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I'm nearing the end of the semester of an introductory-level complex variables class. (Very introductory -- it's the version of the class that's only required for engineering and physics majors, as it doesn't require two semesters of undergrad analysis that are prerequisite to the complex variables class for math majors, at my school.)

One of the many fascinating things I've seen this semester has been, speaking in broad terms, the behavior of analytic function, and the way that a harmonic function and its conjugate 'synchronize' (for lack of a better word) to create analyticity.

Despite the examples I've seen of harmonic functions being steady-state solutions to heat problems and showing up in descriptions of electric fields and whatnot, I feel like I lack any sense of what the 'harmonicity' of harmonic functions is all about.

On a side note: I do recall one day, however, where I was working through an example having to do with the level curves of a harmonic function and its conjugate, where I believe the significance what they points of intersection where always orthogonal. This 'mesh' notion created, for me, a visual image of how the the two functions work together to give an analytic function its synchronized, predictable nature. (But, as with most things, I could be mistaken in my understanding of this; these weren't points being stressed in the book, and it was in a chapter later in the book than what we'll cover in the class.)

So, my question is that of how one ought finish this statement:
"I was considering a problem, and I intuitively knew the solution would need to be a harmonic function because the problem had the property..."

I stress the word 'intuitively,' by the way.

If you feel this misses the point of harmonic functions and how I should think of them, then by all means please answer however you feel is appropriate.

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A harmonic function is a function whose value at a point is always equal to the average of its values on a sphere centered at that point (reference). This is why they show up as steady-state solutions to the heat equation: if this averaging property weren't true, then heat would be flowing either from or to a point.

You might be interested in reading Needham's Visual Complex Analysis.

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  • $\begingroup$ Ah right, I suppose it's the 'steadiness' of the solution itself that suggests that it must be a harmonic function. (I imagine there's symmetry -- of some sort -- to the paths through the region, too...but that's just a guess, at the moment.) Also, I just got my hands on a copy of Needham's book, though I haven't done much more than thumb through it a bit, so far. Thanks for the answer! $\endgroup$ – steve_0804 Apr 15 '14 at 4:04
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If you take a circular piece of wire, and dip it into a soap solution to make a bubble surface across the circle, and then place the circular piece of wire in the $xy$-plane, if the circular wire has small perturbations of order $\epsilon$, then the first order solutions of the $z$-coordinate as functions of $x$ and $y$ are harmonic.

This also intuitively explains the maximum principle - that a harmonic function can not attain its maximum on the interior of the domain.

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  • $\begingroup$ I appreciate you offering something to visualize. It's not immediately apparent to me how perturbations lead to solutions being harmonic, but I'll give it some thought. (Something about breaking radial symmetry, perhaps? Just an initial guess.) Thanks for the answer! $\endgroup$ – steve_0804 Apr 15 '14 at 4:17
  • $\begingroup$ Its the perturbations don't make it harmonic. The perturbations make it interesting. Without perturbations, it would represent the constant function. $\endgroup$ – Stephen Montgomery-Smith Apr 15 '14 at 11:51

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