RemarkRemark: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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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.
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Agreement with the earlier C^2 holomorphic-components theorem

The present definition of harmonicity is exactly the one already reached from holomorphic functions in The $C^{2}$ real and imaginary parts of a holomorphic function satisfy Laplace's equation and form a harmonic-conjugate pair: when $f = u + i v$ is holomorphic and $u, v$ are $C^{2}$ , both real components satisfy $u_{x x} + u_{y y} = 0$ and $v_{x x} + v_{y y} = 0$ . This page keeps that convention and develops the converse direction from harmonic data, rather than introducing a second notion of “harmonic component”.

Depends on

The $C^2$ real and imaginary parts of a holomorphic function satisfy Laplace's equation and form a harmonic-conjugate pair

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)