Talk:Harmonic function

Incorrect statement?

There also exists a seemingly weaker definition that is equivalent. Indeed a function is harmonic if and only if it is weakly harmonic.

Weakly harmonic doesn't require that f be twice continuously differentiable. I think the correct statement is "If function is harmonic then it is also weakly harmonic." The reverse does not seem to be true. That is, weakly harmonic is necessary but not a sufficient condition for a function to be harmonic. However, I could be wrong so I won't edit the page unless someone else agrees. Timhoooey (talk) 23:19, 1 November 2008 (UTC)

You are wrong, sorry. Since the Laplace equation is elliptic, its solutions are automatically smooth (and even real analytic). Tiphareth (talk) 16:25, 2 November 2008 (UTC)

Mean value theorem on a manifold?

Can anyone point out a proof of the claim that harmonic functions on Riemannian manifolds satisfy a mean-value property with respect to metric balls? —Preceding unsigned comment added by Razkupferman (talk • contribs) 12:23, 11 February 2010 (UTC)

Indeed, this claim needs a citation, since it is not mentioned in the references and it certainly is not "general folklore". I remember seeing an article where a somekind of higher order asymptotic mean value property was proved for the Laplace-Beltrami operator, which indicates that the fully analogous mean value theorem might even be false. Lapasotka (talk) 03:33, 30 April 2010 (UTC)

Cauchy's integral theorem?

Do harmonic functions also fulfill an analogon to Cauchy's integral theorem in complex analysis, i.e. that a line integral is independent of the path if there are no singularities? --Roentgenium111 (talk) 17:06, 14 July 2011 (UTC)

Line integrals usually depend on the path, but surface integrals only depend on the boundary of the domain of integration. See Green's identities. 83.14.20.14 (talk) 10:05, 23 April 2012 (UTC)

Dubious: Singularities determine harmonic functions

The article as written gives a vague impression that all harmonic functions have singularities. This statement is not tue. While this seems to be the most common case in applications, mathematically, the most common examples are not singular, like x^2-y^2 or 2xy. Adding such a function to a function with singularities leaves the singularities essentially unchanged while staying harmonic. I know that it says that real and imaginary portions of complex functions are harmonic, but more examples would be nice. — Preceding unsigned comment added by 128.187.97.18 (talk) 16:38, 28 May 2012 (UTC)

Ambiguity

I would like to point out that in some contexts "harmonic function" means the eigen-functions of the laplacian operator: ∇²f = λf. For example, when we say that sin(x) and cos(x) are harmonic functions we refer to this definition.

In this article, "harmonic" is defined as the solutions of the laplace equation ∇²f = 0. This is a special case of the previous definition, for the eigen-value λ=0. According to this definition, sin(x) and cos(x) are not harmonic functions.

Both definitions are currently in use, but they are not equivalent. Maybe this ambiguity should be pointed out in the introduction.

--Juansempere (talk) 08:46, 29 September 2023 (UTC)