Gauss's mean-value theorem
/ GOWS /
Gauss's mean-value theorem is the mean-value property named and credited to Gauss, usually stated for harmonic functions: the value of a harmonic function at the center of a disk equals the average of its values on the boundary circle. It is the same averaging law you meet for holomorphic functions, now phrased for the real-valued solutions of Laplace's equation that govern steady heat, electrostatic potential, and ideal fluid flow.
Precisely: if u is harmonic on and inside the disk of radius r about z_0 (meaning u_xx + u_yy = 0 there), then u(z_0) = (1 / (2 pi)) times the integral from 0 to 2 pi of u(z_0 + r e^(i theta)) dtheta. You can get this directly from Cauchy's formula by noting that a harmonic u on a disk is the real part of some holomorphic f; the mean-value property of f passes to its real part. Physically it says a steady temperature at a point is the average of the temperatures on any circle around it — there are no spontaneous hot spots in equilibrium.
Gauss's version is the gateway to the maximum principle for harmonic functions and to the Poisson integral formula, which upgrades the plain circular average to a weighted average that also handles off-center points. It is worth remembering that the equality is exact, not approximate: equilibrium fields are perfectly self-averaging, which is precisely why the Dirichlet problem (prescribe boundary values, find the interior field) has a unique, smoothing solution.
The harmonic function u(x, y) = x has, on the circle of radius r about the origin, average value (1 / (2 pi)) times the integral of r cos(theta) dtheta from 0 to 2 pi, which is 0 — equal to u(0, 0) = 0.
A harmonic function's center value is the average of its boundary values.
The averaging is over a circle (the boundary), giving the value at the center only; the average over the whole solid disk also equals the center value, but these are two different (both true) statements, not the same one.