Let $C$ be a circle of fixed radius. What is the integral $\int\limits_C u\,dx+v\,dy$ for a complex polynomial $f(z)=u(x,y)+iv(x,y)$ where $z=x+iy$?
I think it should be $0$, but am not getting the exact proof. I think the integral equals $\operatorname{Re} \left(\int\limits_C \overline{f(z)} \, dz \right)$, which equals $\operatorname{Re}\left(\int\limits_C \frac{|f(z)|}{f(z)}\,dz\right)$. But, do the residues cancel out in the end? Any hints? Thanks beforehand.