= Conjugate reflection of a holomorphic function
{title2=$f^*(z)=\overline{f(\overline z)}$}
If $f$ is <holomorphic> on an <open set> $U$, its conjugate reflection $f^*$ is <holomorphic> on $U^*=\{z:\overline z\in U\}$, with $(f^*)'(z)=\overline{f'(\overline z)}$. The identity follows by conjugating the difference quotient with increment $\overline h$; the two conjugations restore <complex differentiability>.
If $U$ is invariant under <complex conjugation> and is <connected>, and $f$ is real at a set of real points accumulating inside $U$, then $f=f^*$ by the <identity theorem>. In particular, an <entire function> real at every $1/n$ has this symmetry. This is a symmetry statement on an already defined domain; <Schwarz reflection> also constructs extensions across a boundary.
Back to article page
