- #1

geoffrey159

- 535

- 72

## Homework Statement

What is the partial fraction decomposition in ##\mathbb{R}[X]## of ##F = \frac{1}{X^{2n} - 1 } ##, ##n\ge 1##.

## Homework Equations

## The Attempt at a Solution

Is this correct ?

## F = \frac{1}{2n}(\frac{1}{X-1} - \frac{1}{X+1} + 2 \sum_{k = 1}^{n-1} \frac{

\cos (\frac{k\pi}{n})X - 1 }{ X^2 - 2\cos (\frac{k\pi}{n}) X +1 })##

I have done the decomposition in ##\mathbb{C}[X]## and grouped the terms with the form

## \frac{a}{X - w} + \frac{b} {X - \bar w } ##, with ##w \in \mathbb{C}-\mathbb{R} ## , in order to get the terms that are in the sum.

The polynomials at the denominator are irreducible in ##\mathbb{R}[X]##, and the numerators have degree strictly less than the degree of the denominators. It looks correct to me but I have a doubt