CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Two harmonic conjugates differ by a real constant

Statement

Let $Ω$ be a complex domain, let $u : Ω \to R$ be harmonic, and let $v_{1}, v_{2}$ be harmonic conjugates of $u$ on $Ω$ . Then $v_{1} - v_{2}$ is a real constant on $Ω$ .

Facts & Assumptions

Given: Harmonic conjugates $v_{1}, v_{2}$ of the same harmonic function $u$ on a domain $Ω$ .

[L1]

By definition, $u + i v_{1}$ and $u + i v_{2}$ are holomorphic on $Ω$ (Harmonic conjugates).

[L2]

Sums, differences, and scalar multiples of holomorphic functions are holomorphic (Linearity, product, reciprocal, and quotient rules for complex derivatives).

[L3]

A real-valued holomorphic function on a domain is constant (A real-valued holomorphic function on a domain is constant).

Proof

technique · direct

1.1L1L2algebra

By [L1], the functions $F_{1} : = u + i v_{1}$ and $F_{2} : = u + i v_{2}$ are holomorphic, so [L2] makes $- i (F_{1} - F_{2}) = v_{1} - v_{2}$ holomorphic on $Ω$ .

2.1step 1.1L3∎

The function $v_{1} - v_{2}$ is real-valued, so [L3] makes it constant on $Ω$ .

Depends on

Used by

Harmonic conjugates and integral logarithmic-pole monodromy on surfaces Lemma

Dependency tree · two levels

9 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)