Mittag-Leffler Expansion for Cotangent Function

Theorem

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

For non-integer $z \in \C$. Where $\cot$ is the cotangent function.

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

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: