TheoremStatement: AI-adaptedProof: Literature-sourcedprecheck 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

1.1givenL1F1

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

1.2givenL1F1

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

2.1step 1.1L2algebra

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

3.1step 1.2L2algebra∎

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 $u, v$ .

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

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