DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

The circle and disc mean-value properties

Definition

Let $Ω \subseteq C$ be open and let $u : Ω \to R$ be continuous.

The circle mean-value property says that for every closed disc $\overline{D (a, r)} \subseteq Ω$ with $r > 0$ , $u (a) = \frac{1}{2 π} \int_{0}^{2 π} u (a + r e^{i t}) d t .$
The disc mean-value property says that for every closed disc $\overline{D (a, r)} \subseteq Ω$ with $r > 0$ , $u (a) = \frac{2}{r^{2}} \int_{0}^{r} (\frac{1}{2 π} \int_{0}^{2 π} u (a + s e^{i t}) d t) s d s .$

When both hold, $u$ is said to satisfy the mean-value property on $Ω$ .

Remarks

The disc formula is written as the average of the concentric circle averages, so it is a genuinely plane-local statement and does not need any separate area integration convention.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

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)