unit disk upper half plane conformal equivalence theorem
Theorem 1.
There is a conformal map from Δ, the unit disk, to UHP, the upper half plane.
Proof.
Define f:ˆℂ→ˆℂ (where ˆℂ 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 transformation.
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 axis to the unit circle
. Since f(i)=0, f maps UHP to Δ and f-1:Δ→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 |