In Joseph Bak and Donald Newman Complex Analysis book , there is generalization of Schwarz reflection across segment on the real line of complex function. The genarlization say that given analytic smooth simple curve $\gamma:(a,b) \mapsto \mathbb{C} $, we can define reflection across that curve.
I'm having troubles with the definition of that idea, and appreciate a reliable source that will help me with that idea.
Schwarz reflection over analytic curve
complex-analysis
Best Answer
Rough sketch of the idea: Let $g$ be some biholomorphic function on an open set that restricted to $\mathbb R$ parameterizes a part of a curve. Let $f$ be defined on an open neighborhood of a real segment, intersected with the closed upper half plane. Suppose that $f$ maps the real segment into the curve parameterized by $g$ and that $g^{-1} \circ f$ satisfies the conditions for Schwarz’ reflection principle. (Note that it maps reals to reals.) Then $g^{-1} \circ f$ extends holomorphically over the real line. Apply $g$ again to get an extension of $f$, whose image extends over (is “reflected in”) the curve.