Mittag-Leffler Expansion for Cotangent Function/Real Domain

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Theorem

Let $\alpha \in \R$ be a real number which is specifically not an integer.

Then:

$\ds \dfrac 1 \alpha + \sum_{n \mathop \ge 1} \dfrac {2 \alpha} {\alpha^2 - n^2} = \pi \cot \pi \alpha$


Proof

From Half-Range Fourier Cosine Series for $\cos \alpha x$ over $\openint 0 \pi$:

$\ds \cos \alpha x \sim \frac {2 \alpha \sin \pi \alpha} \pi \paren {\frac 1 {2 \alpha^2} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {\cos n x} {\alpha^2 - n^2} }$


Setting $x = \pi$:

\(\ds \cos \pi \alpha\) \(=\) \(\ds \frac {2 \alpha \sin \pi \alpha} \pi \paren {\frac 1 {2 \alpha^2} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {\cos n \pi} {\alpha^2 - n^2} }\)
\(\ds \leadsto \ \ \) \(\ds \cos \pi \alpha\) \(=\) \(\ds \frac {2 \alpha \sin \pi \alpha} \pi \paren {\frac 1 {2 \alpha^2} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {\paren {-1}^n} {\alpha^2 - n^2} }\) Cosine of Multiple of Pi
\(\ds \leadsto \ \ \) \(\ds \frac {\pi \cos \pi \alpha} {\sin \pi \alpha}\) \(=\) \(\ds \frac 1 \alpha + \sum_{n \mathop = 1}^\infty \frac {2 \alpha} {\alpha^2 - n^2}\)
\(\ds \leadsto \ \ \) \(\ds \pi \cot \pi \alpha\) \(=\) \(\ds \frac 1 \alpha + \sum_{n \mathop = 1}^\infty \frac {2 \alpha} {\alpha^2 - n^2}\) Definition of Cotangent and rearranging

$\blacksquare$


Sources

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