Why is the Mean Value Theorem (of holomorphic functions) called "Gauss's"?

Question

A handy special case of the Cauchy Integral Formula says that, if a complex function $f$ is analytic on and inside a circle of radius $r$ around $a$,

$$f(a) = \frac{1}{2\pi}\int_0^{2\pi} f(a +re^{it}) dt$$

Some sources, e.g. the Brown/Churchill textbook (sec. 59) or WolframAlpha, call this "Gauss's Mean Value Theorem". I understand "mean value" (although it's unfortunate there is already a different Mean Value Theorem in basic real calculus), but ask the

Question: What did Gauss have to do with this?

The textbook by Remmert, which is delightful in its historical remarks, and mentions Gauss often, does not make any connection of this specific fact to Gauss and just calls it the "mean value equality" (sec. 7.2 p. 203). The textbook by Ahlfors does not even give it a name, just mentions it as special case of the CIF in its section on maximum modulus (III.3.4).

I am aware there is a version of the theorem for harmonic functions. Maybe Gauss wrote on that?

Answer 1

The naming, apparently, derives from the corresponding theorem for harmonic functions, the mean value property. According to Netuka-Veselý's survey, Neumann originally called it der Gauss'sche Satz des arithmetischen Mittels (Gauss's theorem on the arithmetic mean) back in 1877.

The result, in the 3D case, appears in Gauss's Allgemeine Lehrsätze in Beziehung auf die im verkehrtem Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs - und Abstossungs - Kräfte (General theories relating to the forces of attraction and repulsion acting in the inverse ratio of the square of the distance) from 1840 (reprinted in Werke, Bd. 5, 197–242). In rough translation:

"THEOREM. If $V$ is the potential of a mass distributed in any way in the element of a spherical surface $ds$ of radius $R$, then, integrated over the entire spherical surface, $$\int V ds=4\pi(RM^0+RRV^0),$$ where $M^0$ is the entire mass inside the sphere and $V^0$ is the potential of the mass outside the sphere in the center of the sphere, and the masses, which may be distributed continuously on the surface of the sphere, are assigned as desired to the external or internal masses."

One gets the usual formulation when $M^0=0$ and both sides are divided by $4\pi R^2$.

Koebe proved the converse, that the mean value property implies harmonicity, in 1906. Kellogg still called it Neumann's way in Converses of Gauss’ theorem on the arithmetic mean (1934), but Walsh already renamed it into "Gauss's mean value theorem" in Maximal Convergence of Sequences of Harmonic Polynomials, p.346 (1937) and applied the name to all dimensions. Much later, some expositors decided that the result for holomorphic functions was a simple enough consequence (in 2D) to transfer the name, see e.g. Burckel, A Strong Converse to Gauss's Mean-Value Theorem (1980). But this is not universal, most call it "mean value theorem for holomorphic functions" without attaching a name.