OliForum Matematico

How do you prove this $\displaystyle z\cot \left(z\right)=1-2\sum _{k=1}^{\infty}\frac{z^2}{k^2\pi ^2-z^2}$ ?

i’ve seen this formula on a document about Bernoulli’s numbers and their relations with Riemann’s zeta function (http://www.luciocadeddu.com/tesi/Aru_magistrale.pdf corollario 2.5.2) but i really don’t understand why this is true…

I haven't looked at the document you link to, but the standard way to treat this kind of problems is to combine the Cauchy integral formula and the Phragmén–Lindelöf principle to get an appropriate residue theorem for unbounded domains. For this specific identity I believe there also is a more ad hoc technique known as the Herglotz trick.