Formula:DLMF:25.11:E37

<< Formula:DLMF:25.11:E36

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E38 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\HurwitzZeta@{nk}{a}=-n\ln\Gamma\left(a\right)+\ln\left(\prod_{j=0}^{n-1}\Gamma\left(a-{\mathrm{e}^{(2j+1)\pi\mathrm{i}/n}}\right)\right)}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle n=2,3,4,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\Re{a}\geq 1}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\Pi}}}$ : product : http://drmf.wmflabs.org/wiki/Definition:prod
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (37), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E36

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E38 >>