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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    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2adec
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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.m2addec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aedec
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    polylogarithm
    Defining formula
    Li s ⁡ ( z ) polylogarithm 𝑠 𝑧 {\displaystyle{\displaystyle\mathrm{Li}_{\NVar{s}}\left(\NVar{z}\right)}}
    xml-id
    C25.S12.E10.m2abdec
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    Q12148
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C25.S1.XMD3.m1edec
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    Q12149
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C25.S1.XMD7.m1bdec
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    Q12156
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C25.S1.XMD8.m1gdec
    0 references
     
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