10/06/2009, 04:47 AM
(10/05/2009, 11:58 PM)mike3 Wrote: How do you derive the general term for the asymptotic series?The first 5 terms of the expansion are in the preprint
D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, in press.
Preprint, English version: http://www.ils.uec.ac.jp/~dima/PAPERS/2009vladie.pdf
Any, Mathematica or Maple, allow to derrive several coefficients,
but I have not yet got the general expression for them.
If you dig it well, you may do better than I did.
Quote:And I decided to run the graphing procedure using the 811-node approximation with A = 24. It took a very long time (3 hours I think) to generate the graph due to the arbitrary precision math.If you want beautiful portrait of the fractal, I recommend the algotirhm from my preprint mentioned. It returns 14 digits wihtin a hindred operations. You can make a 10^6x zoom, and it still looks perfect.
Quote:I suppose it could have been accelerated by computing fewer points and using an interpolation technique...Better to fit the function with elementary functions, see the preprint above.
For my pics, first, I use the fit, and calculate the function on the mesh.
Then, the linear interpolation is used to draw lines.
You may strip out any of my algorithms posted at Citizendium.
Quote:The graph is very intriguing. It's like the Julia set of the exponential map grows there. I guess the structure gets ever more convoluted and fractal-like the further one goes toward the right. What's up with that? This behavior is really neat yet it's not like the usual special functions I've seen.It gives the fractal. Its behavior is discussed in our preprint
D.Kouznetsov, H.Trappmann. Cut lines of the inverse of the Cauchi-tetrational. preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/2009fractae.pdf
The paper is under consideration in the Fractal Journal.
We can make a movie, "Zooming into the map of the Tetrational". To me, it is the most beautiful fractal among those I even had seen; and it has no parameters; so, it is so fundamental as the Mandelbrot. Would Kazimir Malevich paint the tet(z) for |z|<5,
(instead of his "Black Square"), he would have to live until now, still painting it.
The map of this function shows the mounts, gaps, valleys, rivers, flowers, ets. Play with various visualisations of this function, and you will see them.