Formula:DLMF:25.4:E5

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Formula:DLMF:25.5:E1 >>

${\displaystyle{\displaystyle{\displaystyle(-1)^{k}\RiemannZeta^{(k)}@{1-s}=\frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\left(\Re{(c^{k-m})}\cos\left(\tfrac{1}{2}\pi s\right)+\Im{(c^{k-m})}\sin\left(\tfrac{1}{2}\pi s\right)\right){\Gamma^{(r)}}\left(s\right)\RiemannZeta^{(m-r)}@{s}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle{\displaystyle c=-\ln\left(2\pi\right)-\tfrac{1}{2}\pi\mathrm{i}}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle s\neq 0,1}}}$ &
${\displaystyle{\displaystyle{\displaystyle k=1,2,3,\dots}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2
${\displaystyle{\displaystyle{\displaystyle\Im{z}}}}$ : imaginary part : http://dlmf.nist.gov/1.9#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation (5), Section 25.4 of DLMF.

URL links

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