Satisfying the Cauchy Riemann Equations

Question

Where does the function $f(z)=\bar{z} $ satisfy the Cauchy-Riemann Equations?

Answer 1

There are two forms of the Cauchy-Riemann equation we could look at to obtain the answer. The first form asks where $$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$ is satisfied (note: $f(z)= u+iv$). There is also an equivalent relation we satisfy to show that the Cauchy-Riemann equations are satisfied. This has the form: $$\frac{\partial f(z)}{\partial \bar{z}}=0$$ With either of these relationships, it should be fairly simple to solve.