# Schwarz reflection principle

In this section, take $$\Omega$$ to be a region symmetric about the real axis, so $$z\in \Omega \iff \overline{z} \in \Omega$$. Partition this set as $$\Omega^+ \subseteq {\mathbb{H}}, I \subseteq {\mathbf{R}}, \Omega^- \subseteq \overline{{\mathbb{H}}}$$.

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 $${\mathbf{R}}$$, then the extension defined by $$F^-(z) = \overline{f(\overline{z})}$$ for $$z\in {\mathbb{H}}^-$$ is a well-defined holomorphic function on $${\mathbf{C}}$$.

Apply the symmetry principle.

$${\mathbb{H}}^+, {\mathbb{H}}^-$$ can be replaced with any region symmetric about a line segment $$L\subseteq {\mathbf{R}}$$.

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