# A series expansion of $\cot (\pi z)$

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