Timeline for Summation of series involving $\sinh$ of a square root

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Jun 6, 2016 at 22:05	vote	accept	CommunityBot
Jun 4, 2016 at 13:19	comment	added	Noam Shalev - nospoon		We can generalize this result using the series (with $\theta\in [0,\pi/2]$) $$\displaystyle \sum_{n \geq 1} \frac{(-1)^{n+1}n\sin(\theta n)}{n^2+x^2}=\frac{\pi}{2} \frac{\sinh \theta x}{\sinh \pi x}.$$ For instance, the 2D case is $$\sum_{n,m=1}^{\infty} \frac{(-1)^{n+m}}{n m} \sin(\theta n) \sin(\theta m) \frac{\sinh \theta\sqrt{n^2+m^2}}{\sinh \pi \sqrt{n^2+m^2}} =\frac{\theta^3}{12 \pi}.$$ Note the curious case when $\theta=1$. Again, this easily generalizes to series of higher dimensions.
Jun 3, 2016 at 19:41	history	edited	Noam Shalev - nospoon	CC BY-SA 3.0	A sign error. I apologize for littering the front page.
Jun 2, 2016 at 16:38	review	First posts
Jun 2, 2016 at 16:48
Jun 2, 2016 at 16:35	history	answered	Noam Shalev - nospoon	CC BY-SA 3.0