# 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}, 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 (http://planetmath.org/ConformalMapping) 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 $z$ 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.

Title Schwarz-Christoffel transformation SchwarzChristoffelTransformation 2013-03-22 14:41:02 2013-03-22 14:41:02 pahio (2872) pahio (2872) 15 pahio (2872) Result msc 31A99 msc 30C20 Schwarz’ transformation ConformalMapping