TITLE: A series expansion for $\cot (\pi z)$
QUESTION [15 upvotes]: How to show the following identity holds? 
$$
\displaystyle\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}=\pi\cot \pi z-\dfrac{1}{z}\qquad |z|<1
$$

REPLY [7 votes]: I have found a link which deals with this problem: people.reed.edu/~jerry/311/cotan.pdf