Talk:Harmonic function

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Dumb question: what's "harmonic" about a harmonic function ? It was one of the reasons I came to the page, but I didn't get an answer. unlikely_ending 11:19, 5 November 2006 (UTC)

I understand that at least one reason why harmonic functions are so-called is that if you look for solutions of Laplace's equation (i 3-dim, say) which are products F(x)G(y)H(z) of functions of each variable separately, then you end up with 3 differential equations for x, y, z, whose solution is essentially 'Simple Harmonic Motion'. Madmath789 11:54, 5 November 2006 (UTC)

Historical reason: around 1880 Kelvin introduced the term "Spherical Harmonics" for a class of special functions on the sphere, that are the eigenfunctions of the spherical Laplacian (i.e. the Laplace-Beltrami operator of the standard sphere). In this sense these functions are the natural generalization of the one-variable functions cos(kt) and sin(kt), that describe the harmonic components of sound waves. It turns out that, in cartesian coordinates, the spherical harmonics of the unit sphere of R^n are exactly the homogeneous polynomials in n variables solving the Laplace equation. For this reason, after some decades, people started to use the world "harmonic" for any solution of the Laplace equation, not only for homogeneous polynomials. (I learned these notices in a beautiful book by S.Axler, P.Bourdon, W.Ramey, and I added it as reference in the article. In fact, I would suggest it to anybody interested in the theory of harmonic functions. You can download it for free from the Axler page). PMajer 21:07, 13 November 2007 (UTC)

Can somebody write down how harmonic functions are connected to stochastic processes? or else leave this point away in the description.

This is a tough one, as the connections between harmonic functions and stochastic processes is quite deep - you have to know a lot about both potential theory and measure-theoretic probability theory to understand it, unfortunately. I don't think it would be easy to explain the connction in this article without making ot way too technical (I certainly would not attempt to do it!) You can get an idea of the problem by looking at Doob's book 'Classical Potential Theory and Its Probabilistic Counterpart'. Sorry if this does not really answer your query! Madmath789 11:54, 5 November 2006 (UTC)

[edit] Merge with potential theory

Shouldn't this be merged with potential theory? —Preceding unsigned comment added by 90.229.231.115 (talk) 19:30, 26 October 2007 (UTC)


[edit] Vote for new external link

Here's my site full of PDE's with harmonic functions. Someone please put it in the external links if you think it's helpful!

http://www.exampleproblems.com/wiki/index.php/PDE:Laplaces_Equation

[edit] defintiion

Harmonic functions are often defined as continuous complex valued functions (on an open subset) with zero laplacian,implicit in this is that 2nd partial derivatives w.r.t x and y exist. Should this article include both defintions and a comparison?