I'm trying to differentiate this complex function $$ f(z) = (|z|-1)^2 $$ in order to determine where it satisfies the Cauchy-Riemann equation $ \frac{\partial f}{\partial \bar{z}} = 0$.
So I first differentiate with respect to z. (When I calculate it, I differentiate "as if z is real" -- is that correct?) $$ \frac{\partial f}{\partial z} = 2 (\lvert z\rvert -1) \frac{z}{\lvert z \rvert} $$
And then according to this, I can calculate $\frac{\partial f}{\partial \bar{z}}$ by taking the complex conjugate of $\frac{\partial f}{\partial z}$: $$ \frac{\partial f}{\partial \bar{z}} = 2 (\lvert z\rvert -1) \frac{\bar{z}}{\lvert z \rvert} $$
However, the correct answer is supposed to be $$ \frac{\partial f}{\partial \bar{z}} = 2 (\lvert z\rvert -1) \frac{z}{ \bar{z}} $$
Where am I going wrong? Any advice would be much appreciated.