$\displaystyle\frac{1}{(x-z_k)(x-1/z_k)}=\frac{1}{x^2-1}(\frac{x}{x-z_k}-\frac{1/x}{1/x-z_k})$
$\displaystyle f(x)=\prod_{k=0}^{n-1} (x-z_k)=x^n-1$
$\displaystyle\sum_{k=0}^{n-1}\frac{1}{x-z_k}=\frac{f'(x)}{f(x)}=\frac{nx^{n-1}}{x^n-1}$
$\displaystyle\sum_{k=0}^{n-1}\frac{1}{(x-z_k)(x-1/z_k)}=\frac{1}{x^2-1}(x\frac{nx^{n-1}}{x^n-1}-\frac{1}{x}\frac{nx^{1-n}}{x^{-n}-1})=\frac{n(x^n+1)}{(x^n-1)(x^2-1)}$ |