Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Infinite tetration of the imaginary unit
(06/20/2011, 05:27 AM)sheldonison Wrote: My conjecture is for bases on the Shell Thron boundary, there is an analytic superfunction with a real period, whose structure depends on what the continued fraction representation of the real period is. As long as the period is a real number (with an infinite continued fraction representation), then I suspect the superfunction is analytic. If the period is a rational number, then I don't think there is an analytic superfunction. For example, this base, with a real period=3, probably doesn't have an analytic superfunction, developed from the neutral fixed point, because starting with a point near L, and iterating the function x=B^x three times, doesn't get you back to the initial starting point.
Base= 0.030953557167612060 + 1.7392241043091316i
L= 0.39294655583435517 + 0.46203078407110528i
Hi Sheldon -

I've inserted your base-parameter and got the following plot for the orbit/for the three partial trajectories in the same style of my previous plots. I seem to have problems to understand your comment correctly. For instance, isn't that fixpoint attracting instead of neutral?

[Image: br10.png]

Having seen this I assume, that also with a starting-point near the fixpoint we get something converging to the fixpoint, however slow. But, well, that would be now another job to prove.

[Image: br_01.png]

In my initial plot it seemed, that there is only one base b0, whose orbits are between converging to the fixpoint and diverging, and because the base at 1.71290*I is such a base I assume, that we get either convergence here or divergence to a triplett of cumulation points.

What do you think?


A startingpoint x0=0.41*(1+I)=b^^0, even nearer at the fixpoint L, exhibits now repelling properties of the fixpoint. So I think, that in fact there are three "oscillating" fixpoints in the near of the orbit of the last experiment and the trajectories of the first picture do not approach the fixpoint L but that triplett of accumulation(?) points.

[Image: br_041.png]

Gottfried Helms, Kassel

Messages In This Thread
Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 12:09 AM
RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 01:36 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 1,631 03/20/2018, 12:16 AM
Last Post: tommy1729
  [MO] Is there a tetration for infinite cardinalities? (Question in MO) Gottfried 10 14,511 12/28/2014, 10:22 PM
Last Post: MphLee
  Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 6,226 05/06/2014, 09:47 PM
Last Post: tommy1729
  Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried 5 8,280 07/17/2013, 09:46 AM
Last Post: Gottfried
  Wonderful new form of infinite series; easy solve tetration JmsNxn 1 4,954 09/06/2012, 02:01 AM
Last Post: JmsNxn
  The imaginary tetration unit? ssroot of -1 JmsNxn 2 6,226 07/15/2011, 05:12 PM
Last Post: JmsNxn
  the infinite operator, is there any research into this? JmsNxn 2 6,289 07/15/2011, 02:23 AM
Last Post: JmsNxn
  Tetration and imaginary numbers. robo37 2 6,270 07/13/2011, 03:25 PM
Last Post: robo37
  Infinite Pentation (and x-srt-x) andydude 20 30,488 05/31/2011, 10:29 PM
Last Post: bo198214
  Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e)) Gottfried 91 107,571 03/03/2011, 03:16 PM
Last Post: Gottfried

Users browsing this thread: 1 Guest(s)