Mittag-Leffler Expansion for Hyperbolic Cosecant Function
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Theorem
- $\ds \pi \map \csch {\pi z} = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac z {z^2 + n^2}$
where:
- $z \in \C$ is not an integer multiple of $i$
- $\csch$ is the hyperbolic cosecant function.
Proof
\(\ds \pi \map \csch {\pi z}\) | \(=\) | \(\ds i \pi \map \csc {i \pi z}\) | Hyperbolic Cosecant in terms of Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds i \paren {\dfrac 1 {i z} + 2 i \sum _{n \mathop = 1}^\infty \paren {-1}^n \dfrac z {\paren {i z}^2 - n^2} }\) | Mittag-Leffler Expansion for Cosecant Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 z - 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \dfrac z {-z^2 - n^2}\) | $i^2=-1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 z + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \dfrac z {z^2 + n^2}\) |
$\blacksquare$
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: $37.8$
- 2009: Murray R. Spiegel, Seymour Lipschutz, John Schiller and Dennis Spellman: Complex Variables (2nd ed.): $7.10$: Some Special Expansions