Schwarz--Christoffel transformation

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.



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