DLMF:25.11.E21 (Q7695): Difference between revisions

From DRMF
Jump to navigation Jump to search
Changed an Item: Add constraint
Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
Defining formula:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5acdec

Revision as of 13:40, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:25.11.E21
No description defined

    Statements

    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 𝑘 digamma 2 𝑛 2 𝜋 𝑘 Bernoulli-polynomial-B 2 𝑛 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 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 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:25.11.E21
    0 references
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    C24.S2.SS1.m1addec
    0 references
    B n ( x ) Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle B_{\NVar{n}}\left(\NVar{x}\right)}}
    C24.S2.SS1.m2aadec
    0 references
    ζ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    C25.S11.E1.m2avdec
    0 references
    ζ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    C25.S2.E1.m2acdec
    0 references
    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2acdec
    0 references
    cos z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    C4.S14.E2.m2abdec
    0 references
    ψ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    C5.S2.E2.m2aadec
    0 references
    ! {\displaystyle{\displaystyle!}}
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references