Mittag-Leffler Expansion for Square of Secant Function

Theorem

$\ds \pi^2 \map {\sec^2} {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac {\paren {2 n + 1} + 4 z^2} {\paren {\paren {2 n + 1}^2 - 4 z^2}^2}$

where:

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

$\sec$ denotes the secant 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: $37.5$