DLMF:5.17.E7 (Q2187): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Riemann zeta function / rank
 
Normal rank
Property / Symbols used: Riemann zeta function / qualifier
 
test:

ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}

\Riemannzeta@{\NVar{s}}
Property / Symbols used: Riemann zeta function / qualifier
 
xml-id: C25.S2.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
 
test:

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.m2abdec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
test:

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.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11274 / rank
 
Normal rank
Property / Symbols used: Q11274 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11274 / qualifier
 
xml-id: C5.S1.XMD3.m1adec
Property / Symbols used
 
Property / Symbols used: Q11279 / rank
 
Normal rank
Property / Symbols used: Q11279 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11279 / qualifier
 
xml-id: C5.S1.XMD4.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11297 / rank
 
Normal rank
Property / Symbols used: Q11297 / qualifier
 
test:

C 𝐶 {\displaystyle{\displaystyle C}}

C
Property / Symbols used: Q11297 / qualifier
 
xml-id: C5.S17.XMD2.m1adec

Latest revision as of 00:09, 2 January 2020

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English
DLMF:5.17.E7
No description defined

    Statements

    test
    C = lim n → ∞ ⁡ ( ∑ k = 1 n k ⁢ ln ⁡ k - ( 1 2 ⁢ n 2 + 1 2 ⁢ n + 1 12 ) ⁢ ln ⁡ n + 1 4 ⁢ n 2 ) = γ + ln ⁡ ( 2 ⁢ π ) 12 - ζ ′ ⁡ ( 2 ) 2 ⁢ π 2 = 1 12 - ζ ′ ⁡ ( - 1 ) , 𝐶 subscript → 𝑛 superscript subscript 𝑘 1 𝑛 𝑘 𝑘 1 2 superscript 𝑛 2 1 2 𝑛 1 12 𝑛 1 4 superscript 𝑛 2 2 𝜋 12 diffop Riemann-zeta 1 2 2 superscript 𝜋 2 1 12 diffop Riemann-zeta 1 1 {\displaystyle{\displaystyle C=\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln k-% \left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln n+\tfrac{1}{4}n^% {2}\right)=\frac{\gamma+\ln\left(2\pi\right)}{12}-\frac{\zeta'\left(2\right)}{% 2\pi^{2}}=\frac{1}{12}-\zeta'\left(-1\right),}}
    0 references
    DLMF id
    DLMF:5.17.E7
    0 references
    Symbols used
    Euler’s constant
    test
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2aadec
    0 references
    Riemann zeta function
    test
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2abdec
    0 references
    Q11274
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C5.S1.XMD3.m1adec
    0 references
    Q11279
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1bdec
    0 references
    Q11297
    test
    C 𝐶 {\displaystyle{\displaystyle C}}
    xml-id
    C5.S17.XMD2.m1adec
    0 references
     
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