DLMF:25.11.E21 (Q7695): 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.m2acdec
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.m2acdec
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
test:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2abdec
Property / Symbols used
 
Property / Symbols used: psi (or digamma) function / rank
 
Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
 
test:

ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}

\digamma@{\NVar{z}}
Property / Symbols used: psi (or digamma) function / qualifier
 
xml-id: C5.S2.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
test:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5acdec
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.m2acdec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
test:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2adec
Property / Symbols used
 
Property / Symbols used: Q12148 / rank
 
Normal rank
Property / Symbols used: Q12148 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12148 / qualifier
 
xml-id: C25.S1.XMD3.m1kdec
Property / Symbols used
 
Property / Symbols used: Q12163 / rank
 
Normal rank
Property / Symbols used: Q12163 / qualifier
 
test:

h ℎ {\displaystyle{\displaystyle h}}

h
Property / Symbols used: Q12163 / qualifier
 
xml-id: C25.S11.XMD1.m1dec
Property / Symbols used
 
Property / Symbols used: Q12162 / rank
 
Normal rank
Property / Symbols used: Q12162 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q12162 / qualifier
 
xml-id: C25.S11.XMD2.m1bdec

Latest revision as of 12:40, 2 January 2020

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DLMF:25.11.E21
No description defined

    Statements

    test
    ζ ′ ⁡ ( 1 - 2 ⁢ n , h k ) = ( ψ ⁡ ( 2 ⁢ n ) - ln ⁡ ( 2 ⁢ π ⁢ k ) ) ⁢ B 2 ⁢ n ⁡ ( h / k ) 2 ⁢ n - ( ψ ⁡ ( 2 ⁢ n ) - ln ⁡ ( 2 ⁢ π ) ) ⁢ B 2 ⁢ n 2 ⁢ n ⁢ k 2 ⁢ n + ( - 1 ) n + 1 ⁢ π ( 2 ⁢ π ⁢ k ) 2 ⁢ n ⁢ ∑ r = 1 k - 1 sin ⁡ ( 2 ⁢ π ⁢ r ⁢ h k ) ⁢ ψ ( 2 ⁢ n - 1 ) ⁡ ( r k ) + ( - 1 ) n + 1 ⁢ 2 ⋅ ( 2 ⁢ n - 1 ) ! ( 2 ⁢ π ⁢ k ) 2 ⁢ n ⁢ ∑ r = 1 k - 1 cos ⁡ ( 2 ⁢ π ⁢ r ⁢ h k ) ⁢ ζ ′ ⁡ ( 2 ⁢ n , r k ) + ζ ′ ⁡ ( 1 - 2 ⁢ n ) k 2 ⁢ n , diffop Hurwitz-zeta 1 1 2 𝑛 ℎ 𝑘 digamma 2 𝑛 2 𝜋 𝑘 Bernoulli-polynomial-B 2 𝑛 ℎ 𝑘 2 𝑛 digamma 2 𝑛 2 𝜋 Bernoulli-number-B 2 𝑛 2 𝑛 superscript 𝑘 2 𝑛 superscript 1 𝑛 1 𝜋 superscript 2 𝜋 𝑘 2 𝑛 superscript subscript 𝑟 1 𝑘 1 2 𝜋 𝑟 ℎ 𝑘 digamma 2 𝑛 1 𝑟 𝑘 ⋅ superscript 1 𝑛 1 2 2 𝑛 1 superscript 2 𝜋 𝑘 2 𝑛 superscript subscript 𝑟 1 𝑘 1 2 𝜋 𝑟 ℎ 𝑘 diffop Hurwitz-zeta 1 2 𝑛 𝑟 𝑘 diffop Riemann-zeta 1 1 2 𝑛 superscript 𝑘 2 𝑛 {\displaystyle{\displaystyle\zeta'\left(1-2n,\frac{h}{k}\right)=\frac{(\psi% \left(2n\right)-\ln\left(2\pi k\right))B_{2n}\left(h/k\right)}{2n}-\frac{(\psi% \left(2n\right)-\ln\left(2\pi\right))B_{2n}}{2nk^{2n}}+\frac{(-1)^{n+1}\pi}{(2% \pi k)^{2n}}\sum_{r=1}^{k-1}\sin\left(\frac{2\pi rh}{k}\right){\psi^{(2n-1)}}% \left(\frac{r}{k}\right)+\frac{(-1)^{n+1}2\cdot(2n-1)!}{(2\pi k)^{2n}}\sum_{r=% 1}^{k-1}\cos\left(\frac{2\pi rh}{k}\right)\zeta'\left(2n,\frac{r}{k}\right)+% \frac{\zeta'\left(1-2n\right)}{k^{2n}},}}
    0 references
    DLMF id
    DLMF:25.11.E21
    0 references
    Symbols used
    Q11109
    test
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1addec
    0 references
    Q11288
    test
    B n ⁡ ( x ) Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle B_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C24.S2.SS1.m2aadec
    0 references
    Hurwitz zeta function
    test
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2avdec
    0 references
    Riemann zeta function
    test
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2aadec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2acdec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q12148
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C25.S1.XMD3.m1kdec
    0 references
    Q12163
    test
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C25.S11.XMD1.m1dec
    0 references
    Q12162
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C25.S11.XMD2.m1bdec
    0 references
     
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