Schwarz reflection principle

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This article is about the reflection principle in complex analysis. For reflection principles of set theory, see Reflection principle.

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

\overline{F(\bar{z})}

or

F(\bar{z})=\overline{F(z)}.

That is, we make the definition that agrees along the real axis.

The result proved by H. A. Schwarz is as follows. Suppose that F is a continuous function on the closed upper half plane \left\{ z \in \mathbb{C}\ |\ \mathrm{Im}(z) \geq 0 \right\} , holomorphic on the upper half plane \left\{ z \in \mathbb{C}\ |\ \mathrm{Im}(z) > 0 \right\} , which takes real values on the real axis. Then the extension formula given above is an analytic continuation to the whole complex plane.

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 weaked. In fact Morera's theorem is well adapted to proving such statements. Contour integrals 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 functions.

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