Mean value property

Statement

Let $U \subseteq \mathbb{C}$ be an open set, and identify $\mathbb{C}$ with $\mathbb{R}^2$ by $x + iy \leftrightarrow (x, y)$. Let $u : U \to \mathbb{R}$ be harmonic: $u$ is twice continuously differentiable and $u_{xx} + u_{yy} = 0$ on $U$. Let $c \in \mathbb{C}$ and $r > 0$ be such that the closed disc $\{z : |z - c| \le r\}$ is contained in $U$. Then $$u(c) = \frac{1}{2\pi} \int_0^{2\pi} u(c + r e^{i\theta}) \, d\theta.$$

Remarks

The value of $u$ at the center of the circle is the average of $u$ on the circle. For holomorphic functions, the same formula holds, and it follows from Cauchy's integral formula with $a = c$. On a disc, $u$ is the real part of a holomorphic function, which gives the harmonic case. The property also holds for averages over the solid disc. A continuous function with the mean value property on all small circles is harmonic.

Explanations

L1 human

If $u$ is harmonic, its value at a point is the average of its values on each circle around that point. This is true as long as the closed disc stays in the domain. For example, $u = x^2 - y^2$ at $0$ is $0$, and on the unit circle it is $\cos 2\theta$, whose average is $0$.