View source for Schwarz reflection principle

{{short description|Mathematics principle in complex analysis}}
{{about|the reflection principle in complex analysis|reflection principles of set theory|Reflection principle}}

In [[mathematics]], the '''Schwarz reflection principle''' is a way to extend the domain of definition of a [[complex analytic function]], i.e., it is a form of [[analytic continuation]]. It states that if an analytic function is defined on the [[upper half-plane]], and has well-defined (non-singular) real values on the [[real axis]], then it can be extended to the conjugate function on the lower half-plane. In notation, if <math>F(z)</math> is a function that satisfies the above requirements, then its extension to the rest of the [[complex plane]] is given by the formula,
<math display="block">F(\bar{z}) = \overline{F(z)}.</math>

That is, we make the definition that agrees along the real axis.
[[File:Hermann Amand Schwarz (1843-1921) by Louis Zipfel.jpg|thumb|[[Hermann Schwarz]]]]

The result proved by [[Hermann Schwarz]] is as follows. Suppose that ''F'' is a [[continuous function]] on the closed upper half plane <math>\left\{ z \in \Complex \mid \operatorname{Im}(z) \geq 0 \right\} </math>, [[Holomorphic function|holomorphic]] on the upper half plane <math>\left\{ z \in \Complex \mid \operatorname{Im}(z) > 0 \right\} </math>, which takes real values on the real axis. Then the extension formula given above is an [[analytic continuation]] to the whole complex plane.<ref>{{Cite book |last=Cartan |first=Henri |title=Elementary theory of analytic functions of one or several variables |page=75}}</ref>

In practice it would be better to have a theorem that allows ''F'' certain singularities, for example ''F'' a [[meromorphic function]]. To understand such extensions, one needs a proof method that can be weakened. In fact [[Morera's theorem]] is well adapted to proving such statements. [[Contour integral]]s involving the extension of ''F'' clearly split into two, using part of the real axis. So, given that the principle is rather easy to prove in the special case from Morera's theorem, understanding the proof is enough to generate other results.

The principle also adapts to apply to [[harmonic function]]s.

==See also==
* [[Kelvin transform]]
* [[Method of image charges]]
* [[Schwarz function]]

==References==
{{Reflist}}

==External links==
* {{springer|title=Riemann-Schwarz principle|id=p/r081990}}
* {{mathworld|SchwarzReflectionPrinciple|author=Todd Rowland}}

[[Category:Harmonic functions]]
[[Category:Theorems in complex analysis]]
[[Category:Mathematical principles]]