TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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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.
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A plane harmonic function that vanishes on a nonempty open set vanishes everywhere on the domain

Statement

Let $Ω$ be a complex domain and let $u : Ω \to R$ be harmonic. If $u = 0$ on some nonempty open subset of $Ω$ , then $u = 0$ on all of $Ω$ .

Facts & Assumptions

Given: A complex domain $Ω$ , a harmonic function $u$ on $Ω$ , and a nonempty open subset $U \subseteq Ω$ on which $u = 0$ .

[L1]

Near every point of $Ω$ , the function $u$ is the real part of a holomorphic function (Every plane harmonic function is locally the real part of a holomorphic function).

[L2]

If the real part of a holomorphic function has an interior local maximum, then the function is constant (Maximum principle for the real part of a holomorphic function).

[L3]

A complex domain is connected (A complex domain is a nonempty connected open subset of $C$ ).

Proof

technique · direct

1.1

Let $S : = {z \in Ω : u = 0 on some neighbourhood of z}$ . Then $U \subseteq S$ , so $S$ is nonempty, and $S$ is open by definition.

given

2.1

Let $a \in \overline{S}$ . By [L1], choose a disc $D \subseteq Ω$ around $a$ and a holomorphic function $F$ on $D$ with $Re F = u$ there. Since $S \cap D$ contains a nonempty open set on which $Re F = 0$ , both $Re F$ and $Re (- F)$ have interior local maxima $0$ on $D$ ; [L2] therefore makes both $F$ and $- F$ constant, so $u = Re F$ vanishes on all of $D$ . Hence $a \in S$ .

step 1.1L1L2

3.1

Thus $S$ is closed in $Ω$ , and [L3] makes the nonempty clopen set $S$ equal to all of $Ω$ . Therefore $u = 0$ everywhere on $Ω$ .

step 1.1step 2.1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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)