I would like to compute the contour integral
$ \int z^{2n-1} \exp\left(- i k \log \left(\frac{1-z}{1+z} \right) \right) dz $
over the unit circle in $\mathbb C$ for $n \in \mathbb Z$ and $k \in \mathbb R$, where $i^2 = -1$. The contribution from the pole at $z=0$ is clear. The problem is that the function is not really well defined at $z = \pm 1$ and I would like to avoid those points with a contour deformation (which I can justify in the context of the problem). If the residue at $z = \pm 1$ would be zero by some argument, I wouldn't matter how I avoid those points, showing that the result is unique.
Does anyone know how to argue this?
Thanks a lot!