Mittag-Leffler Expansion for Cosecant Function

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Theorem

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

where:

$z \in \C$ is not an integer
$\cosec$ is the cosecant function.


Real Domain

Let $\alpha \in \R$ be a real number which is specifically not an integer.

Then:

$\pi \cosec \pi \alpha = \dfrac 1 \alpha + \ds 2 \sum_{n \mathop \ge 1} \paren {-1}^n \dfrac {\alpha} {\alpha^2 - n^2}$


Proof


Source of Name

This entry was named for Magnus Gustaf Mittag-Leffler.


Sources