Mittag-Leffler Expansion for Cotangent Function

Theorem

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

where:
 * $z \in \C$ is not an integer
 * $\cot$ is the cotangent function.

Outline of proof
Informally, we can say:
 * $\pi \cot \pi z = \dfrac {\mathrm d} {\mathrm dz} \left({\ln \sin \left({\pi z}\right)}\right)$.

We then use the Euler Formula for Sine Function to write $\sin \left({\pi z}\right)$ as an infinite product and differentiate its logarithm.

Formally, we work with logarithmic derivatives and use Logarithmic Derivative of Infinite Product of Analytic Functions.

Proof
Let $\mathcal L$ denote the logarithmic derivative.

On the open set $\C \setminus \Z$ we have: