A study on complex integrals involving absolute values (Q1366836)

Using the residue theorem, the author evaluates \(\int_{|z|=a} z^{n-1}|z^n-k|^{-1} dz\), where \(k\in \{\pm b^n,\pm ib^n\}\) and \(1\leq b<a\). The key, of course, is that on the unit circle complex conjugates are reciprocals, so after a change of variable the integrand becomes a rational function, whose poles are clearly visible.