DLMF:2.5.E46 (Q822)

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DLMF:2.5.E46
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    Statements

    Defining formula
    res z = k ⁡ [ - ζ z - 1 ⁢ Γ ⁡ ( 1 - z ) ⁢ π ⁢ csc ⁡ ( π ⁢ z ) ] = ( - ln ⁡ ζ + ψ ⁡ ( k ) ) ⁢ ζ k - 1 ( k - 1 ) ! , subscript residue 𝑧 𝑘 delimited-[] superscript 𝜁 𝑧 1 Euler-Gamma 1 𝑧 𝜋 𝜋 𝑧 𝜁 digamma 𝑘 superscript 𝜁 𝑘 1 𝑘 1 {\displaystyle{\displaystyle\Residue_{z=k}\left[-\zeta^{z-1}\Gamma\left(1-z% \right)\pi\csc\left(\pi z\right)\right]=\left(-\ln\zeta+\psi\left(k\right)% \right)\dfrac{\zeta^{k-1}}{(k-1)!},}}
    0 references
    DLMF id
    DLMF:2.5.E46
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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.m2ahdec
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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.m2aldec
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    cosecant function
    Defining formula
    csc ⁡ z 𝑧 {\displaystyle{\displaystyle\csc\NVar{z}}}
    xml-id
    C4.S14.E5.m2adec
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    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2aadec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
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