Schwarz reflection principle

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

Theorem.

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

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

References

• 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
• 2 John B. Conway. . Springer-Verlag, New York, New York, 1995.
Title Schwarz reflection principle SchwarzReflectionPrinciple 2013-03-22 14:17:58 2013-03-22 14:17:58 jirka (4157) jirka (4157) 7 jirka (4157) Theorem msc 30C35 Schwarz reflection theorem reflection principle symmetric region