 Solution

The sector contains exactly one simple pole, at $z_0=e^{i\pi /n}$. All roots of $1+z^n$ are simple, and its residue issince $z_0^n=-1$. For the positively oriented boundary, the residue theorem givesThe vertex at zero is harmless: $f$ is analytic there for nonnegative integer $m$, or a small indent may be removed because its integral is $O({\rho }^{m+1})$.