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.

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?
I do not know how helpful my post would be but there is a generalization of the Schwarz reflection principle. First of all if $\displaystyle f$, the conformal map, takes the real line to a straight line in $\displaystyle \mathbb{C}$ then $\displaystyle z\mapsto e^{-i\theta} f(z)$ will map the real axis to the real axis where $\displaystyle \theta$ 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 $\displaystyle \gamma: I \to \mathbb{C}$ be a curve where $\displaystyle I$ is an interval. We say that $\displaystyle \gamma$ is regular iff $\displaystyle \gamma$ is a restriction of some one-to-one analytic function $\displaystyle g$ on an open set $\displaystyle S$ containing $\displaystyle I$ with $\displaystyle g'\not = 0$. If the set $\displaystyle S$ happens to be symmetric i.e. $\displaystyle S = \overline{S}$ (complex conjugation) then we can define "reflection across $\displaystyle \gamma$". Let $\displaystyle w \in f(S)$, define $\displaystyle w^* = \gamma( \overline{\gamma^{-1}(w)})$. We see that points on $\displaystyle \gamma$ are invariant under this reflection and that $\displaystyle (w^*)^* = w$, exactly how we expect a reflection to behave. In general what happens is that if $\displaystyle f$ is analytic function on a region $\displaystyle \Omega$ that contains a regular curve $\displaystyle \gamma_1$ and that $\displaystyle f(\gamma_1)$ is part of another regular curve $\displaystyle \gamma_2$, define $\displaystyle z^*$ as reflection of $\displaystyle z$ across $\displaystyle \gamma_1$, and define $\displaystyle w^\#$ as reflection of $\displaystyle w$ across $\displaystyle \gamma_2$, then we can extend $\displaystyle f$ to $\displaystyle \Omega^*$ by defining $\displaystyle (f(z^*))^\#$ for all $\displaystyle z\in \Omega^*$.

3. Thanks a lot.
The first argument would be enough here. Didn't know about the generalised reflection principle.

Thanks!

### schwarz reflection principle conformal

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