How would one compute $\frac{1}{2\pi i }\oint_{|z| = \frac{1}{2}} \frac{\Phi_{n}(z)}{z^{k + 1}} dz$ in terms of k and n. If this is not possible, how would someone find a good approximation for this.
Edit: I have worked on this and have found a simplification: $\frac{1}{2\pi i }\oint_{|z| = \frac{1}{2}} \frac{\Phi_{n}(z)}{z^{k + 1}} dz = \frac{\Phi_n^{(k)}(0)}{k!} = \int_0^1 \Phi_{n}(e(\alpha))e(n\alpha) d\alpha$ by the orthogonality relation $\int_0^1e(nx) dx = 1$ if $n = 0$ and equals $0$ otherwise.
($\Phi_{n}(x)$ is the nth cyclotomic polynomial)
I, for one would use the explicit formula in terms of the Möbius function., followed by the Residue theorem.