Schwarz-Christoffel transformation



where the aj’s are real numbers satisfying  a1<a2<<an, the kj’s are real numbers satisfying  |kj|1;  the integral expression means a complex antiderivative, c and C are complex constants.

The transformation  zw  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 - to , w moves along the broken line so that the direction turns the amount kjπ anticlockwise every z passes a point aj.  If the broken line closes to a polygon, then  k1+k2++kn=2.

This transformation is used in solving two-dimensional potential problems.  The parameters aj and kj are chosen such that the given polygonal domain in the complex w-plane can be obtained.

A half-trivial example of the transformation is


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