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 is analytic on $G_{\mathrm{+}}$ and further that $f\mathit{}\mathrm{(}x\mathrm{)}$ is real for $x\mathrm{\in }{G}_{\mathrm{0}}$ (that is for 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 $\mathrm{\partial }\mathit{}\mathrm{\Omega }$. Suppose that $f\mathrm{:}G\mathrm{\cup }\gamma \mathrm{\to }\mathrm{C}$ is a continuous function that is 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 $G\mathrm{\cup }\kappa$.

Title	Schwarz reflection principle
Canonical name	SchwarzReflectionPrinciple
Date of creation	2013-03-22 14:17:58
Last modified on	2013-03-22 14:17:58
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	7
Author	jirka (4157)
Entry type	Theorem
Classification	msc 30C35
Synonym	Schwarz reflection theorem
Synonym	reflection principle
Defines	symmetric region