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Does anyone know the Hausdorff dimension of - or at least a program to calculate it - the plot of the a function? For example, the functions y=sin(1/x), y=x*sin(1/x),  y=1/(1+1/(1-sinc(1/x)))) etc. all have fractal dimensions, I'm pretty sure. When you think of things like the Minkowski ? function, or the Blancmange function (the Takagi Curve), which both have fractal dimension, I wonder about analytic functions as well... clearly they aren't fractal everywhere, do they may be one of the weird "multifractals", in which case I just mean at the origin. Anybody?

Well, I suppose it wouldn't have to be the Hausdorff dimension exactly - if it's something easier to calculate, that would be fine too - assuming it gives the same result, which in this case I'm not at all sure it would. Wolfram Alpha seems to always interpret lim (ln(abs(...))) x->0 as lim (ln(x)) x-> 0 * abs(...). That multiplication doesn't make any sense. When I try leaving off the log, it still goes nuts, taking the limit of the abs. Bad Alpha. Btw, kram, do you use the preview version? You seem to use Alpha a lot, you probably should. There's a link somewhere.

When I say the "preview version", I mean the beta version. It's just like the regular, public version, but it has slightly more experimental features. It's not hard to get in, just send a message saying you would like to help, and they'll give you a username and password to the URL. Also, you get an e-mail every 3 weeks or so talking about the new features (very briefly). Oh, and I sent a bug report to Wolfram about the weird way it was replacing the absolute function with multiplication... maybe a solution is on the way.

Back on topic, that function does give nice patterns in the complex plane. I'm thinking that maybe it will be hard to calculate the dimension with box-counting because immediately near the origin, the number of boxed grows quadratically, and far away it grows linearly, and in between it's becoming linear. To get an idea of how it behaves as the size shrinks, one would need an idea of how quickly the "quadratic zone" shrinks. Because the function is not strictly self similar, and the oscillations become much more compressed near the origin, I reckon that the dimension of x*sin(1/x) is actually a multifractal with a dimension of . 1 is for 'most' of it, 2 for the singularity. Perhaps more interesting would be looking at how the local dimension at X changes as the box size Epsilon changes.

Wolfram Alpha Preview. Find it here: http://www.wolframalpha.com/participate/test.html.

When I said it grows quadratically, I meant that right at the origin is infinitely dense. Yes, the frequency grows inverse linearly. I've been trying to come up with a function that keeps the height to wavelength ratio near constant and still goes to infinite frequency at zero, because this would be more classically self-similar and thus fractal. I'm having trouble, though...