Mittag-Leffler Expansion for Tangent Function

Theorem

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

where:

$z \in \C$ is not a half-integer

$\tan$ is the tangent 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: $37.4$
• 2009: Murray R. Spiegel, Seymour Lipschutz, John Schiller and Dennis Spellman: Complex Variables (2nd ed.): $7.10$: Some Special Expansions