Maybe this is a little contrived; I'm trying to understand how a change of variables will change how to compute a residue or contour integral. Anyway, suppose we have complex contour integral
$$\frac{1}{2 \pi i} \int_\gamma \frac{1}{1-ez}\frac{1}{z^{n+1}}dz$$
where $n \in \mathbb{Z}$ and $\gamma$ is a circle of radius $\epsilon > 0$ centered around $0$. Now suppose we change variables via $z=e^{-\zeta}$, the above becomes $$\frac{-1}{2\pi i}\int_{\gamma'} \frac{1}{1-e^{1-\zeta}}e^{n\zeta}d\zeta.$$
For the first integral it would be sufficient to compute the residue of the integrand. After the change of variables, the integrand now has a pole at $\zeta=1$ but the curve $\gamma'$ isn't closed.
Question: Can we evaluate the first integral by first making the change of variables and then considering the residue of the integrand of the second integral at $\zeta=1$?