Let $C$ be a circle in the complex plane with center $a$ and radius $R$. I am trying to evaluate $\oint_{C} P(z) d \bar{z}$.
If I set $z=\bar{u}$ then we have $\bar{z}=u$ and $d\bar{z}=du$. Thus we may write $\oint_{C} P(z) d \bar{z} = \oint_{\bar{C}} P(\bar{u}) du$ where $\bar{C}$ is circle with radius $R$ and center $\bar{a}$. If $P(\bar{u})$ was an analytic function of $u$ I could apply Cauchy's Theorem and get 0. Does it matter that the conjugation function is not analytic?
Thank you for any help or hints.