Real boundary values let a holomorphic function reflect across the real axis by complex conjugation.Suggest a slogan
Let \(\Omega\subseteq\CC\) be an open set with \(\overline{z}\in\Omega\) for every \(z\in\Omega\), and put \(\Omega^+\coloneqq\{z\in\Omega : \operatorname{Im}z>0\}\), \(\Omega^-\coloneqq\{z\in\Omega : \operatorname{Im}z<0\}\), and \(I\coloneqq\Omega\cap\RR\). Let \(f\) be holomorphic on \(\Omega^+\), continuous on \(\Omega^+\cup I\), and real-valued on \(I\). Then the function \[ F(z)\coloneqq\begin{cases}f(z), & z\in\Omega^+\cup I,\\ \overline{f(\overline{z})}, & z\in\Omega^-,\end{cases} \] is holomorphic on \(\Omega\). In particular, for \(\Omega=\CC\): if \(f\) is holomorphic on the upper half-plane, continuous on its closure, and real-valued on \(\RR\), then \(F\) is entire.