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

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