TheoremStatement: Literature-sourcedProof: 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.

Plane harmonic functions are smooth and real analytic

Statement

Every plane harmonic function is of class $C^{\infty}$ and is real analytic in the two real coordinates.

Facts & Assumptions

Given: A harmonic function $u$ on an open subset $Ω \subseteq C$ .

[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]

Holomorphic functions are smooth and real analytic in their two real coordinates (Holomorphic functions are real analytic and smooth in their two real coordinates).

Proof

technique · direct

1.1

Fix $a \in Ω$ . By [L1], some disc $D (a, r) \subseteq Ω$ and some holomorphic $F = U + i V$ on that disc satisfy $u = U$ there.

L1choose

2.1

By [L2], the coordinate map $(U, V)$ is smooth and real analytic on $D (a, r)$ , so its first coordinate $U = u$ is smooth and real analytic there.

step 1.1L2

3.1

Since $a$ was arbitrary, $u$ is smooth and real analytic on all of $Ω$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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)