Formula:DLMF:25.11:E20

<< Formula:DLMF:25.11:E19

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E21 >>

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \opminus^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@{x+a})^k} {(x+a)^{s+2}} \diff{x} + k (2s+1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x} (\ln@{x+a})^{k-1}} {(x+a)^{s+2}} \diff{x} - k (k-1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x} (\ln@{x+a})^{k-2}} {(x+a)^{s+2}} \diff{x} }}

Constraint(s)

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

Note(s)

primes on ${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ 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 >>