TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-26

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Every plane harmonic function is locally the real part of a holomorphic function

Statement

Let $Ω \subseteq C$ be open, let $u : Ω \to R$ be harmonic (Plane harmonic functions), and let $a \in Ω$ . Then some radius $r > 0$ and some holomorphic function $F$ on the disc $D (a, r)$ satisfy

$u (z) = Re F (z) (z \in D (a, r)) .$

Facts & Assumptions

Given: An open set $Ω$ , a harmonic function $u$ on $Ω$ , and a point $a \in Ω$ .

[L1]

If a $C^{2}$ real function is harmonic, then $u_{x} - i u_{y}$ has continuous first partials and satisfies the Cauchy-Riemann equations, because $u_{x x} = - u_{y y}$ and $u_{x y} = u_{y x}$ (Clairaut--Schwarz theorem for continuous second partial derivatives, Continuous first partial derivatives and the Cauchy–Riemann equations imply complex differentiability pointwise, and holomorphy when they hold throughout an open set).

[L2]

Every holomorphic function on a homologically simply connected complex domain has a primitive, and every star-shaped plane domain is homologically simply connected (Every holomorphic function on a homologically simply connected domain has a primitive, Star-shaped plane domains are homologically simply connected).

[L3]

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

[L4]

A complex-valued function with continuous first partials satisfying the Cauchy-Riemann equations is holomorphic (Continuous first partial derivatives and the Cauchy–Riemann equations imply complex differentiability pointwise, and holomorphy when they hold throughout an open set).

Proof

technique · direct

1.1givenL1choose

Choose $r > 0$ with $D (a, r) \subseteq Ω$ , and define $g : = u_{x} - i u_{y}$ on $D (a, r)$ . Since $u$ is harmonic and $C^{2}$ , [L1] makes $g$ holomorphic on $D (a, r)$ .

2.1step 1.1L2

The disc $D (a, r)$ is star-shaped, hence homologically simply connected by [L2], so $g$ has a primitive $G$ there with $G^{'} = g$ .

3.1step 2.1L3L4algebra

Write $G = U + i V$ . Because $G^{'} = g = u_{x} - i u_{y}$ , one has $U_{x} = u_{x}$ and $U_{y} = u_{y}$ , so the real-valued function $H : = u - U$ has continuous first partials with $H_{x} = H_{y} = 0$ on $D (a, r)$ . Hence the Cauchy-Riemann equations hold for $H$ , [L4] makes $H$ holomorphic there, and [L3] makes it constant.

4.1step 3.1algebra∎

If $H \equiv c$ on $D (a, r)$ , then $F : = G + c$ is holomorphic there and $Re F = U + c = u$ .

Depends on

Used by

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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)