Mittag-Leffler Expansion for Hyperbolic Cotangent Function

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Theorem

$\displaystyle \pi \, \map \coth {\pi z} = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 + n^2}$

where:

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


Proof

\(\displaystyle \pi \, \map \coth {\pi z}\) \(=\) \(\displaystyle \pi i \, \map \cot {\pi i z}\) Hyperbolic Cotangent in terms of Cotangent
\(\displaystyle \) \(=\) \(\displaystyle i \paren {\frac 1 {i z} + 2 i \sum_{n \mathop = 1}^\infty \frac z {\paren {i z}^2 - n^2} }\) Mittag-Leffler Expansion for Cotangent Function
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \frac z {-z^2 - n^2}\) $i^2 = -1$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 + n^2}\)

$\blacksquare$


Source of Name

This entry was named for Magnus Gustaf Mittag-Leffler.


Sources