---
tag: Y7A1TP40MB
kind: concept
name: Circle average
msc: [31A05, 28A25]
sources:
  - url: https://en.wikipedia.org/wiki/Mean_value_theorem
---

## Summary

The circle average of a function is its mean value over a circle, taken with respect to normalized arc length.
It is the basic quantity in the mean value property of harmonic functions and in the sub-mean value property of subharmonic functions.
Many results of planar potential theory, such as Jensen's formula, are identities between circle averages.

## Explanations

### L1 human

Walk once around a circle at constant speed and record the value of a function along the way.
The circle average is the average of all the recorded values.
For $f(z) = \abs{z}^2$ and the circle of radius $r$ around $0$, every recorded value is $r^2$, so the average is $r^2$.
For $f(z) = z$ and the same circle, the values point in all directions equally and cancel, so the average is $0$, the value at the centre.