Talk:PlanetPhysics/Schwarz Christoffel Transformation

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Schwarz--Christoffel transformation %%% Primary Category Code: 02.30.-f %%% Filename: SchwarzChristoffelTransformation.tex %%% Version: 1 %%% Owner: pahio %%% Author(s): pahio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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Let $$w = f(z) = c\int\frac{dz}{(z-a_1)^{k_1}(z-a_2)^{k_2}\ldots(z-a_n)^{k_n}}+C,$$ where the $a_j$'s are real numbers satisfying\, $a_1 < a_2 < \ldots < a_n$, the $k_j$'s are real numbers satisfying\, $|k_j| \leqq 1$;\, the integral expression means a complex antiderivative, $c$ and $C$ are complex constants.

The transformation\, $z \mapsto w$\, maps the real axis and the upper half-plane conformally onto the closed area bounded by a broken line.\, Some vertices of this line may be in the infinity (the corresponding angles are = 0).\, When $z$ moves on the real axis from $-\infty$ to $\infty$, $w$ moves along the broken line so that the direction turns the amount $k_j\pi$ anticlockwise every time $z$ passes a point $a_j$.\, If the broken line closes to a polygon, then\, $k_1\!+\!k_2\!+\!\ldots\!+\!k_n = 2$.

This transformation is used in solving \htmladdnormallink{two-dimensional}{http://planetphysics.us/encyclopedia/CoriolisEffect.html} potential problems.\, The \htmladdnormallink{parameters}{http://planetphysics.us/encyclopedia/Parameter.html} $a_j$ and $k_j$ are chosen such that the given polygonal \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} in the complex $w$-plane can be obtained.

A half-trivial example of the transformation is $$w = \frac{1}{2}\int\frac{dz}{(z-0)^{\frac{1}{2}}} = \sqrt{z},$$ which maps the upper half-plane onto the first quadrant of the complex plane.

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