Let $0 < |\alpha| < 1$. Find $\displaystyle{\int_{\gamma}\frac{\mathrm{Re}(z)}{z-\alpha}dz}$
in terms of $\alpha$, where $\gamma$ is the circle $|z| = 1$ oriented in the counterclockwise direction.
I tried to express $\mathrm{Re}(z)$ as $\frac{z+\overline{z}}{2}$. Then how should I proceed on? Any idea/suggestion is welcomed!:D
Hint: Note that when $|z|=1$, $\bar{z}=\frac{1}{z}$.