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# Schwarz-Christoffel transformation

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 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

$w=\frac{1}{2}\int\frac{dz}{(z-0)^{{\frac{1}{2}}}}=\sqrt{z},$ |

which maps the upper half-plane onto the first quadrant of the complex plane.

## Mathematics Subject Classification

31A99*no label found*30C20

*no label found*

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## Corrections

anticlockwise? by CWoo ✘

use \cdots by Mathprof ✘

use \cdots by Mathprof ✘

limits on integral by Mathprof ✘