Let $G \subseteq \mathbb C$ be a region and $f : G \longrightarrow \mathbb C$ be an analytic function. Let $\overline G : = \left \{z\ |\ \overline {z} \in G \right \}.$ Then by C-R equations show that the function $f^* : \overline G \longrightarrow \mathbb C$ defined by $f^* (z) = \overline {f (\overline {z})}$ is analytic on $\overline G.$
I can able to show that the C-R equations are satisfied by the function $f^*$ by writing $f^*(z) = u(x,-y) - i\ v(x,-y)$ where $f(z) = u(x,y) + i\ v(x,y).$ But will that suffice to show the analyticity of $f^*$ on $\overline G\ $? Instead I tried to show the analyticity by the power series expansion of $f^*$ around some $a \in G.$ If $f(z) = \sum\limits_{n=0}^{\infty} a_n (z-a)^n$ is the power series expansion of $f(z)$ around $a$ then $f^*(z) = \sum\limits_{n=0}^{\infty} \overline {a_n} (z - \overline {a})^n$ is the power series expansion of $f^*(z)$ around $z = \overline {a}.$ This definitely proves that $f^*$ is analytic on $\overline {G}.$ But I am not quite sure as to how C-R equations will be able to conclude the same. Could anybody provide some suggestion regarding this?
Thanks a bunch!
Since $f$ is analytic, $u$ and $v$ are of class $C^1$. It follows from this and from the Cauchy-Riemann equations that $f^*$ is holomorphic. And every holomorphic function is analytic.