Let \(S \coloneqq \{z \in \mathbb{D} \mid \operatorname{Im}(z) \ge 0\}\) be the closed upper half of the unit disk \(\mathbb{D}\). Suppose \(f: S \to \mathbb{C}\) is continuous on \(S\), real-valued on \(S \cap \mathbb{R} = (-1, 1)\), and holomorphic on the interior \(S^\circ = \{z \in \mathbb{D} \mid \operatorname{Im}(z) > 0\}\).
Prove that \(f\) extends to a holomorphic function on the entire unit disk \(\mathbb{D}\) (the Schwarz Reflection Principle).
Solution
Define \[ F(z)= \begin{cases} f(z),&\Im z\ge0,\\ \overline{f(\overline z)},&\Im z<0. \end{cases} \] Because \(f(x)\in\mathbb R\) for \(-1<x<1\), the two formulas agree continuously on the real diameter. Thus \(F\) is continuous on \(\mathbb D\).
On the lower half-disk, \(z\mapsto\overline{f(\overline z)}\) is holomorphic. Indeed, if \(w=\overline z\), then its derivative is \[ \overline{f'(w)}. \] Hence \(F\) is holomorphic off the real diameter.
To cross the diameter, apply Morera’s theorem. For any triangle \(T\Subset\mathbb D\), split \(T\) along the real axis into finitely many polygons lying in the upper and lower half-disks. Approximate any boundary segment on the real axis by parallel segments at heights \(\pm\varepsilon\). Cauchy’s theorem applies away from the axis, and continuity of \(F\) lets \(\varepsilon\to0\). Thus \[ \oint_{\partial T}F(z)\,dz=0. \] Morera’s theorem implies that \(F\) is holomorphic on all of \(\mathbb D\). By construction \(F=f\) on the upper half-disk.