Mittag-Leffler Expansion for Secant Function
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Theorem
- $\ds \pi \map \sec {\pi z} = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 - 4 z^2}$
where:
- $z \in \C$ is not a half-integer
- $\sec$ is the secant function.
Proof
\(\ds \pi \map \sec {\pi z}\) | \(=\) | \(\ds \pi \map \csc {\frac \pi 2 - \pi z}\) | Secant and Cosecant are Cofunctions in radians | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {1/2 - z} + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {1/2 - z} {\paren {1/2 - z}^2 - n^2}\) | Mittag-Leffler Expansion for Cosecant Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {1/2 - z} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \paren {\frac 1 {n + 1/2 - z} - \frac 1 {n - 1/2 + z} }\) | Partial Fractions Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {1/2 - z} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac 1 {n + 1/2 - z} + \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac 1 {n - 1/2 + z}\) | splitting into two convergent series | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {n + 1/2 - z} + \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {n + 1/2 + z}\) | rewriting both sums starting from $n=0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sum_{n \mathop = 0}^\infty \paren {-1}^n \paren {\frac 1 {2 n - 2 z + 1} + \frac 1 {2 n + 2 z + 1} }\) | combining convergent series | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 - 4 z^2}\) | Difference of Two Squares |
$\blacksquare$
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: $37.3$
- 2009: Murray R. Spiegel, Seymour Lipschutz, John Schiller and Dennis Spellman: Complex Variables (2nd ed.): $7.10$: Some Special Expansions