A holomorphic function on the upper half of a symmetric region that is real on the real axis extends to the whole region by reflection. The proof glues two holomorphic functions along a segment and applies Morera’s theorem.
Throughout, \(\Omega\) is an open set symmetric about the real axis, so \(z\in \Omega \iff \bar{z} \in \Omega\), and \(\Omega^+ \coloneqq \Omega\cap\ts{\Im z>0}\), \(I\coloneqq\Omega\cap\RR\), and \(\Omega^- \coloneqq \Omega\cap\ts{\Im z<0}\).
The symmetry principle
Symmetry principle (Tag T-5SKNT)
Let \(\Omega\subseteq\CC\) be open, and put \(\Omega^+\coloneqq\Omega\cap\ts{\Im z>0}\), \(\Omega^-\coloneqq\Omega\cap\ts{\Im z<0}\), and \(I\coloneqq\Omega\cap\RR\). Let \(f^+\) be holomorphic on \(\Omega^+\) and \(f^-\) holomorphic on \(\Omega^-\), and suppose both extend continuously to \(I\) with \(f^+(x)=f^-(x)\) for all \(x\in I\). Then the function \[ f(z)\coloneqq \begin{cases} f^+(z) & z\in\Omega^+,\\ f^-(z) & z\in\Omega^-,\\ f^+(z)=f^-(z) & z\in I, \end{cases} \] is holomorphic on \(\Omega\).
(Symmetry principle, by Morera)
The glued function \(f\) is holomorphic on \(\Omega^\pm\) by hypothesis, so it remains to check holomorphy along \(I\). Let \(T\subset\Omega\) be a closed triangle. If \(T\) misses \(I\), then \(T\) lies in \(\Omega^+\) or in \(\Omega^-\) and \(\int_{\partial T} f=0\) by Goursat.
If \(T\) meets \(I\), let \(T^+\coloneqq T\cap\ts{\Im z\geq 0}\) and \(T^-\coloneqq T\cap\ts{\Im z\leq 0}\), convex polygons with positively oriented boundaries. The boundaries share the segment \(T\cap\RR\) with opposite orientations, so \(\int_{\partial T} f = \int_{\partial T^+} f + \int_{\partial T^-} f\). For small \(\varepsilon>0\), the convex polygon \(T^+_\varepsilon \coloneqq T^+\cap\ts{\Im z\geq\varepsilon}\) lies in \(\Omega^+\), so \(\int_{\partial T^+_\varepsilon} f = 0\) by Goursat’s theorem applied to a triangulation of \(T^+_\varepsilon\). As \(\varepsilon\to 0\) the boundaries \(\partial T^+_\varepsilon\) converge to \(\partial T^+\), and since \(f\) is uniformly continuous on the compact set \(T\), \(\int_{\partial T^+} f = \lim_{\varepsilon\to 0}\int_{\partial T^+_\varepsilon} f = 0\); likewise \(\int_{\partial T^-} f=0\). Thus \(\int_{\partial T} f=0\), and Morera’s theorem gives holomorphy on \(\Omega\).
The reflection
Schwarz reflection principle (Tag T-Q3GGF)
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.
(Schwarz reflection)
Define \(F(z)\coloneqq f(z)\) for \(z\in \Omega^+\cup I\), and \(F(z)\coloneqq \overline{f(\bar z)}\) for \(z\in \Omega^-\). On \(\Omega^-\) the difference quotient is \[ \frac{F(z+h)-F(z)}{h} = \overline{\frac{f(\bar z+\bar h)-f(\bar z)}{\bar h}} ,\] using \(\overline{A}/h = \overline{A/\bar h}\). As \(h\to 0\) we have \(\bar h\to 0\), so \(F'(z)=\overline{f'(\bar z)}\) and \(F\) is holomorphic on \(\Omega^-\). On \(I\) the function \(f\) is real valued, so \(\overline{f(\bar x)}=\overline{f(x)}=f(x)\) for \(x\in I\), and \(\overline{f(\bar z)}\to f(x)\) as \(z\to x\) from \(\Omega^-\). Thus \(F\) is continuous on \(\Omega\), its restrictions to \(\Omega^\pm\) are holomorphic, and the symmetry principle Symmetry principle gives holomorphy on \(\Omega\).
Composing with Möbius transformations in the domain and the target, the same argument extends a function holomorphic on one side of a line segment or circular arc and continuous up to it, with boundary values on a line or circle, by reflection in the arc and in the target line or circle.
(The real-valued hypothesis is used)
The constant function \(f(z) = i\) on \(\Omega^+\) is continuous up to \(I\) but not real valued there. Its reflection is \(\overline{f(\bar z)} = -i\) on \(\Omega^-\), and the glued function is discontinuous on \(I\).