Schwarz reflection

In mathematics, applying 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

or

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 ${\displaystyle \left\{z\in \mathbb {C} \ |\ \mathrm {Im} (z)\geq 0\right\}}$, holomorphic on the upper half plane ${\displaystyle \left\{F(z)\in \mathbb {C} \ |\ \mathrm {Im} (z)>0\right\}}$, which takes real values on the real axis. Then the extension formula given above is an analytic continuation to the whole complex plane.

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