Formula:DLMF:25.11:E20

<< Formula:DLMF:25.11:E19

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E21 >>

${\displaystyle{\displaystyle{\displaystyle(-1)^{k}\HurwitzZeta^{(k)}@{s}{a}=\frac{(\ln a)^{k}}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum_{r=0}^{k-1}\frac{(\ln a)^{r}}{r!(s-1)^{k-r+1}}-s(s+1)\int_{0}^{\infty}\frac{\PeriodicBernoulliB{2}@{x}(\ln\left(x+a\right))^{k}}{(x+a)^{s+2}}\mathrm{d}x+k(2s+1)\int_{0}^{\infty}\frac{\PeriodicBernoulliB{2}@{x}(\ln\left(x+a\right))^{k-1}}{(x+a)^{s+2}}\mathrm{d}x-k(k-1)\int_{0}^{\infty}\frac{\PeriodicBernoulliB{2}@{x}(\ln\left(x+a\right))^{k-2}}{(x+a)^{s+2}}\mathrm{d}x}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\Re{s}>-1}}}$ &
${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle a>0}}}$

Note(s)

primes on ${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta}}}$ denote derivatives with respect to ${\displaystyle{\displaystyle{\displaystyle s}}}$

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\widetilde{B}_{n}}}}$ : periodic Bernoulli functions : http://dlmf.nist.gov/24.2#iii
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (20), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E19

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E21 >>