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

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Harmonic conjugates

Definition

Let $Ω \subseteq C$ be open, let $u : Ω \to R$ be harmonic (Plane harmonic functions), and let $v : Ω \to R$ . Then $v$ is a harmonic conjugate of $u$ on $Ω$ when the complex-valued function

$u + i v : Ω \to C$

is holomorphic on $Ω$ .

Remarks

The definition is asymmetric on purpose: it singles out $v$ as a conjugate of $u$ , even though later the same holomorphic function shows that $u$ is also a harmonic conjugate of $- v$ .

Depends on

Plane harmonic functions

Used by

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Sources

Jeremy Orloff, MIT 18.04 Topic 5: Introduction to Harmonic Functions (standard reference, not scraped)