Difference between revisions of "Z coth(z) = 2 Sum of (-1)^(n+1) zeta(2n) z^(2n)/pi^(2n)"
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(Created page with "==Theorem== The following formula holds for $|z|<\pi$: $$z \mathrm{coth}(z) = 2 \displaystyle\sum_{k=0}^{\infty} (-1)^{k+1} \zeta(2k) \dfrac{z^{2k}}{\pi^{2k}},$$ where $\mathr...") |
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Latest revision as of 06:05, 4 March 2018
Theorem
The following formula holds for $|z|<\pi$: $$z \mathrm{coth}(z) = 2 \displaystyle\sum_{k=0}^{\infty} (-1)^{k+1} \zeta(2k) \dfrac{z^{2k}}{\pi^{2k}},$$ where $\mathrm{coth}$ denotes hyperbolic cotangent, $\zeta$ denotes Riemann zeta, and $\pi$ denotes pi.
Proof
References
- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.20 (2)$
