Mittag-Leffler Expansion for Cotangent Function/Proof 2

Proof
Let $\map \zeta s$ be the Riemann zeta function.

Let $\ds \map g z = \sum_{n \mathop = 1}^\infty z^n \map \zeta {2 n}$ be the generating function of $\map \zeta {2 n}$

By Power Series Expansion for Cotangent Function, for $\size z < 1$:

By Riemann Zeta Function at Even Integers:


 * $\map \zeta {2 n} = \dfrac {\paren {-1}^{n + 1} \pi^{2 n} 2^{2 n - 1} B_{2 n} } {\paren {2 n}!}$

Thus:

By Analytic Continuation of Generating Function of Dirichlet Series and Uniqueness of Analytic Continuation:
 * $\ds \dfrac {\pi z \map \cot {\pi z} - 1} {-2} = \sum_{n \mathop = 1}^\infty \dfrac {z^2} {n^2 - z^2}$

for all of $\C$, as this is the overlap of their domains.

Thus: