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 (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
Canonical name	SchwarzChristoffelTransformation
Date of creation	2013-03-22 14:41:02
Last modified on	2013-03-22 14:41:02
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	15
Author	pahio (2872)
Entry type	Result
Classification	msc 31A99
Classification	msc 30C20
Synonym	Schwarz’ transformation
Related topic	ConformalMapping