DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Stolz approach regions at the boundary point 1 of the unit disc

Definition

For $C \geq 1$ , the Stolz approach region at $1$ is $S_{C} : = {z \in C : ∣ z ∣ < 1 and ∣ 1 - z ∣ \leq C (1 - ∣ z ∣)} .$ A net or sequence approaches $1$ within a Stolz region if it converges to $1$ and all sufficiently late points lie in one fixed $S_{C}$ . Convergence to $1$ is part of the definition and is not implied by the membership condition: the constant sequence $z_{n} = 0$ satisfies $∣ 0 ∣ < 1$ and $∣ 1 - 0 ∣ = 1 \leq 1 \cdot (1 - ∣ 0 ∣)$ , so it lies in $S_{1}$ at every index while converging to $0$ . The definition uses the complex modulus of Real and imaginary parts, complex conjugation, and modulus. Radial approach through $0 \leq r < 1$ lies in $S_{1}$ .

Depends on

Real and imaginary parts, complex conjugation, and modulus

Used by

Abel's limit theorem: a convergent complex series is recovered by its power series along every Stolz approach to 1 Theorem

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

L. Ahlfors, Complex Analysis, 3rd ed., Ch. 2, Abel's theorem (standard reference, not scraped)