# unit disk upper half plane conformal equivalence theorem

###### Theorem 1.

There is a conformal map from $\mathrm{\Delta}$, the unit disk^{}, to $U\mathit{}H\mathit{}P$, the upper half plane.

###### Proof.

Define $f:\widehat{\u2102}\to \widehat{\u2102}$ (where $\widehat{\u2102}$ denotes the Riemann Sphere) to be $f(z)={\displaystyle \frac{z-i}{z+i}}$. Notice that ${f}^{-1}(w)=i{\displaystyle \frac{1+w}{1-w}}$ and that $f$ (and therefore ${f}^{-1}$) is a Mobius transformation^{}.

Notice that $f(0)=-1$, $f(1)={\displaystyle \frac{1-i}{1+i}}=-i$ and $f(-1)={\displaystyle \frac{-1-i}{-1+i}}=i$. By the Mobius Circle Transformation Theorem, $f$ takes the real axis^{} to the unit circle^{}. Since $f(i)=0$, $f$ maps $UHP$ to $\mathrm{\Delta}$ and ${f}^{-1}:\mathrm{\Delta}\to UHP$. ∎

Title | unit disk upper half plane conformal equivalence theorem |
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Canonical name | UnitDiskUpperHalfPlaneConformalEquivalenceTheorem |

Date of creation | 2013-03-22 13:37:52 |

Last modified on | 2013-03-22 13:37:52 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 12 |

Author | CWoo (3771) |

Entry type | Theorem |

Classification | msc 30C20 |

Related topic | UnitDisk |

Related topic | UpperHalfPlane |

Related topic | MobiusTransformation |

Related topic | MobiusCircleTransformationTheorem |

Related topic | ConvertingBetweenThePoincareDiscModelAndTheUpperHalfPlaneModel |