Mittag-Leffler Expansion for Hyperbolic Tangent Function
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Theorem
- $\ds \pi \map \tanh {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {4 z^2 + \paren {2 n + 1}^2}$
where:
- $z \in \C$ is not a half-integer multiple of $i$
- $\tanh$ is the hyperbolic 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: $37.10$
- 2009: Murray R. Spiegel, Seymour Lipschutz, John Schiller and Dennis Spellman: Complex Variables (2nd ed.): $7.10$: Some Special Expansions