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

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