# Series Expansion for Pi Cotangent of Pi Lambda

 It has been suggested that this page or section be merged into Mittag-Leffler Expansion for Cotangent Function. (Discuss)
 It has been suggested that this page or section be merged into Partial Fractions Expansion of Cotangent. (Discuss)

## Theorem

Let $\lambda \in \R \setminus \Z$ be a real number which is not an integer.

Then:

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

## Proof

Let $\map f x$ be the real function defined on $\openint 0 \pi$ as:

$\map f x = \cos \lambda x$

From Half-Range Fourier Cosine Series: $\cos \lambda x$ over $\openint 0 \pi$ its Fourier series can be expressed as:

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

Because of the nature of this expansion, we have that:

$\map f \pi = \map f {-\pi}$

and so the expansion holds for $x = \pi$.

Also note that because $\lambda$ is not an integer, $\sin \lambda \pi \ne 0$ and so $\cot \pi \lambda$ is defined.

So, setting $x = \pi$:

 $\ds \cos \lambda \pi$ $=$ $\ds \frac {2 \lambda \sin \lambda \pi} \pi \paren {\frac 1 {2 \lambda^2} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {\cos n \pi} {\lambda^2 - n^2} }$ $\ds \leadsto \ \$ $\ds \pi \cot \pi \lambda$ $=$ $\ds 2 \lambda \paren {\frac 1 {2 \lambda^2} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {\cos n \pi} {\lambda^2 - n^2} }$ Definition of Real Cotangent Function $\ds$ $=$ $\ds \frac 1 \lambda + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {2 \lambda \cos n \pi} {\lambda^2 - n^2}$ simplification $\ds$ $=$ $\ds \frac 1 \lambda + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {2 \lambda \paren {-1}^n} {\lambda^2 - n^2}$ Cosine of Multiple of Pi $\ds$ $=$ $\ds \frac 1 \lambda + \sum_{n \mathop = 1}^\infty \frac {2 \lambda} {\lambda^2 - n^2}$ simplification

$\blacksquare$