DLMF:25.6.E12 (Q7649): Difference between revisions

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‎Changed claim: Defining formula (P1): \Riemannzeta''@{0}=-\tfrac{1}{2}(\ln@{2\pi})^{2}+\tfrac{1}{2}\EulerConstant^{2}-\tfrac{1}{24}\pi^{2}+\StieltjesConstants{1}
 
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Property / Defining formulaProperty / Defining formula

ζ ′′ ⁡ ( 0 ) = - 1 2 ⁢ ( ln ⁡ ( 2 ⁢ π ) ) 2 + 1 2 ⁢ γ 2 - 1 24 ⁢ π 2 + γ 1 , diffop Riemann-zeta 2 0 1 2 superscript 2 𝜋 2 1 2 2 1 24 superscript 𝜋 2 Stieltjes-constants 1 {\displaystyle{\displaystyle\zeta''\left(0\right)=-\tfrac{1}{2}(\ln\left(2\pi% \right))^{2}+\tfrac{1}{2}{\gamma^{2}}-\tfrac{1}{24}\pi^{2}+\gamma_{1},}}

\Riemannzeta''@{0}=-\tfrac{1}{2}(\ln@{2\pi})^{2}+\tfrac{1}{2}\EulerConstant^{2}-\tfrac{1}{24}\pi^{2}+\StieltjesConstants{1},

ζ ′′ ⁡ ( 0 ) = - 1 2 ⁢ ( ln ⁡ ( 2 ⁢ π ) ) 2 + 1 2 ⁢ γ 2 - 1 24 ⁢ π 2 + γ 1 diffop Riemann-zeta 2 0 1 2 superscript 2 𝜋 2 1 2 2 1 24 superscript 𝜋 2 Stieltjes-constants 1 {\displaystyle{\displaystyle\zeta''\left(0\right)=-\tfrac{1}{2}(\ln\left(2\pi% \right))^{2}+\tfrac{1}{2}{\gamma^{2}}-\tfrac{1}{24}\pi^{2}+\gamma_{1}}}

\Riemannzeta''@{0}=-\tfrac{1}{2}(\ln@{2\pi})^{2}+\tfrac{1}{2}\EulerConstant^{2}-\tfrac{1}{24}\pi^{2}+\StieltjesConstants{1}
Property / Symbols used
 
Property / Symbols used: Riemann zeta function / rank
 
Normal rank
Property / Symbols used: Riemann zeta function / qualifier
 
Defining formula:

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.m2akdec
Property / Symbols used
 
Property / Symbols used: Stieltjes constants / rank
 
Normal rank
Property / Symbols used: Stieltjes constants / qualifier
 
Defining formula:

γ n Stieltjes-constants 𝑛 {\displaystyle{\displaystyle\gamma_{\NVar{n}}}}

\StieltjesConstants{\NVar{n}}
Property / Symbols used: Stieltjes constants / qualifier
 
xml-id: C25.S2.E5.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.m2addec
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.m2aadec

Latest revision as of 16:07, 2 January 2020

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DLMF:25.6.E12
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    Statements

    Defining formula
    ζ ′′ ⁡ ( 0 ) = - 1 2 ⁢ ( ln ⁡ ( 2 ⁢ π ) ) 2 + 1 2 ⁢ γ 2 - 1 24 ⁢ π 2 + γ 1 diffop Riemann-zeta 2 0 1 2 superscript 2 𝜋 2 1 2 2 1 24 superscript 𝜋 2 Stieltjes-constants 1 {\displaystyle{\displaystyle\zeta''\left(0\right)=-\tfrac{1}{2}(\ln\left(2\pi% \right))^{2}+\tfrac{1}{2}{\gamma^{2}}-\tfrac{1}{24}\pi^{2}+\gamma_{1}}}
    0 references
    DLMF id
    DLMF:25.6.E12
    0 references
    Symbols used
    Euler’s constant
    Defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2adec
    0 references
    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2akdec
    0 references
    Stieltjes constants
    Defining formula
    γ n Stieltjes-constants 𝑛 {\displaystyle{\displaystyle\gamma_{\NVar{n}}}}
    xml-id
    C25.S2.E5.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.m2addec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
     
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