Proving the Complex Conjugate is Analytic

Question

Let G be a region and define $G^∗ = \{z : z̅ ∈ G\}$. If $f : G \to \mathbb{C}$ is analytic, prove that $f^* : G^∗ → C$, defined by $f^*(z) = \overline{f(\overline{z})}$ is also analytic.

Answer 1

Another solution, different to the ones on the duplicates, is by using the Wirtinger Operators:

Let $f^*(z)=\overline{f(\overline{z})}$. Now since $f$ is analytic in $G$ iff $$\frac{\partial}{\partial \overline{z}} \ f=0$$ in $G$, and in this case one can see that the Cauchy Riemann equations for $f$ gives that also $$\frac{\partial}{\partial \overline{z}}f^*=0$$ in $G^*$, thus indeed $f^*$ is analytic in $G^*$.