Integrating $\bar{z}^n$ over a circle Evaluate: $$\int_{C(0,1)} \bar z^n dz $$ I am not quite sure how to approach this problem. I am trying to see conceptually what is going on and I am unsure how to approach the integration. 

If you want to avoid parametrising the curve, you can exploit that $z\bar z = |z|^2$, so on the curve $\bar z = \frac{1}{z}$. Hence
$$
\int_C \bar z^n\,dz = \int_C \frac{1}{z^n}\,dz
$$
which is $0$ for $n\neq 1$, and $2\pi i$ for $n=1$ by any of your favourite tools to compute complex curve integrals (Cauchy's integral formula, residues or the complex version of the fundamental theorem of calculus).
