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Math Help - Schwarz reflection principle

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by Different View Post
    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 cannot help you with this, sorry .

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

    Thanks!
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