DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-26

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.

Harmonic conjugates

Definition

Let $Ω \subseteq C$ be open, let $u : Ω \to R$ be harmonic (Plane harmonic functions), and let $v : Ω \to R$ . Then $v$ is a harmonic conjugate of $u$ on $Ω$ when the complex-valued function

$u + i v : Ω \to C$

is holomorphic on $Ω$ .

Remarks

The definition is asymmetric on purpose: it singles out $v$ as a conjugate of $u$ , even though later the same holomorphic function shows that $u$ is also a harmonic conjugate of $- v$ .

Harmonic conjugates

Definition

Remarks

Depends on

Dependency tree · 1 level

Sources