I'm currently working on the derivation of Bethe ansatz equations.
I have an equation like this:
$ z^{-1}\prod_{m=1,m\neq l}^{Q-1}\frac{q^2z-z_{m}}{z-z_{m}}+q^{-2}z \prod_{m=1,m\neq l}^{Q-1}\frac{q^{-2}z-z_{m}}{z-z_{m}}-(z+z^{-1})=E,$
where $z\in \mathbb{C},q\in\mathbb{C}$ and $E$ is an energy eigenvalue.
Obviously, the poles exist at $z=0, z=\infty , z=z_{m}$. Since the lhs is meromorphic function and the rhs is a constant, to make them equal, I have to calculate the residues and null them.
For $z=0$ the problem is solved. But I have problem with the two other cases.
I tried to calculate the residue at $z=\infty$ and at $z=z_{m}$ via
$Res(f(z),z=\infty)=-Res(-z^{-2}f(1/z),z=0)$ and $ Res(f(z),z=z_{m})=\lim_{z\to z_{m}}(z-z_{m})\cdots$
The vanishing of the resiude at $z=z_{m}$ should give equations like this:
$z^2_{l}=q^{Q}\prod_{m=1,m\neq l}^{Q-1} \frac{q^2 z_{l} -z_{m}}{z_l-q^2z_{m}}$
But in both cases I failed. Please, can anybody help. I am grateful for any ideas.