Mittag-Leffler Expansion for Cotangent Function/Proof 1

Outline of proof
Informally, we can say:
 * $\pi \cot \pi z = \map {\dfrac \d {\d z} } {\ln \map \sin {\pi z} }$.

We then use the Euler Formula for Sine Function to write $\map \sin {\pi z}$ 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 $\LL$ denote the logarithmic derivative.

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