the partial fraction of ${\pi}\cot{\pi}z$ from the partial fraction of $\frac{\pi^{2}}{\sin^{2}{\pi}z}$

Asked Apr 05 '14 at 14:57

Active Apr 05 '14 at 15:51

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I want to deduce the equation

$${\pi}\cot{\pi}z=\frac{1}{z}+ \sum_{n=1}^{\infty} \frac{2z}{z^{2}-n^{2}}$$

where the convergence is uniform on compact subsets of $\mathbb{C}-\mathbb{Z}$ from the partial fraction expression $\frac{\pi^{2}}{\sin^{2}{\pi}z}$. Any help would be great. Thank you.

edited Apr 05 '14 at 15:51

mle

2,287

asked Apr 05 '14 at 14:57

user135582

What's the derivative of $\pi \cot \pi z$? – Daniel Fischer Apr 05 '14 at 14:58
It is $\frac{\pi^{2}}{sin^{2}{\pi}z}$, I have noticed that before and I tried to integrate the series term by term but I didn't get the right equation. Thank you. – user135582 Apr 05 '14 at 15:20
In the partial fraction decomposition of $\frac{\pi^2}{\sin^2\pi z}$, group the terms corresponding to $n$ and $-n$. Since you know where you want to arrive, that should help. – Daniel Fischer Apr 05 '14 at 15:30
See ZZ's upvoted comment on this post. – Lucian Apr 05 '14 at 16:01

the partial fraction of ${\pi}\cot{\pi}z$ from the partial fraction of $\frac{\pi^{2}}{\sin^{2}{\pi}z}$

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