Hausdorff dimension of a function

[Timeroot]

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?

[kram1032]

they surely have as they repeat a certain pattern at different scales.

here would be the lyapunov exponent for sin(1/x): http://www.wolframalpha.com/input/?i=limit%281%2Fn*ln%28abs%28d^n%2Fdx^n+sin%28pi%2Fx%29%29%29%2Cn+-%3E+infinity%29
but wolfram (of course in this case^^) Isn't capable for finding a solution...

For Hausdorff, you'll need some scale values...
http://www.wolframalpha.com/input/?i=sin%281%2Fx%29%3D0
here you can see, the periodicy is 1/(2*pi*n) which you could kinda use as a scale factor...
The other value is harder to find. I'm not sure right now of how to explicitely get that value (might be possible though): a function of the number of circles (for sin, a circle would be from 2pin to 2pi(n+1) ) in an aribitary two boundaries...

[Timeroot]

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.

[kram1032]

I messed up the period: It's actually 1/(pi*n) - the 2 is for zeros happening twice as often

Hmm... Looking at the complex version, the 1/(1+1/(1-sinc(1/x)))) one gets a nice pattern - a fractal wavelet
real
imaginary
absolute
argument

Which preview version do you mean?

The Lyapunov limit can't be solved explicitly, I guess... You'd need a program for that...

[Timeroot]

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.

[kram1032]

they rather seem to grow inverse linear for the whole thing...


gotta look into wolfram beta or is it wolfram alpha beta?
Thanks for the hint

[Timeroot]

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...

[kram1032]

Thanks for the link


EDIT: Damn, fractals indeed have a nice tan today

real
imag
abs
-arg (negative for better visibillity - for some reason, the one where it's the most reasonable does not get a contour plot...)