Holomorphic functions on opposite sides of the real axis glue holomorphically when their continuous boundary values agree.Suggest a slogan
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 \(\Omega\).