TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge 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 exist on homologically simply connected plane domains

Statement

Let $Ω$ be a homologically simply connected complex domain and let $u : Ω \to R$ be harmonic. Then $u$ has a harmonic conjugate on $Ω$ (Harmonic conjugates).

Equivalently, there is a holomorphic function $F : Ω \to C$ with $Re F = u$ on $Ω$ .

Facts & Assumptions

Given: A homologically simply connected complex domain $Ω$ and a harmonic function $u : Ω \to R$ .

[L1]

If $u$ is harmonic, then $g : = u_{x} - i u_{y}$ has continuous first partials and satisfies the Cauchy-Riemann equations, because $u_{x x} = - u_{y y}$ and $u_{x y} = u_{y x}$ (Clairaut--Schwarz theorem for continuous second partial derivatives, Continuous first partial derivatives and the Cauchy–Riemann equations imply complex differentiability pointwise, and holomorphy when they hold throughout an open set).

[L2]

Every holomorphic function on a homologically simply connected complex domain has a primitive (Every holomorphic function on a homologically simply connected domain has a primitive).

[L3]

A real-valued holomorphic function on a domain is constant (A real-valued holomorphic function on a domain is constant).

[L4]

A complex-valued function with continuous first partials satisfying the Cauchy-Riemann equations is holomorphic (Continuous first partial derivatives and the Cauchy–Riemann equations imply complex differentiability pointwise, and holomorphy when they hold throughout an open set).

Proof

technique · direct

1.1

Define $g : = u_{x} - i u_{y}$ on $Ω$ . Since $u$ is harmonic, [L1] makes $g$ holomorphic on $Ω$ .

givenL1

2.1

By [L2], the holomorphic function $g$ has a primitive $G$ on $Ω$ with $G^{'} = g$ .

step 1.1L2

3.1

Write $G = U + i V$ . From $G^{'} = g = u_{x} - i u_{y}$ one gets $U_{x} = u_{x}$ and $U_{y} = u_{y}$ , so $H : = u - U$ has continuous first partials with $H_{x} = H_{y} = 0$ on $Ω$ . Hence the Cauchy-Riemann equations hold for $H$ , [L4] makes $H$ holomorphic, and [L3] makes it constant.

step 2.1L3L4algebra

4.1

If $H \equiv c$ on $Ω$ , then $F : = G + c$ is holomorphic on $Ω$ and $Re F = U + c = u$ . Writing $F = u + i v$ defines a harmonic conjugate $v$ of $u$ on $Ω$ .

step 3.1algebra∎

Depends on

Used by

Dependency tree · two levels

20 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

Jeremy Orloff, MIT 18.04 Topic 5: Introduction to Harmonic Functions (standard reference, not scraped)
Sigurdur Helgason, MIT 18.112 Lecture 16: Harmonic Functions (standard reference, not scraped)