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How to show the following identity holds?

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

  • 0
    Search-engining "Herglotz Trick" will be interesting in this context.2010-08-28
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    Rasmus: Thanks for your information! I see it is in Chapter 23 of *Proofs From The Book* by Martin Aigner and Günter M. Ziegler.2010-08-28
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    many proofs appeared at SE : [here](http://math.stackexchange.com/questions/159752/proof-about-z-cot-z-1-2-sum-k-ge1z2-k2-pi2-z2), [here](http://math.stackexchange.com/questions/141470/find-the-sum-of-sum-1-k2-a2-when-0a1), [here](http://math.stackexchange.com/questions/110494/possibility-to-simplify-sum-limits-k-infty-infty-frac-left/110495#110495) and so on...2012-11-04
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    @Raymond Manzoni Many thanks.2012-11-04
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    Glad it helped Américo. Cheers !2012-11-04

1 Answers 1

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I have found a link which deals with this problem: people.reed.edu/~jerry/311/cotan.pdf

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    Why a negative vote! I don't Understand :x)2010-08-28
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    Maybe they are expecting you to give a short description of the content in the paper, instead of just citing it. Regards.2013-06-11