Schwarz reflection principle

Remarks

Proof idea: $F$ is holomorphic on $\Omega \setminus \R$, since $z \mapsto \overline{f(\bar z)}$ satisfies the Cauchy–Riemann equations when $f$ does. $F$ is continuous on $\Omega$, because $f$ is real on $\sigma$, so the two definitions agree there. The integral of $F$ over each small triangle in $\Omega$ is $0$: for triangles meeting $\R$, approximate by triangles slightly off the axis. Morera's theorem then gives that $F$ is holomorphic. Example: $f(z) = e^{z}$ on the upper half-plane is real on $\R$, and its reflection is $\overline{e^{\bar z}} = e^{z}$, as expected. The hypothesis that $f$ is real on $\sigma$ is needed: $f(z) = i$ is continuous up to $\R$ but its reflection is $-i$ below the axis, which is not even continuous. By a Möbius transformation the same principle holds for reflection across a circle, with $\bar z$ replaced by the inverse point.

Going deeper

For a student

The reflection principle is a basic tool of analytic continuation. It is used to extend conformal maps whose boundary goes to a line or a circle, as for the Schwarz–Christoffel formula. It also shows that a conformal map between domains bounded by analytic arcs extends across those arcs. There is a version for harmonic functions that vanish on a line, which is the main step in some proofs.