DLMF:25.12.E13 (Q7732): Difference between revisions

From DRMF
Jump to navigation Jump to search
‎Created a new Item: Wikidata Toolkit example test item creation
 
‎Changed an Item: Add constraint
 
(9 intermediate revisions by the same user not shown)
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
Defining formula:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2abdec
Property / Symbols used
 
Property / Symbols used: Hurwitz zeta function / rank
 
Normal rank
Property / Symbols used: Hurwitz zeta function / qualifier
 
Defining formula:

ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}

\Hurwitzzeta@{\NVar{s}}{\NVar{a}}
Property / Symbols used: Hurwitz zeta function / qualifier
 
xml-id: C25.S11.E1.m2adec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2aedec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2acdec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2acdec
Property / Symbols used
 
Property / Symbols used: Q10777 / rank
 
Normal rank
Property / Symbols used: Q10777 / qualifier
 
Defining formula:

( a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}

(\NVar{a},\NVar{b})
Property / Symbols used: Q10777 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r29.m6aadec
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.m2acdec
Property / Symbols used
 
Property / Symbols used: Q12161 / rank
 
Normal rank
Property / Symbols used: Q12161 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q12161 / qualifier
 
xml-id: C25.S1.XMD6.m1dec
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.m1cdec
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.m1hdec

Latest revision as of 13:50, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:25.12.E13
No description defined

    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
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    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
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    Q10777
    Defining formula
    ( a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}
    xml-id
    introduction.Sx4.p1.t1.r29.m6aadec
    0 references
    polylogarithm
    Defining formula
    Li s ⁡ ( z ) polylogarithm 𝑠 𝑧 {\displaystyle{\displaystyle\mathrm{Li}_{\NVar{s}}\left(\NVar{z}\right)}}
    xml-id
    C25.S12.E10.m2acdec
    0 references
    Q12161
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C25.S1.XMD6.m1dec
    0 references
    Q12149
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C25.S1.XMD7.m1cdec
    0 references
    Q12156
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C25.S1.XMD8.m1hdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q7732&oldid=40604"