DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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.

Orientation-preserving conformality for a real-differentiable complex map at a point

Definition

For vectors $p = (p_{1}, p_{2})$ and $q = (q_{1}, q_{2})$ in the oriented Euclidean plane, put

$ω (p, q) : = p_{1} q_{2} - p_{2} q_{1} .$

A real-linear map $L : R^{2} \to R^{2}$ is a similarity of ratio $λ > 0$ when

$⟨ L h, L k ⟩ = λ^{2} ⟨ h, k ⟩$

for all $h, k \in R^{2}$ , using the inner product of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ . It is orientation-preserving when $ω (L e_{1}, L e_{2}) > 0$ and orientation-reversing when that quantity is negative.

Let $U \subseteq C$ be open and let $f : U \to C$ be real totally differentiable at $a$ . The map $f$ is orientation-preserving conformal at $a$ when $D f (a)$ is an orientation-preserving similarity. This is a pointwise condition on the real derivative. It asserts neither local nor global injectivity of $f$ .

Depends on

Used by

Dependency tree · two levels

25 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

J. Lebl, Guide to Cultivating Complex Analysis, §2.2.4 (standard reference, not scraped)
R. Howell and J. Mathews, Complex Analysis, §9.1 (standard reference, not scraped)