Schwarz reflection principle

For a region $G\subset ℂ$ define $G^{*}:=\{z:\overline{z}\in G\}$ (where $\overline{z}$ is the complex conjugate of $z$). If $G$ is a symmetric region, that is $G=G^{*}$, then we define$G_{+}:=\{z\in G:\mathrm{Im}z>0\}$, and$G_0:=\{z\in G:\mathrm{Im}z=0\}$.

Theorem.

Let $G\mathrm{\subset }\mathrm{C}$ be a region such that $G\mathrm{=}{G}^{\mathrm{*}}$ and suppose that$f\mathrm{:}{G}_{\mathrm{+}}\mathrm{\cup }{G}_{\mathrm{0}}\mathrm{\to }\mathrm{C}$ is a continuous functions that isanalytic on $G_{\mathrm{+}}$ and further that $f\mathit{}\mathrm{(}x\mathrm{)}$ is real for $x\mathrm{\in }{G}_{\mathrm{0}}$ (that isfor real $x$), then there is an analytic function $g\mathrm{:}G\mathrm{\to }\mathrm{C}$such that $g\mathit{}\mathrm{(}z\mathrm{)}\mathrm{=}f\mathit{}\mathrm{(}z\mathrm{)}$ for $z\mathrm{\in }{G}_{\mathrm{+}}\mathrm{\cup }{G}_{\mathrm{0}}$.

That is you can “reflect” an analytic function across the real axis. Note that by composing with various conformal mappings you could generalize the above to reflection across an analytic curve.So loosely stated, the theorem says that if an analytic function is defined in a region with some “nice” boundary and the function behaves “nice” on this boundary, then we can extend the function to a larger domain. Let us make this statement precise with the following generalization.

Theorem.

Let $G\mathrm{,}\mathrm{\Omega }\mathrm{\subset }\mathrm{C}$ be regions and let $\gamma$ and $\omega$be free analytic boundary arcs in $\mathrm{\partial }\mathit{}G$ and $\mathrm{\partial }\mathit{}\mathrm{\Omega }$. Supposethat $f\mathrm{:}G\mathrm{\cup }\gamma \mathrm{\to }\mathrm{C}$ is a continuous function thatis analytic on $G$, $f\mathit{}\mathrm{(}G\mathrm{)}\mathrm{\subset }\mathrm{\Omega }$ and $f\mathit{}\mathrm{(}\gamma \mathrm{)}\mathrm{\subset }\omega$, then for any compact set $\kappa \mathrm{\subset }\gamma$, $f$ has an analytic continuation to an open set containing $G\mathrm{\cup }\kappa$.