For a contour integral in a closed path we have the Residue Theorem:
$$ \int_\gamma f(z) dz = 2\pi i \sum_k Res(f, z_k) $$
If $\gamma$ is not closed, but, for example, is defined as $ \gamma(\theta) = Re^{i \theta}$, $\theta \in [0, \pi]$. Do we have $$ \int_\gamma f(z) dz = \pi i \sum_k Res(f, z_k) ?$$
For some context: when evaluating a real integral from $-\infty$ to $\infty$, with a singularity at $z=0$, we can use a path that excludes this singularity from the interior of the curve, however we still have its contribution in the integral.