DLMF:25.12.E12 (Q7731)

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DLMF:25.12.E12
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    Statements

    Defining formula
    Li s ⁡ ( z ) = Γ ⁡ ( 1 - s ) ⁢ ( ln ⁡ 1 z ) s - 1 + ∑ n = 0 ∞ ζ ⁡ ( s - n ) ⁢ ( ln ⁡ z ) n n ! , polylogarithm 𝑠 𝑧 Euler-Gamma 1 𝑠 superscript 1 𝑧 𝑠 1 superscript subscript 𝑛 0 Riemann-zeta 𝑠 𝑛 superscript 𝑧 𝑛 𝑛 {\displaystyle{\displaystyle\mathrm{Li}_{s}\left(z\right)=\Gamma\left(1-s% \right)\left(\ln\frac{1}{z}\right)^{s-1}+\sum_{n=0}^{\infty}\zeta\left(s-n% \right)\frac{(\ln z)^{n}}{n!},}}
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    DLMF id
    DLMF:25.12.E12
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    constraint
    | ln ⁡ z | < 2 ⁢ π 𝑧 2 𝜋 {\displaystyle{\displaystyle|\ln z|<2\pi}}
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    s ≠ 1 , 2 , 3 , … 𝑠 1 2 3 … {\displaystyle{\displaystyle s\neq 1,2,3,\dots}}
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    | ln ⁡ z | < 2 ⁢ π 𝑧 2 𝜋 {\displaystyle{\displaystyle|\ln z|<2\pi}}
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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.m2aadec
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