Lesson plan
Schwarz function
The Schwarz function of a curve in the complex plane is an analytic function which maps the points of the curve to their complex conjugates. It can be used to generalize the Schwarz reflection principle to reflection across arbitrary analytic curves, not just across the real axis. The Schwarz function exists for analytic curves. More precisely, for every non-singular, analytic Jordan arc \Gamma in the complex plane, there is an open neighborhood \Omega of \Gamma and a unique analytic function S on \Omega such that S(z) = \overline{z} for every z \in \Gamma . The "Schwarz function" was named by Philip J. Davis and Henry O. Pollak (1958) in honor of Hermann Schwarz, who introduced the Schwarz reflection principle for analytic curves in 1870. However, the Schwarz function does not explicitly appear in Schwarz's works.
Article summary
The Schwarz function of a curve in the complex plane is an analytic function which maps the points of the curve to their complex conjugates. It can be used to generalize the Schwarz reflection principle to reflection across arbitrary analytic curves, not just across the real axis. The Schwarz function exists for analytic curves. More precisely, for every non-singular, analytic Jordan arc \Gamma in the complex plane, there is an open neighborhood \Omega of \Gamma and a unique analytic function S on \Omega such that S(z) = \overline{z} for every z \in \Gamma . The "Schwarz function" was named by Philip J. Davis and Henry O. Pollak (1958) in honor of Hermann Schwarz, who introduced the Schwarz reflection principle for analytic curves in 1870. However, the Schwarz function does not explicitly appear in Schwarz's works.
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Discussion questions
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