---
tag: 60TNVD82Q5
kind: concept
name: Harmonic at a point
msc: [31B05]
sources:
  - url: https://en.wikipedia.org/wiki/Harmonic_function
---

## Summary

A function on an open subset of $\R^n$ is harmonic at a point when it is harmonic on some neighbourhood of that point.
This local notion lets one say where a function is harmonic without fixing a domain in advance.
A function is a [[harmonic function|0DEB0JMN0W]] on an open set exactly when it is harmonic at each point of the set.

## Explanations

### L1 human

Being harmonic at a point is not a condition on the value at that single point.
It asks that the function solves Laplace's equation in a small ball around the point.
For example, $\log(x^2 + y^2)$ is harmonic at every point of the plane except the origin, where it is not even defined.
The function $x^3$ in the plane has Laplacian $6x$, which vanishes on the line $x = 0$ but on no ball, so it is harmonic at no point.