DLMF:25.12.E13 (Q7732)

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DLMF:25.12.E13
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    Statements

    Defining formula
    Li s ⁡ ( e 2 ⁢ π ⁢ i ⁢ a ) + e π ⁢ i ⁢ s ⁢ Li s ⁡ ( e - 2 ⁢ π ⁢ i ⁢ a ) = ( 2 ⁢ π ) s ⁢ e π ⁢ i ⁢ s / 2 Γ ⁡ ( s ) ⁢ ζ ⁡ ( 1 - s , a ) , polylogarithm 𝑠 superscript 𝑒 2 𝜋 𝑖 𝑎 superscript 𝑒 𝜋 𝑖 𝑠 polylogarithm 𝑠 superscript 𝑒 2 𝜋 𝑖 𝑎 superscript 2 𝜋 𝑠 superscript 𝑒 𝜋 𝑖 𝑠 2 Euler-Gamma 𝑠 Hurwitz-zeta 1 𝑠 𝑎 {\displaystyle{\displaystyle\mathrm{Li}_{s}\left(e^{2\pi ia}\right)+e^{\pi is}% \mathrm{Li}_{s}\left(e^{-2\pi ia}\right)=\frac{(2\pi)^{s}e^{\pi is/2}}{\Gamma% \left(s\right)}\zeta\left(1-s,a\right),}}
    0 references
    DLMF id
    DLMF:25.12.E13
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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.m2abdec
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    Hurwitz zeta function
    Defining formula
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
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