harmonic function

调和函数

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A real or complex C^2 function u on an open subset of R^n is harmonic when its Laplacian, the sum of the coordinate second derivatives, vanishes throughout the domain.

欧氏空间开集上的二阶连续可微函数，其各坐标二阶偏导数之和处处为零时，称为调和函数。

Example

$u(x,y)=3x^2-3y^2$ is harmonic because $u_{xx}+u_{yy}=6-6=0$.

Usage and distinctions

A harmonic function is a solution of the homogeneous Laplace equation; a nonzero source gives a Poisson equation.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John K. Hunter — Notes on Partial Differential EquationsChapter 2 opening, printed p.19; §2.1, Theorem 2.1 · Checked 2026-10-14

Translation sources

1. 北京大学数学科学学院 — 数学分析（实验班）课程介绍第三部分，级数理论：六、调和函数和Laplace方程初步 · Checked 2026-10-14

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