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

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Plane harmonic functions

Definition

Let $Ω \subseteq C$ be open, and write $z = x + i y$ for the real coordinates on $C = R^{2}$ . A real-valued function $u : Ω \to R$ is harmonic on $Ω$ when $u$ is of class $C^{2}$ and satisfies Laplace's equation

$u_{x x} + u_{y y} = 0$

throughout $Ω$ .

Remarks

This page is plane-specific: the Laplacian is the two-variable operator $\partial_{x}^{2} + \partial_{y}^{2}$ , and later pages generalize the theory to higher dimensions.

The function is real-valued by convention. Complex-valued harmonic maps are handled componentwise by asking both real coordinates to be harmonic.

Used by

A harmonic function can vanish on a line without being zero everywhere Counterexample
The product of two harmonic functions need not be harmonic Counterexample
Harmonic conjugates Definition
log|z| is harmonic on the punctured plane Example
A bounded harmonic function near an isolated puncture extends harmonically Theorem
Every plane harmonic function is locally the real part of a holomorphic function Theorem
Harmonic conjugates exist on homologically simply connected plane domains Theorem
Maximum and minimum principles for plane harmonic functions Theorem

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)