Difference between revisions of "Z coth(z) = sum of 2^(2n)B (2n) z^(2n)/(2n)!"
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(Created page with "==Theorem== The following formula holds for $|z| < \pi$: $$z \mathrm{coth}(z) = \displaystyle\sum_{k=0}^{\infty} 2^{2k} B_{2k} \dfrac{z^{2k}}{(2k)!},$$ where $\mathrm{coth}$...") |
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==References== | ==References== | ||
| − | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=Z coth(z) = 2z/(e^(2z)-1) + z|next= | + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=Z coth(z) = 2z/(e^(2z)-1) + z|next=Z coth(z) = 2 Sum of (-1)^(n+1) zeta(2n) z^(2n)/pi^(2n)}}: $\S 1.20 (2)$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 06:03, 4 March 2018
Theorem
The following formula holds for $|z| < \pi$: $$z \mathrm{coth}(z) = \displaystyle\sum_{k=0}^{\infty} 2^{2k} B_{2k} \dfrac{z^{2k}}{(2k)!},$$ where $\mathrm{coth}$ denotes hyperbolic cotangent, $B_{2k}$ denotes Bernoulli numbers, and $(2k)!$ denotes factorial.
Proof
References
- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.20 (2)$
