DLMF:25.12.E12 (Q7731): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
 
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Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
Defining formula:

ln z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2aedec
Property / Symbols used
 
Property / Symbols used: polylogarithm / rank
 
Normal rank
Property / Symbols used: polylogarithm / qualifier
 
Defining formula:

Li s ( z ) polylogarithm 𝑠 𝑧 {\displaystyle{\displaystyle\mathrm{Li}_{\NVar{s}}\left(\NVar{z}\right)}}

\polylog{\NVar{s}}@{\NVar{z}}
Property / Symbols used: polylogarithm / qualifier
 
xml-id: C25.S12.E10.m2abdec
Property / Symbols used
 
Property / Symbols used: Q12148 / rank
 
Normal rank
Property / Symbols used: Q12148 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12148 / qualifier
 
xml-id: C25.S1.XMD3.m1edec
Property / Symbols used
 
Property / Symbols used: Q12149 / rank
 
Normal rank
Property / Symbols used: Q12149 / qualifier
 
Defining formula:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q12149 / qualifier
 
xml-id: C25.S1.XMD7.m1bdec
Property / Symbols used
 
Property / Symbols used: Q12156 / rank
 
Normal rank
Property / Symbols used: Q12156 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q12156 / qualifier
 
xml-id: C25.S1.XMD8.m1gdec

Latest revision as of 13:49, 2 January 2020

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DLMF:25.12.E12
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    Statements

    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!},}}
    0 references
    DLMF:25.12.E12
    0 references
    | 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}}
    0 references
    | ln z | < 2 π 𝑧 2 𝜋 {\displaystyle{\displaystyle|\ln z|<2\pi}}
    0 references
    Γ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    C5.S2.E1.m2aadec
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    ζ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    C25.S2.E1.m2adec
    0 references
    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2addec
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    ! {\displaystyle{\displaystyle!}}
    introduction.Sx4.p1.t1.r15.m5adec
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    ln z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    C4.S2.E2.m2aedec
    0 references
    Li s ( z ) polylogarithm 𝑠 𝑧 {\displaystyle{\displaystyle\mathrm{Li}_{\NVar{s}}\left(\NVar{z}\right)}}
    C25.S12.E10.m2abdec
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    n 𝑛 {\displaystyle{\displaystyle n}}
    C25.S1.XMD3.m1edec
    0 references
    s 𝑠 {\displaystyle{\displaystyle s}}
    C25.S1.XMD7.m1bdec
    0 references
    z 𝑧 {\displaystyle{\displaystyle z}}
    C25.S1.XMD8.m1gdec
    0 references