Formula:DLMF:25.8:E8

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Failed to parse (unknown function "\hiderel"): {\displaystyle {\displaystyle \sum_{k \hiderel{=} 1}^\infty \frac{\RiemannZeta@{2k}}{k} z^{2k} = \ln@{\frac{\cpi z}{\sin@{\cpi z}}} }}

Constraint(s)


Proof

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Divide by in

Failed to parse (unknown function "\hiderel"): {\displaystyle {\displaystyle \sum_{k \hiderel{=} 0}^\infty \RiemannZeta@{2k} z^{2k} = - \tfrac{1}{2} \cpi z \cot@{\cpi z} }}

and integrate.


Symbols List

 : sum : http://drmf.wmflabs.org/wiki/Definition:sum
 : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
 : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
 : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
 : sine function : http://dlmf.nist.gov/4.14#E1

Bibliography

Equation (8), Section 25.8 of DLMF.

URL links

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