1. ## Schwarz reflection principle

I am reading a proof that any conformal map from upper half plane H to polygon P is a linear function of Schwarz-Christoffel integral.

In the proof the Schwarz reflection principle is applied to a situation where function is not real valued on the real line segment, but it maps this segment to a straight line in C.
Why can we apply a reflection principle here, which assumes that function must be real valued?

2. Originally Posted by Different
I am reading a proof that any conformal map from upper half plane H to polygon P is a linear function of Schwarz-Christoffel integral.
I do not know how helpful my post would be but there is a generalization of the Schwarz reflection principle. First of all if $f$, the conformal map, takes the real line to a straight line in $\mathbb{C}$ then $z\mapsto e^{-i\theta} f(z)$ will map the real axis to the real axis where $\theta$ is the angle the line makes with the positive x-axis. Maybe this realization is all you need in whatever problem you are working on. Second of all there is a generalization. Let $\gamma: I \to \mathbb{C}$ be a curve where $I$ is an interval. We say that $\gamma$ is regular iff $\gamma$ is a restriction of some one-to-one analytic function $g$ on an open set $S$ containing $I$ with $g'\not = 0$. If the set $S$ happens to be symmetric i.e. $S = \overline{S}$ (complex conjugation) then we can define "reflection across $\gamma$". Let $w \in f(S)$, define $w^* = \gamma( \overline{\gamma^{-1}(w)})$. We see that points on $\gamma$ are invariant under this reflection and that $(w^*)^* = w$, exactly how we expect a reflection to behave. In general what happens is that if $f$ is analytic function on a region $\Omega$ that contains a regular curve $\gamma_1$ and that $f(\gamma_1)$ is part of another regular curve $\gamma_2$, define $z^*$ as reflection of $z$ across $\gamma_1$, and define $w^\#$ as reflection of $w$ across $\gamma_2$, then we can extend $f$ to $\Omega^*$ by defining $(f(z^*))^\#$ for all $z\in \Omega^*$.