DLMF:25.11.E21 (Q7695)

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

Statements

Defining formula

{\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}},}}

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DLMF:25.11.E21

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Defining formula

{\displaystyle{\displaystyle B_{\NVar{n}}}}

C24.S2.SS1.m1addec

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Defining formula

{\displaystyle{\displaystyle B_{\NVar{n}}\left(\NVar{x}\right)}}

C24.S2.SS1.m2aadec

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Hurwitz zeta function

Defining formula

{\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}

C25.S11.E1.m2avdec

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Riemann zeta function

Defining formula

{\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}

C25.S2.E1.m2acdec

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the ratio of the circumference of a circle to its diameter

Defining formula

{\displaystyle{\displaystyle\pi}}

C3.S12.E1.m2acdec

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cosine function

Defining formula

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

C4.S14.E2.m2abdec

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psi (or digamma) function

Defining formula

{\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}

C5.S2.E2.m2aadec

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Defining formula

{\displaystyle{\displaystyle!}}

introduction.Sx4.p1.t1.r15.m5acdec

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principal branch of logarithm function

Defining formula

{\displaystyle{\displaystyle\ln\NVar{z}}}

C4.S2.E2.m2acdec

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Defining formula

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

C4.S14.E1.m2adec

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Defining formula

{\displaystyle{\displaystyle n}}

C25.S1.XMD3.m1kdec

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Defining formula

{\displaystyle{\displaystyle h}}

C25.S11.XMD1.m1dec

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Defining formula

{\displaystyle{\displaystyle k}}

C25.S11.XMD2.m1bdec

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