$E$ be a domain, define $E^*=\{z \in C: \overline{z}\in E\}$ $f: E \to C$ is analytic, then $f^*(z)=\overline{f(\overline{z})}$ analytic on $E*$.
How do I rigorously show $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is?