Schwarz function

The Schwarz function, named for Hermann Schwarz, was introduced in a 1974 book by Philip J. Davis. It defines a method by which the Schwarz reflection principle may be generalized to reflection through arbitrary analytic curves in the complex plane, not just through the real axis. Given a non-singular, analytic Jordan arc $\Gamma$ in the complex plane, the Schwarz function $S$ is a uniquely determined analytic function on a neighborhood of $\Gamma$ with the property that $S(z)=z^{*}$ for every point $z\in \Gamma$.