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| {{about|the reflection principle in complex analysis|reflection principles of set theory|Reflection principle}}
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| In [[mathematics]], the '''Schwarz reflection principle''' is a way to extend the domain of definition of an [[analytic function]] of a [[complex variable]] ''F'', which is defined on the [[upper half-plane]] and has well-defined and [[real number]] boundary values on the [[real axis]]. In that case, the putative extension of ''F'' to the rest of the [[complex plane]] is
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| :<math>\overline{F(\bar{z})}</math>
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| or
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| :<math>F(\bar{z})=\overline{F(z)}.</math>
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| That is, we make the definition that agrees along the real axis.
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| The result proved by [[Hermann Schwarz|H. A. Schwarz]] is as follows. Suppose that ''F'' is a [[continuous function]] on the closed upper half plane <math>\left\{ z \in \mathbb{C}\ |\ \mathrm{Im}(z) \geq 0 \right\} </math>, [[holomorphic]] on the upper half plane <math>\left\{ z \in \mathbb{C}\ |\ \mathrm{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.
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| 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 tweaked. 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.
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| The principle also adapts to apply to [[harmonic function]]s.
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| ==See also==
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| *[[Kelvin transform]]
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| *[[Method of image charges]]
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| ==External links==
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| * {{springer|title=Riemann-Schwarz principle|id=p/r081990}}
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| *{{mathworld|SchwarzReflectionPrinciple}}
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| [[Category:Complex analysis]]
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| [[Category:Harmonic functions]]
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| [[Category:Theorems in complex analysis]]
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| [[Category:Mathematical principles]]
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