Mittag-Leffler Expansion for Cotangent Function

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Theorem

$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$

where:

$z \in \C$ is not an integer
$\cot$ is the cotangent function.


Real Domain

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

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


Proof 1

Let $\mathcal L$ denote the logarithmic derivative.

On the open set $\C \setminus \Z$ we have:

\(\ds \pi \cot \pi z\) \(=\) \(\ds \mathcal L \left({\sin \left({\pi z}\right)}\right)\) Primitive of Cotangent Function, or a complex version thereof
\(\ds \) \(=\) \(\ds \mathcal L \left({\pi z \prod_{n \mathop = 1}^\infty \left({1 - \frac {z^2} {n^2} }\right)}\right)\) Euler Formula for Sine Function
\(\ds \) \(=\) \(\ds \mathcal L \left({\pi z}\right) + \sum_{n \mathop = 1}^\infty \mathcal L \left({1 - \frac {z^2} {n^2} }\right)\) Logarithmic Derivative of Infinite Product of Analytic Functions
\(\ds \) \(=\) \(\ds \frac \pi {\pi z} + \sum_{n \mathop = 1}^\infty \frac 1 {1 - \frac {z^2} {n^2} } \cdot \frac {\mathrm d} {\mathrm d z} \left({1 - \frac {z^2} {n^2} }\right)\) Definition of Logarithmic Derivative of Meromorphic Function
\(\ds \) \(=\) \(\ds \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \frac z {n^2 - z^2}\) Derivative of Power
\(\ds \) \(=\) \(\ds \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}\)

$\blacksquare$


Proof 2

Let $\map \zeta s$ be the Riemann zeta function.

Let $\ds \map g z = \sum_{n \mathop = 1}^\infty z^n \map \zeta {2 n}$ be the generating function of $\map \zeta {2 n}$


By Power Series Expansion for Cotangent Function, for $\size z < 1$:

\(\ds \pi \map \cot {\pi z}\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \pi^{2 n} 2^{2 n} B_{2 n} \, z^{2 n - 1} } {\paren {2 n}!}\)
\(\ds \leadsto \ \ \) \(\ds \pi z \map \cot {\pi z}\) \(=\) \(\ds 1 + \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \pi^{2 n} 2^{2 n} B_{2 n} \, z^{2 n} } {\paren {2 n}!}\)
\(\ds \leadsto \ \ \) \(\ds \frac {\pi z \map \cot {\pi z} - 1} {-2}\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} \pi^{2 n} 2^{2 n - 1} B_{2 n} \, z^{2 n} } {\paren {2 n}!}\)

By Riemann Zeta Function at Even Integers:

$\map \zeta {2 n} = \dfrac {\paren {-1}^{n + 1} \pi^{2 n} 2^{2 n - 1} B_{2 n} } {\paren {2 n}!}$

Thus:

\(\ds \dfrac {\pi z \map \cot {\pi z} - 1} {-2}\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n}\)
\(\ds \) \(=\) \(\ds \map g {z^2}\)

By Analytic Continuation of Generating Function of Dirichlet Series and Uniqueness of Analytic Continuation:

$\ds \dfrac {\pi z \map \cot {\pi z} - 1} {-2} = \sum_{n \mathop = 1}^\infty \dfrac {z^2} {n^2 - z^2}$

for all of $\C$, as this is the overlap of their domains.

Thus:

\(\ds \pi z \map \cot {\pi z} - 1\) \(=\) \(\ds 2 \sum_{n \mathop = 1}^\infty \frac {z^2} {z^2 - n^2}\)
\(\ds \leadsto \ \ \) \(\ds \pi z \map \cot {\pi z}\) \(=\) \(\ds 1 + 2 \sum_{n \mathop = 1}^\infty \frac {z^2} {z^2 - n^2}\)
\(\ds \leadsto \ \ \) \(\ds \pi \map \cot {\pi z}\) \(=\) \(\ds \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}\)

$\blacksquare$


Also see


Source of Name

This entry was named for Magnus Gustaf Mittag-Leffler.


Sources