧 1 2 superscript subscript 𝑘 0 𝐾 1 subscript 𝑔 𝑘 superscript 𝑧 𝑘 subscript 𝑅 𝐾 𝑧 {\displaystyle{\displaystyle\Gamma\left(z\right)=e^{-z}z^{z}\left(\frac{2\pi}{% z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right),}}
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    DLMF id
    DLMF:5.11.E10
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    constraint
    K = 1 , 2 , 3 , … 𝐾 1 2 3 … {\displaystyle{\displaystyle K=1,2,3,\dots}}
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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.m2aedec
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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.m2aedec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
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    Q11279
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1fdec
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    Q11273
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C5.S1.XMD7.m1edec
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    Q11286
    test
    g k subscript 𝑔 𝑘 {\displaystyle{\displaystyle g_{k}}}
    xml-id
    C5.S11.XMD1.m1cdec
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    remainder (locally)
    test
    R K ⁢ ( z ) subscript 𝑅 𝐾 𝑧 {\displaystyle{\displaystyle R_{K}(z)}}
    xml-id
    C5.S11.XMD3.m1dec
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