# simple example of composed conformal mapping

Let’s consider the mapping

$$f:\u2102\to \u2102\mathit{\hspace{1em}}\mathrm{with}\mathit{\hspace{1em}}f(z)=az+b,$$ |

where $a$ and $b$ are complex and $a\ne 0$.

Because ${f}^{\prime}(z)\equiv a\ne 0$, the mapping is conformal in the whole $z$-plane. Denote $a:=\varrho {e}^{i\alpha}$ (where $\varrho ,\alpha \in \mathbb{R}$) and

${z}_{1}:=\varrho z,$ | (1) |

${z}_{2}:={e}^{i\alpha}{z}_{1},$ | (2) |

$w:={z}_{2}+b.$ | (3) |

Then the mapping $z\mapsto {z}_{1}$ means a dilation in the complex plane, the mapping ${z}_{1}\mapsto {z}_{2}$ a rotation^{} by the angle $\alpha $ and the mapping ${z}_{2}\mapsto w$ a translation determined by the vector from the origin to the point $b$. Thus $f$ is composed of these three consecutive mappings which all are conformal.

Title | simple example of composed conformal mapping |
---|---|

Canonical name | SimpleExampleOfComposedConformalMapping |

Date of creation | 2013-03-22 16:47:25 |

Last modified on | 2013-03-22 16:47:25 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Example |

Classification | msc 30E20 |

Classification | msc 53A30 |