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How to show the following identity holds?

$$ \displaystyle\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}=\pi\cot \pi z-\dfrac{1}{z}\qquad |z|<1 $$

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Search-engining "Herglotz Trick" will be interesting in this context. – Rasmus Aug 28 '10 at 8:38
Rasmus: Thanks for your information! I see it is in Chapter 23 of Proofs From The Book by Martin Aigner and Günter M. Ziegler. – Américo Tavares Aug 28 '10 at 8:48
many proofs appeared at SE : here, here, here and so on... – Raymond Manzoni Nov 4 '12 at 15:44
@Raymond Manzoni Many thanks. – Américo Tavares Nov 4 '12 at 15:49
Glad it helped Américo. Cheers ! – Raymond Manzoni Nov 4 '12 at 15:51

I have found a link which deals with this problem:

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Why a negative vote! I don't Understand :x) – anonymous Aug 28 '10 at 13:32
Maybe they are expecting you to give a short description of the content in the paper, instead of just citing it. Regards. – awllower Jun 11 '13 at 13:04

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