Mittag-Leffler Expansion for Hyperbolic Cotangent Function
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Theorem
- $\ds \pi \map \coth {\pi z} = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 + n^2}$
where:
- $z \in \C$ is not an integer multiple of $i$
- $\coth$ is the hyperbolic cotangent function.
Proof
\(\ds \pi \map \coth {\pi z}\) | \(=\) | \(\ds \pi i \map \cot {\pi i z}\) | Hyperbolic Cotangent in terms of Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds i \paren {\frac 1 {i z} + 2 i \sum_{n \mathop = 1}^\infty \frac z {\paren {i z}^2 - n^2} }\) | Mittag-Leffler Expansion for Cotangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \frac z {-z^2 - n^2}\) | $i^2 = -1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 + n^2}\) |
$\blacksquare$
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: $37.7$
- 2009: Murray R. Spiegel, Seymour Lipschutz, John Schiller and Dennis Spellman: Complex Variables (2nd ed.): $7.10$: Some Special Expansions