# DLMF:25.11.E20 (Q7694)

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DLMF:25.11.E20
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## Statements

${\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(s,a\right)=\frac{(\ln a% )^{k}}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum_{r=0}^{k-1}% \frac{(\ln a)^{r}}{r!(s-1)^{k-r+1}}-\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(% \widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x+a\right))^{k}}{(x+a)^{s+2}}% \mathrm{d}x+\frac{k(2s+1)}{2}\int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x% \right)-B_{2})(\ln\left(x+a\right))^{k-1}}{(x+a)^{s+2}}\mathrm{d}x-\frac{k(k-1% )}{2}\int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x% +a\right))^{k-2}}{(x+a)^{s+2}}\mathrm{d}x,}}$
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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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${\displaystyle{\displaystyle B_{\NVar{n}}}}$
C24.S2.SS1.m1acdec
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${\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}$
C25.S11.E1.m2audec
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${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$
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${\displaystyle{\displaystyle!}}$
introduction.Sx4.p1.t1.r15.m5abdec
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${\displaystyle{\displaystyle\int}}$
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${\displaystyle{\displaystyle\ln\NVar{z}}}$
C4.S2.E2.m2abdec
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${\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}$
C24.S2.SS3.m1acdec
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${\displaystyle{\displaystyle\Re}}$
C1.S9.E2.m1ahdec
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${\displaystyle{\displaystyle x}}$
C25.S1.XMD5.m1fdec
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${\displaystyle{\displaystyle a}}$
C25.S1.XMD6.m1rdec
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${\displaystyle{\displaystyle s}}$
C25.S1.XMD7.m1odec
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${\displaystyle{\displaystyle k}}$