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Tetra-series
#11
andydude Wrote:Have you tried computing or plotting or yet? I would do this but I don't have any code yet for AS(x), and I'm lazy.

Andrew Robbins

Yes, I fiddled with this a bit.

First, it is obvious, that in

Sb(x) = x - b^x + b^b^x - ... + ...

there is some "self-similarity", if x itself is a powertower of base b

Sb(0) = s0
Sb(1) = -s0
Sb(b) = 1 - s0
Sb(b^b) = b - 1 + s0
and so on.

For the fixpoints, let t0,t1,t2,... be the fixpoints for base b
t_k = h_k(b)

then
Sb(t) = t - t + t - t +... -... = t*(eta(0)) = t/2

I've added some graphs for different bases for varying x, marking x=t0,t1 and x=0, x=1, x=b

It would be interesting to show graphs of negative heights, aka iterated logarithms...

Gottfried
   
   
   
   
   
Gottfried Helms, Kassel
Reply


Messages In This Thread
Tetra-series - by Gottfried - 11/20/2007, 12:47 PM
RE: Tetra-series - by andydude - 11/21/2007, 07:14 AM
RE: Tetra-series - by Gottfried - 11/22/2007, 07:04 AM
RE: Tetra-series - by andydude - 11/21/2007, 07:51 AM
RE: Tetra-series - by Gottfried - 11/21/2007, 09:41 AM
RE: Tetra-series - by Ivars - 11/21/2007, 03:58 PM
RE: Tetra-series - by Gottfried - 11/21/2007, 04:37 PM
RE: Tetra-series - by Gottfried - 11/21/2007, 06:59 PM
RE: Tetra-series - by andydude - 11/21/2007, 07:24 PM
RE: Tetra-series - by Gottfried - 11/21/2007, 07:49 PM
RE: Tetra-series - by andydude - 11/21/2007, 08:39 PM
RE: Tetra-series - by Gottfried - 11/23/2007, 10:47 AM
RE: Tetra-series - by Gottfried - 12/26/2007, 07:39 PM
RE: Tetra-series - by Gottfried - 02/18/2008, 07:19 PM
RE: Tetra-series - by Gottfried - 06/13/2008, 07:15 AM
RE: Tetra-series - by Gottfried - 06/22/2008, 05:25 PM
Tetra-series / Inverse - by Gottfried - 06/29/2008, 09:41 PM
RE: Tetra-series / Inverse - by Gottfried - 06/30/2008, 12:11 PM
RE: Tetra-series / Inverse - by Gottfried - 07/02/2008, 11:01 AM
RE: Tetra-series / Inverse - by andydude - 10/31/2009, 10:38 AM
RE: Tetra-series / Inverse - by andydude - 10/31/2009, 11:01 AM
RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 01:25 PM
RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 02:40 PM
RE: Tetra-series / Inverse - by andydude - 10/31/2009, 09:37 PM
RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 10:33 PM
RE: Tetra-series / Inverse - by Gottfried - 11/01/2009, 07:45 AM
RE: Tetra-series / Inverse - by andydude - 11/03/2009, 03:56 AM
RE: Tetra-series / Inverse - by andydude - 11/03/2009, 04:12 AM
RE: Tetra-series / Inverse - by andydude - 11/03/2009, 05:04 AM
RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 12:58 PM

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