Let $f$ be a holomorphic function on the disk
$$ D_{r_0} =\{z\in\ {C} : |z|
$$ f(z) =\frac{1}{2πi}\ \int_{|ζ|=R} \frac{f(ζ)}{ζ−z}\ dζ. $$
$$ 0 =\frac{1}{2πi}\ \int_{|ζ|=R} \frac{f(ζ)}{ ζ−\frac{R^2}{\bar{z}} }\ dζ. $$
My lecture note says that the second equation holds by Cauchy Theorem. But I don't know why the second equation is equal to zero.$\frac{R^2}{\bar{z}}$ could be on the disk which means ${|ζ|=R}$. Am I wrong?.