Mittag-Leffler Expansion for Hyperbolic Cosecant Function

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Theorem

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

where:

$z \in \C$ is not an integer multiple of $i$
$\csch$ is the hyperbolic cosecant function.


Proof

\(\ds \pi \map \csch {\pi z}\) \(=\) \(\ds i \pi \map \csc {i \pi z}\) Hyperbolic Cosecant in terms of Cosecant
\(\ds \) \(=\) \(\ds i \paren {\dfrac 1 {i z} + 2 i \sum _{n \mathop = 1}^\infty \paren {-1}^n \dfrac z {\paren {i z}^2 - n^2} }\) Mittag-Leffler Expansion for Cosecant Function
\(\ds \) \(=\) \(\ds \dfrac 1 z - 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \dfrac z {-z^2 - n^2}\) $i^2=-1$
\(\ds \) \(=\) \(\ds \dfrac 1 z + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \dfrac z {z^2 + n^2}\)

$\blacksquare$


Source of Name

This entry was named for Magnus Gustaf Mittag-Leffler.


Sources