Laurent Series Expansion for Cotangent Function

Theorem

 * $\displaystyle \pi \cot \pi z = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}$

where:
 * $z \in \C$ such that $\cmod z < 1$
 * $\zeta$ is the Riemann Zeta function.

Proof
From Mittag-Leffler Expansion for Cotangent Function:


 * $\displaystyle \pi \cot \pi z = \frac 1 z + 2 \sum_{k \mathop = 1}^\infty \frac z {z^2 - k^2}$

Factoring $-\dfrac 1 {k^2}$:


 * $\displaystyle \pi \cot \pi z = \frac 1 z - 2 \sum_{k \mathop = 1}^\infty \frac z {k^2} \cdot \frac 1 {1 - \frac {z^2} {k^2} }$

Taking $\cmod z < 1$, and noting that $k \ge 1$, we have, by Sum of Infinite Geometric Sequence:


 * $\displaystyle \pi \cot \pi z = \frac 1 z - 2 \sum_{k \mathop = 1}^\infty \frac z {k^2} \cdot \sum_{n \mathop = 1}^\infty \paren {\frac {z^2} {k^2} }^{n - 1}$

from which: