I'm Nina. I have a really tough homework that counts as a test and I couldn't do it. It's really urgent. Help please!
Let $\alpha \in \Bbb{R}^+$. Let $F$ be the function defined in $\Bbb{R}$ by $$F(x) = \frac{1}{\cosh\alpha - \cos x}.$$ What is the decomposition into simple elements of the fraction? $$ -\frac{2X}{X^2 - (e^{\alpha}+ e^{-\alpha} )X+1} $$
Give an other expression for $\displaystyle -\frac{e^{\alpha}}{e^{ix} - e^{\alpha}}$.
And for $\displaystyle -\frac{e^{\alpha}}{e^{ix} - e^{-\alpha}}$.
And big thanks.