Mittag-Leffler Expansion for Hyperbolic Secant Function

Theorem

$\ds \pi \map \sech {\pi z} = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 + 4 z^2}$

where:

$z \in \C$ is not a half-integer multiple of $i$

$\ds \pi \map \sech {\pi z}$

$=$

$\ds \pi \map \sec {i \pi z}$

$\ds $

$=$

$\ds 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 - 4 \paren {i z}^2}$

$\ds $

$=$

$\ds 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 + 4 z^2}$

$i^2 = -1$

$\blacksquare$