DLMF:25.11.E20 (Q7694)

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Language	Label	Description	Also known as
English	DLMF:25.11.E20	No description defined

Statements

Defining formula

{\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(s,a\right)=\frac{(\ln a% )^{k}}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum_{r=0}^{k-1}% \frac{(\ln a)^{r}}{r!(s-1)^{k-r+1}}-\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(% \widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x+a\right))^{k}}{(x+a)^{s+2}}% \mathrm{d}x+\frac{k(2s+1)}{2}\int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x% \right)-B_{2})(\ln\left(x+a\right))^{k-1}}{(x+a)^{s+2}}\mathrm{d}x-\frac{k(k-1% )}{2}\int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x% +a\right))^{k-2}}{(x+a)^{s+2}}\mathrm{d}x,}}

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DLMF:25.11.E20

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{\displaystyle{\displaystyle\Re s>-1}}

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{\displaystyle{\displaystyle s\neq 1}}

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{\displaystyle{\displaystyle a>0}}

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Defining formula

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

C24.S2.SS1.m1acdec

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Hurwitz zeta function

Defining formula

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

C25.S11.E1.m2audec

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Defining formula

{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

C1.S4.SS4.m1addec

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Defining formula

{\displaystyle{\displaystyle!}}

introduction.Sx4.p1.t1.r15.m5abdec

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