unit disk upper half plane conformal equivalence theorem

Theorem 1.

There is a conformal map from Δ, the unit diskMathworldPlanetmath, to UHP, the upper half plane.


Define f:^^ (where ^ denotes the Riemann Sphere) to be f(z)=z-iz+i. Notice that f-1(w)=i1+w1-w and that f (and therefore f-1) is a Mobius transformationMathworldPlanetmath.

Notice that f(0)=-1, f(1)=1-i1+i=-i and f(-1)=-1-i-1+i=i. By the Mobius Circle Transformation Theorem, f takes the real axisMathworldPlanetmath to the unit circleMathworldPlanetmath. Since f(i)=0, f maps UHP to Δ and f-1:ΔUHP. ∎

Title unit disk upper half plane conformal equivalence theorem
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