So I have this formula $$\cot(z)=\sum_{n\in Z} \frac{1}{z-n\pi}=\frac{1}{z}+2z\sum_{k=1}^{\infty}\frac{1}{z^2-(k\pi)^2}$$ from wikipedia. How do you find this formula with Mittag-Leffler theorem? Or is this formula correct?
Thank you
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Just recall the statement of the ML theorem and compute the residue of $\cot z$ at any $z\in\pi\mathbb{Z}$. See also https://math.stackexchange.com/questions/581162/how-does-the-herglotz-trick-work – Jack D'Aurizio Apr 19 '18 at 19:40