Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
the inconsistency depending on fixpoint-selection
GFR Wrote:I think that the two real h's (of the t's) corresponding to the same b = sqrt(2), this time only with the + sign (sorry, Bo, ... it's the age), i.e. h = 2 and h = 4 must be different indeed. Any ... serious serial development should indeed show this situation. We cannot think that we may start from b = sqrt(2) and than, ... bingo! ... we suddely have two different values. The "strange object" that we may call h or t is the result of the application of a "two-valued function". But, two-valued "functions" are not politically correct animals.

But, perhaps, I don't understand what you precisely said. I didn't sleep well, last night. I shall improve! Tomorrow is another day.

Hi Gianfranco -

I've begun a short consideration of the multivalued log(1+x)-function in the context of matrix-operations (extended version of ContinuousIteration, which may of interest.
It seems interesting, but it exhibits, that we also need a more general notion of divergent summation, especially for the complex case. Since Euler-Summation (although principally able) is not well suited (and apparently not much studied) for the complex case, I'm always at the edge of possibilities. At least I cannot proceed much more without finding a reliable base for such summation-concepts. It seems, that all (or nearly all) fractional iterations of tetration, (if based on powerseries) produce hypergeometric divergent powerseries with convergence radius zero, not only the x->exp(x)-1 version. These series cannot be Euler-summed (principally) and thus we need this concept of assigning valued to such divergent series.
The powerseries for log(1+x) = x - x^2/2 + x^3/3 - + ... may be configured for multivaluedness by log_k(1+x) = k*2 Pi i + x - x^2/2 + x^3/3 - + ... and the matrix-operator is then square and has all the nasty properties of divergent series.

Disclaimer: all this is merely more or less speculation, and only motivated to find any usble entry-point to access the problem, which I focus in this thread.

Well - have a good night, I'll stop soon, too; I've to be prepared for an exam of my statistics-class tomorrow. I'll need hawk's eyes... Smile

Kind regards -

Gottfried Helms, Kassel

Messages In This Thread
RE: the inconsistency depending on fixpoint-selection - by Gottfried - 02/07/2008, 09:49 PM

Possibly Related Threads…
Thread Author Replies Views Last Post
  Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... Gottfried 23 56,230 10/20/2017, 08:32 PM
Last Post: Gottfried
  (Again) fixpoint outside Period tommy1729 2 5,968 02/05/2017, 09:42 AM
Last Post: tommy1729
  Polygon cyclic fixpoint conjecture tommy1729 1 4,886 05/18/2016, 12:26 PM
Last Post: tommy1729
  The " outside " fixpoint ? tommy1729 0 3,145 03/18/2016, 01:16 PM
Last Post: tommy1729
  2 fixpoint pairs [2015] tommy1729 0 3,478 02/18/2015, 11:29 PM
Last Post: tommy1729
  [2014] The secondary fixpoint issue. tommy1729 2 6,872 06/15/2014, 08:17 PM
Last Post: tommy1729
  Simple method for half iterate NOT based on a fixpoint. tommy1729 2 6,852 04/30/2013, 09:33 PM
Last Post: tommy1729
  2 fixpoint failure tommy1729 1 5,081 11/13/2010, 12:25 AM
Last Post: tommy1729
  abs f ' (fixpoint) = 0 tommy1729 2 7,785 09/09/2010, 10:13 PM
Last Post: tommy1729
  [Regular tetration] [Iteration series] norming fixpoint-dependencies Gottfried 11 24,798 08/31/2010, 11:55 PM
Last Post: tommy1729

Users browsing this thread: 1 Guest(s)