Laurent Series Expansion for Cotangent Function

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

$\blacksquare$

Also see

• Power Series Expansion for Cotangent Function