# Mittag-Leffler Expansion for Hyperbolic Cotangent Function

## 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.