TheoremStatement: AI-adaptedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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

Statement

Let $f = u + i v$ be holomorphic on an open set $U \subseteq C$ , and assume $u, v \in C^{2} (U)$ . Then

$u_{x x} + u_{y y} = 0, v_{x x} + v_{y y} = 0.$

A $C^{2}$ real function satisfying this equation is called harmonic. Thus $u$ and $v$ are harmonic, and $v$ is a harmonic conjugate of $u$ in the sense that $u + i v$ is holomorphic. No automatic $C^{2}$ regularity or global existence of harmonic conjugates is asserted.

Facts & Assumptions

Given: A holomorphic $f = u + i v$ on $U$ with $u, v \in C^{2} (U)$ .

[L1]

Holomorphy gives $u_{x} = v_{y}$ and $u_{y} = - v_{x}$ (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations).

[F1]

A $C^{2}$ function has continuous iterated partial derivatives through order two ( $C^{k}$ maps and multi-index derivative notation in Euclidean space).

[L2]

If a function is $C^{2}$ on an open subset of $R^{m}$ , then its mixed second partial derivatives agree (Clairaut--Schwarz theorem for continuous second partial derivatives).

Proof

technique · direct

1.1

Differentiating $u_{x} = v_{y}$ in $x$ and $u_{y} = - v_{x}$ in $y$ gives $u_{x x} = v_{y x}$ and $u_{y y} = - v_{x y}$ .

givenL1F1

1.2

Differentiating $u_{x} = v_{y}$ in $y$ and $u_{y} = - v_{x}$ in $x$ gives $u_{x y} = v_{y y}$ and $u_{y x} = - v_{x x}$ .

givenL1F1

2.1

By [L2], $v_{y x} = v_{x y}$ , so step 1.1 gives $u_{x x} + u_{y y} = 0$ .

step 1.1L2algebra

3.1

By [L2], $u_{x y} = u_{y x}$ , so step 1.2 gives $v_{x x} + v_{y y} = 0$ . The terminology in the Statement now applies to the given pair $u, v$ .

step 1.2L2algebra∎

Depends on

Used by

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Sources

J. Lebl, Guide to Cultivating Complex Analysis, Exercises 2.1.6–2.1.7 (standard reference, not scraped)
R. Howell and J. Mathews, Complex Analysis, Theorem 3.3.1 (standard reference, not scraped)

The C2C2 real and imaginary parts of a holomorphic function satisfy Laplace's equation and form a harmonic-conjugate pair