DLMF:25.8.E8 (Q7665): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q12150 / rank
 
Normal rank
Property / Symbols used: Q12150 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q12150 / qualifier
 
xml-id: C25.S1.XMD1.m1gdec

Revision as of 13:34, 2 January 2020

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DLMF:25.8.E8
No description defined

    Statements

    Defining formula
    ∑ k = 1 ∞ ζ ⁡ ( 2 ⁢ k ) k ⁢ z 2 ⁢ k = ln ⁡ ( π ⁢ z sin ⁡ ( π ⁢ z ) ) , superscript subscript 𝑘 1 Riemann-zeta 2 𝑘 𝑘 superscript 𝑧 2 𝑘 𝜋 𝑧 𝜋 𝑧 {\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}z% ^{2k}=\ln\left(\frac{\pi z}{\sin\left(\pi z\right)}\right),}}
    0 references
    DLMF id
    DLMF:25.8.E8
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    constraint
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
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    Symbols used
    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2abdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
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    Q12150
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C25.S1.XMD1.m1gdec
    0 references
     
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