# Schwarz reflection principle

For a region $G\subset \u2102$ 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 $.

## 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 |
---|---|

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 |