You integrate in a loop around a singularity $z$ and get $2\pi i \text{Res}(z)$. Is there a path of integration such that the result is $\gamma 2\pi i \text{Res}(z)$ with $\gamma\in (0,1)$?
If it helps, I know that if the singularity is simple, then it is easy: approach and exit the singularity in rays $\gamma/2\pi$ apart.