Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
[Regular tetration] [Iteration series] norming fixpoint-dependencies
Just came across an older subject and thought it would fit into this "norming"-thread.
As older fellows here may remember, nearly my first contact with tetration was the question of alternating iteration-series for which I worked out some interesting heuristics. (see [1] and [2])

[update] I should explain, that for convenient ascii-notation of the tetration I "misuse" here the common notation. With z^^h I mean in the context of a given fixed base b, the value of z^^h := [/update]

Using base b=sqrt(2) we have the real-valued interval 2..4 for which we may find iteration heights from -inf to +inf if we start at some value z, say z=3, in this interval. Because in both direction the values of z^^h are finite we can compute a value for the alternating series of that values. So using Pari/Gp we can compute

f(z) = sumalt(h=0,(-1)^h*iter(z,h)) + sumalt(h=0,(-1)^h*iter(z,-h) ) - z

to evaluate the alternating iteration-series with center at the chosen z.

It is clear that this series is periodic for z in the interval z..z^^2 . But what's interesting is, that in general the f(z) is "small" and even we find f(z)=0

Because this is a remarkable result (and matches, for instance, the analogue problem when applied to a doubly-infinite geometric series by analytic continuation) this value z (where f(z)=0) introduces itself gently as candidate for a norm-value, at which the height is defined to be zero or at least an integer.

Here is a picture of the sinusoidal curve f(z) when z is moved from z to z^^2 beginning at some arbitrary value z0:


We see that astonishing approximation to a sine-curve, where the amplitude should be normed. Actually the deviance from the sine-curve is of the order of 1e-3 : I mean, if the height-parameter of this curve is compared with the abscissa of the sine-curve after the two curves are matched (for instance by binary search of the same y-values).
I'm not experienced with Fourier-analysis, but I think, it would be profitable to try to describe the f(z)-function by a fourier-decomposition. Analoguously this could be done for the other bases 1<b<eta.


[update]: obviously this provides also a "fixpoint-independent" definition for the real fractional tetration: just match the values of the sin-curve with that of f(z) and define the height h for the representation of the z according to the found abszissa of the sine (though this provides only approximation). [/update]

(both articles are *very* freshman-like and need being improved...)

[1] Short article of magazine-type

[2] longer version

Gottfried Helms, Kassel

Messages In This Thread
RE: [Regular tetration] norming fixpoint-dependencies - by Gottfried - 08/29/2010, 10:31 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Reducing beta tetration to an asymptotic series, and a pull back JmsNxn 2 501 07/22/2021, 03:37 AM
Last Post: JmsNxn
  Perhaps a new series for log^0.5(x) Gottfried 3 4,108 03/21/2020, 08:28 AM
Last Post: Daniel
Question Taylor series of i[x] Xorter 12 22,246 02/20/2018, 09:55 PM
Last Post: Xorter
  Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... Gottfried 23 50,426 10/20/2017, 08:32 PM
Last Post: Gottfried
  (Again) fixpoint outside Period tommy1729 2 5,162 02/05/2017, 09:42 AM
Last Post: tommy1729
  An explicit series for the tetration of a complex height Vladimir Reshetnikov 13 23,279 01/14/2017, 09:09 PM
Last Post: Vladimir Reshetnikov
  Complaining about MSE ; attitude against tetration and iteration series ! tommy1729 0 3,158 12/26/2016, 03:01 AM
Last Post: tommy1729
  2 fixpoints , 1 period --> method of iteration series tommy1729 0 3,215 12/21/2016, 01:27 PM
Last Post: tommy1729
  Taylor series of cheta Xorter 13 24,263 08/28/2016, 08:52 PM
Last Post: sheldonison
  Polygon cyclic fixpoint conjecture tommy1729 1 4,246 05/18/2016, 12:26 PM
Last Post: tommy1729

Users browsing this thread: 1 Guest(s)