Compute $\int_\gamma\overline{\zeta} \, d\zeta$ using Cauchy’s Integral Formula

Question

Let $\gamma$ be the circle of radius $1$ and centre $0$, equipped with the counterclockwise orientation. Compute $$\int_\gamma\overline{\zeta} \, d\zeta$$ using Cauchy’s integral formula.

Any hints on how to proceed?

Answer 1

Just write $$\overline{\zeta}=\frac{|\zeta|^2}{\zeta}$$ and then realize that $|\zeta|^2$ is constant on $\gamma$. Now you are integrating something of the form $$\frac{f(\zeta)}{\zeta-0}$$ with $f$ holomorphic (constant, indeed) and so the Cauchy's integral formula applies.