In this section, take \(\Omega\) to be a region symmetric about the real axis, so \(z\in \Omega \iff \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu \in \Omega\). Partition this set as \(\Omega^+ \subseteq {\mathbb{H}}, I \subseteq {\mathbb{R}}, \Omega^- \subseteq \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu\).
Suppose that \(f^+\) is holomorphic on \(\Omega^+\) and \(f^-\) is holomorphic on \(\Omega^-\), and \(f\) extends continuously to \(I\) with \(f^+(x) = f^-(x)\) for \(x\in I\). Then the following piecewise-defined function is holomorphic on \(\Omega\): \begin{align*} f(z) \coloneqq \begin{cases} f^+(z) & z\in \Omega^+ \\ f^-(z) & z\in \Omega^- \\ f^+(z) = f^-(z) & z\in I. \end{cases} \end{align*}
Apply Morera?
If \(f\) is continuous and holomorphic on \({\mathbb{H}}^+\) and real-valued on \({\mathbb{R}}\), then the extension defined by \(F^-(z) = \mkern 1.5mu\overline{\mkern-1.5muf(\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)\mkern-1.5mu}\mkern 1.5mu\) for \(z\in {\mathbb{H}}^-\) is a well-defined holomorphic function on \({\mathbb{C}}\).
Apply the symmetry principle.
\({\mathbb{H}}^+, {\mathbb{H}}^-\) can be replaced with any region symmetric about a line segment \(L\subseteq {\mathbb{R}}\).