Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Infinite tetration of the imaginary unit
#22
(06/20/2011, 01:36 PM)Gottfried Wrote:
(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?

The definition for the Shell-Thron region boundary is , which is the case. But when the period is an integer (or a fraction), the equations misbehave. At the Shell-Thron boundary, the period is always a real number, and the fixed point is neither attracting nor repelling. At first, I thought the idea of a superfunction with a real period was nonsense in this post, but then I was able to get it to work, except for the cases when the period was an integer, or a fraction with a small denominator. So that experimentation is where my conjecture came from.

For example, here is another case, on the Shell-Thron boundary, that should work fine because the period is a real number, with a period just a little bit bigger than 3. With a sufficient number of iterations, it generates a very nice plot, that appears to lead to an analytic superfunction. But, in the plot, you can see the influence of the base being just a litle bit bigger than an integer. By the way, these Shell-Thron boundary bases are easy to generate. and

base= 0.036314759343852642170871708751 + 1.7435957010705633826865464522i
L= 0.39309905520386861718874315414 + 0.46286165860913191074862913970i
Period= 3.0019951097271885263233102180
- Sheldon
Reply


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 sheldonison - 06/20/2011, 02:46 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 930 03/20/2018, 12:16 AM
Last Post: tommy1729
  [MO] Is there a tetration for infinite cardinalities? (Question in MO) Gottfried 10 10,864 12/28/2014, 10:22 PM
Last Post: MphLee
  Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 4,711 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 6,298 07/17/2013, 09:46 AM
Last Post: Gottfried
  Wonderful new form of infinite series; easy solve tetration JmsNxn 1 4,088 09/06/2012, 02:01 AM
Last Post: JmsNxn
  The imaginary tetration unit? ssroot of -1 JmsNxn 2 5,030 07/15/2011, 05:12 PM
Last Post: JmsNxn
  the infinite operator, is there any research into this? JmsNxn 2 5,126 07/15/2011, 02:23 AM
Last Post: JmsNxn
  Tetration and imaginary numbers. robo37 2 5,002 07/13/2011, 03:25 PM
Last Post: robo37
  Infinite Pentation (and x-srt-x) andydude 20 24,259 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 83,723 03/03/2011, 03:16 PM
Last Post: Gottfried



Users browsing this thread: 1 Guest(s)