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Schwarz--Christoffel transformation (Topic)

Let

$\displaystyle w = f(z) = c\int\frac{dz}{(z-a_1)^{k_1}(z-a_2)^{k_2}\ldots(z-a_n)^{k_n}}+C,$
where the $a_j$'s are real numbers satisfying  $a_1 < a_2 < \ldots < a_n$, the $k_j$'s are real numbers satisfying  $\vert k_j\vert \leqq 1$;  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 two-dimensional potential problems.  The parameters $a_j$ and $k_j$ are chosen such that the given polygonal domain in the complex $w$-plane can be obtained.

A half-trivial example of the transformation is

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



"Schwarz--Christoffel transformation" is owned by pahio.
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Other names:  Schwarz-Christoffel transform

Cross-references: domain, parameters, two-dimensional

This is version 1 of Schwarz--Christoffel transformation, born on 2009-05-01.
Object id is 714, canonical name is SchwarzChristoffelTransformation.
Accessed 1086 times total.

Classification:
Physics Classification: 02.30.-f (Function theory, analysis)
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