Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-28
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The Cayley transform carries the upper half-plane biholomorphically onto the unit disc

Example

The Möbius map C(z):=z−iz+i carries the upper half-plane {Im⁡z>0} biholomorphically onto D.

Facts & Assumptions

Given: The Cayley transform C(z)=(z−i)/(z+i).

[L1]

Every Möbius transformation is a sphere biholomorphism (Every Möbius transformation is a biholomorphism of the Riemann sphere).

Verification

technique · direct
1.1

The determinant condition is ad−bc=2i≠0, so [L1] makes C Möbius. For real x, one has ∣C(x)∣=1, and for z=x+iy with y>0 the identity ∣z+i∣2−∣z−i∣2=4y>0 gives ∣C(z)∣<1.

L1givenalgebra
2.1

The inverse formula is C−1(w)=i(1+w)/(1−w), and the same calculation in reverse shows that ∣w∣<1 implies Im⁡C−1(w)>0. Hence C carries the upper half-plane biholomorphically onto the unit disc.

givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources