I am reading the notes "Elliptic modular forms and their applications" by Don Zagier.
The problem is with the proof of proposition 2 on page 10, which is about integrating $d(log(f))$ over the boundary of a region $D$, where $f$ is a nonzero modular form of weight k on $\Gamma_1$, and $D$ is the closure of the fundamental domain of $\Gamma_1$ minus small $\epsilon$ nbds around the zeros of $f$.
The boundary of $D$ contains a half of a circle around the point $i$ and a third of a circle around the point $\omega = e^{2\pi i/3}$. The text claims that the integral around the partial circles is $2\pi i $ord$(f)/n_P$, where $P$ is the centre of the partial circle, $n_P$ is the order of the stabilizer in $\Gamma_1/\{±1\}$ of any point on the upper half plane representing $P$.
I understand that $2\pi i $ord$(f)$ comes from integrating along an entire circle around a zero of $f$ using the Cauchy integral theorem. But I don't know why the integral along the partial circles would be that along the full circle divided by $n_P$. Can someone help me out?