# DLMF:25.11.E20 (Q7694)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
No description defined
Language Label Description Also known as
English
DLMF:25.11.E20
No description defined

## Statements

$\displaystyle (-1)^{k}\Hurwitzzeta^{(k)}@{s}{a}=\frac{(\ln@@{a})^{k}}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum_{r=0}^{k-1}\frac{(\ln@@{a})^{r}}{r!(s-1)^{k-r+1}}-\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(\perBernoulliB{2}@{x}-\BernoullinumberB{2})(\ln@{x+a})^{k}}{(x+a)^{s+2}}\diff{x}+\frac{k(2s+1)}{2}\int_{0}^{\infty}\frac{(\perBernoulliB{2}@{x}-\BernoullinumberB{2})(\ln@{x+a})^{k-1}}{(x+a)^{s+2}}\diff{x}-\frac{k(k-1)}{2}\int_{0}^{\infty}\frac{(\perBernoulliB{2}@{x}-\BernoullinumberB{2})(\ln@{x+a})^{k-2}}{(x+a)^{s+2}}\diff{x},$
0 references
DLMF:25.11.E20
0 references
${\displaystyle{\displaystyle\Re s>-1}}$
0 references
${\displaystyle{\displaystyle s\neq 1}}$
0 references
${\displaystyle{\displaystyle a>0}}$
0 references
${\displaystyle{\displaystyle B_{\NVar{n}}}}$
C24.S2.SS1.m1acdec
0 references
${\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}$
C25.S11.E1.m2audec
0 references
${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$
0 references
${\displaystyle{\displaystyle!}}$
introduction.Sx4.p1.t1.r15.m5abdec
0 references
${\displaystyle{\displaystyle\int}}$
0 references
${\displaystyle{\displaystyle\ln\NVar{z}}}$
C4.S2.E2.m2abdec
0 references
${\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}$
C24.S2.SS3.m1acdec
0 references
${\displaystyle{\displaystyle\Re}}$
C1.S9.E2.m1ahdec
0 references
${\displaystyle{\displaystyle x}}$
C25.S1.XMD5.m1fdec
0 references
${\displaystyle{\displaystyle a}}$
C25.S1.XMD6.m1rdec
0 references
${\displaystyle{\displaystyle s}}$
C25.S1.XMD7.m1odec
0 references
${\displaystyle{\displaystyle k}}$