### Computations on U and V when they do not commute.

We have $f= U + U^{-1} + V + V^{-1}$. Exp is the exponential map: $e^{f}=\Sigma_{m=0}^{\infty} \frac{f^{n}}{n!} $. U and V do not commute and we have $U^{m}V^{n}= e^{imn}V^{n}U^{m}$ for any m,n integer. I want to find the constant term for the expression $\partial_{2}e^{-f/2} \partial_{2}e^{f}$. We have $\partial_{2}f= V - V^{-1}$.