Laurent Series Expansion for Cotangent Function
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Theorem
\(\ds \pi \cot \pi z\) | \(=\) | \(\ds \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 z - 2 \paren {\dfrac {\pi^2 } 6 z + \dfrac {\pi^4 } {90 } z^3 + \dfrac {\pi^6 } {945 } z^5 + \cdots}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 z - \dfrac {\pi^2 } 3 z - \dfrac {\pi^4 } {45 } z^3 - \dfrac {2 \pi^6 } {945 } z^5 - \cdots\) |
where:
- $z \in \C$ such that $\cmod z < 1$
- $\zeta$ is the Riemann Zeta function.
Proof
From Mittag-Leffler Expansion for Cotangent Function:
\(\ds \pi \cot \pi z\) | \(=\) | \(\ds \frac 1 z + 2 \sum_{k \mathop = 1}^\infty \frac z {z^2 - k^2}\) |
Factoring $-\dfrac 1 {k^2}$:
\(\ds \pi \cot \pi z\) | \(=\) | \(\ds \frac 1 z + 2 \sum_{k \mathop = 1}^\infty \paren {\frac z {k^2} } \paren {\frac 1 {\frac {z^2} {k^2} - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 z - 2 \sum_{k \mathop = 1}^\infty \paren {\frac z {k^2} } \paren {\frac 1 {1 - \frac {z^2} {k^2} } }\) |
Taking $\cmod z < 1$, and noting that $k \ge 1$, we have, by Sum of Infinite Geometric Sequence:
- $\ds \pi \cot \pi z = \frac 1 z - 2 \sum_{k \mathop = 1}^\infty \frac z {k^2} \cdot \sum_{n \mathop = 1}^\infty \paren {\frac {z^2} {k^2} }^{n - 1}$
from which:
\(\ds \pi \cot \pi z\) | \(=\) | \(\ds \frac 1 z - 2 \sum_{k \mathop = 1}^\infty \sum_{n \mathop = 1}^\infty \frac {z^{2 n - 2} \cdot z} {k^{2 n - 2} \cdot k^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 z - 2 \sum_{k \mathop = 1}^\infty \sum_{n \mathop = 1}^\infty \frac 1 {k^{2 n} } \cdot z^{2 n - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} } \cdot z^{2 n - 1}\) | Product of Absolutely Convergent Series | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}\) | Definition of Riemann Zeta Function |
$\blacksquare$