# Schwarz-Christoffel transformation

Let

$$w=f(z)=c\int \frac{dz}{{(z-{a}_{1})}^{{k}_{1}}{(z-{a}_{2})}^{{k}_{2}}\mathrm{\dots}{(z-{a}_{n})}^{{k}_{n}}}+C,$$ |

where the ${a}_{j}$’s are real numbers satisfying $$, 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 $-\mathrm{\infty}$ to $\mathrm{\infty}$, $w$ 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 ${k}_{1}+{k}_{2}+\mathrm{\dots}+{k}_{n}=2$.

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 |