Suppose $f$ is analytic in a domain $\Omega$ that contains the closed unit disk. Show that for every $z$ in the unit disk, $f(z)=\frac{1}{2\pi}\int_0^{2\pi} f(\frac{e^{i\theta}+z}{1+\bar{z}e^{i\theta}}) d\theta$.
I believe that we need to apply Cauchy's integral formula and the conformal self maps of the unit disk. But I have no idea how to put them together. Can someone help me? Thanks