ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-29

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A horizontal strip is mapped biholomorphically to the disc by an exponential and a Cayley transform

Example

Let

$S : = {z \in C : 0 < Im z < π} .$

The map

$Φ (z) : = \frac{e^{z} - i}{e^{z} + i}$

is a biholomorphism from $S$ onto the unit disc $D$ .

Facts & Assumptions

Given: The strip $S$ and the map $Φ$ above.

[F1]

The exponential is holomorphic on $C$ , and in particular on $S$ (The exponential is the inverse biholomorphism from the principal strip to the slit plane).

[F2]

The upper half-plane is the domain $H = {w \in C : Im w > 0},$ and Möbius maps with real coefficients give its automorphisms (Automorphisms of the upper half-plane are real Mobius maps).

Verification

1.1F1given

If $z = x + i y \in S$ , then $0 < y < π$ , so $e^{z} = e^{x} (\cos y + i \sin y)$ has imaginary part $e^{x} \sin y > 0$ ; hence $e^{z} \in H$ .

2.1step 1.1algebra

For $w = u + i v \in H$ one has $∣ w - i ∣^{2} = u^{2} + (v - 1)^{2} < u^{2} + (v + 1)^{2} = ∣ w + i ∣^{2}$ , so $C (w) : = (w - i) / (w + i)$ satisfies $∣ C (w) ∣ < 1$ and maps $H$ into $D$ .

3.1F2step 2.1algebra∎

The inverse Möbius map is $C^{- 1} (ζ) = i (1 + ζ) / (1 - ζ)$ ; for $∣ ζ ∣ < 1$ , the identity $Im ⁣ (i \frac{1 + ζ}{1 - ζ}) = \frac{1 - ∣ ζ ∣^{2}}{∣ 1 - ζ ∣^{2}} > 0$ shows $C^{- 1} (ζ) \in H$ , so [F2] confirms that this Cayley map is exactly the standard upper-half-plane automorphism sending $H$ biholomorphically to $D$ . Together with step 1.1, this makes $Φ = C \circ e^{z} : S \to D$ biholomorphic.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 3 §4.2 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 §1 (standard reference, not scraped)