Laurent Series Expansion for Cotangent Function

Corollary to Laurent Series Expansion for Cotangent Function

 * $\displaystyle \pi \cot \pi z = \frac 1 z - 2\sum_{n=1}^\infty \zeta\left(2n\right)z^{2n-1}$

For $|z| < 1$, where $\zeta$ is the Riemann Zeta function.

Proof
From Laurent Series Expansion for Cotangent Function,


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

Factoring $\displaystyle - \frac 1 {k^2}$,


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

Taking $|z| < 1$, and noting that $k \ge 1$, we have, by Sum of Infinite Geometric Progression,


 * $\displaystyle \pi \cot \pi z = \frac 1 z - 2 \sum_{k=1}^\infty \frac z {k^2} \cdot \sum_{n=1}^\infty \left(\frac {z^2} {k^2}\right)^{n-1}$

From which,