Evaluate a complex integral using Cauchy formula Evaluate 
$$\int_\gamma \frac{\overline{w-z}}{w-z} dw,$$
where where $\gamma$ is the unit circle traversed once in the counterclockwise direction, and z is a point inside the unit disk.
I tried rewriting it into 
$$\int_\gamma \frac{|{w-z}|^2}{{(w-z)}^2} dw,$$
But I'm not sure how that will help.

Hint: Since $\gamma$ is the unit circle and $w$ is on $\gamma$ you have 
$$\bar{w}=\frac{1}{w}$$
Your integral is then
$$\int_\gamma \frac{\overline{w-z}}{w-z} dw,=\int_\gamma \frac{1}{w(w-z)} dw - \bar{z}\int_\gamma \frac{1}{(w-z)} dw,$$
Now the second integral is known, while the first is done either by Partial fraction decomposition, or by putting small nonintersecting curves around $0$ and $z$. Note hat with this approach you should disscuss the case $z=0$ separatelly...
