---
tag: 4X0A0CQ0B0
kind: definition
name: Harmonic at a point
msc: [31B05]
defines: 60TNVD82Q5
uses: [0E0B6RA8K0, 0DEB0JMN0W, 02BH3MCBQB]
sources:
  - url: https://en.wikipedia.org/wiki/Harmonic_function
---

## Definition

Let $U \subseteq \R^n$ be an [[open set|02BH3MCBQB]], let $f \colon U \to \R$ and let $a \in U$.
The function $f$ is *harmonic at $a$* if there is an open set $V$ with $a \in V \subseteq U$ with two properties.

1. The function $f$ is twice continuously differentiable on $V$.
2. Its [[Laplacian|0E0B6RA8K0]] satisfies $\Delta f = 0$ at every point of $V$.

## Remarks

Example: $f(x, y) = \log(x^2 + y^2)$ on $U = \R^2 \setminus \set{0}$ is harmonic at $a = (1, 0)$.
Indeed, on $V = U$ one computes $\partial_x^2 f + \partial_y^2 f = \frac{2(y^2 - x^2)}{(x^2 + y^2)^2} + \frac{2(x^2 - y^2)}{(x^2 + y^2)^2} = 0$.
Non-example: $f(x, y) = x^3$ on $\R^2$ is not harmonic at $(0, 0)$.
Its Laplacian $6x$ vanishes at $(0, 0)$, but it is nonzero at points $(t, 0)$ with $t \ne 0$ arbitrarily close, so no neighbourhood $V$ works.
A function is [[harmonic|0DEB0JMN0W]] on $U$ if and only if it is harmonic at every point of $U$.
In Mathlib the condition reads: $f$ is $C^2$ at $a$ and $\Delta f = 0$ on a neighbourhood of $a$.
This is the same condition, since a function that is $C^2$ at a point is $C^2$ on a neighbourhood of it.