Complex integration along a circle not centered at the origin

Question

I need to evaluate $∮_C\overline{z}^2dz$ around |$z-1|=1$. I understand that this is a circle with radius $1$ centered at $(1,0)$. I know how to do this if the circle is centered at the origin. $\overline{z}^2=e^{-2i\theta}$ and $dz=-2ie^{-2i\theta}$. $$\int_0^{2\pi}e^{-2i\theta}\left(-2ie^{-2i\theta}\right)d\theta=-2i\left[\frac{e^{-4i\theta}}{-3i}\right]_0^{2\pi}=-2i\left(\frac{1}{-3i}-\frac{1}{-3i}\right)=0$$ So now, how can I modify this to account for the circle shifting?

Answer 1

Hint: The parametric equation of a circle centered at $z_0$ is just $z(t) = z_0 + re^{it}$ (where $0\leq t \leq 2\pi$).