Mittag-Leffler Expansion for Square of Cosecant Function

Theorem

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

where:

$z$ is a complex number that is not a integer

$\csc$ denotes the cosecant function.

Proof

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Source of Name

This entry was named for Magnus Gustaf Mittag-Leffler.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 37$: Partial Fraction Expansions: $37.6$