I'm doing practice physics qualifying exam problems and came across this one I didn't know how to solve:
Show that if $f(x)$ is bounded and analytic for $|z|=|x+iy|<1$, then $f(\zeta)=\frac{1}{\pi}\int_{|z|<1}\frac{f(z)\,dx\,dy}{(1-\bar{z}\zeta)^2}$ Hint: First express the area integral in polar coordinates, then transform one of the integrals to a suitable line integral of a rational function that can be evaluated using the calculus of residues.
I tried using $z=re^{i\theta}$ and messing around with the integral, but after a long writeout I am left with a puddle of muddled thoughts. Could someone explain the next steps?