Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
tetration from alternative fixed point
#21
That's a very weird looking sexp!

Does this mean we're going to have to clarify uniqueness based on the infinite list of fix points?

Already, by looking at the new sexp it doesn't seem as fitting as the one based off of the primary fix point. But I don't know how you'd express this aesthetic requirement in mathematical notation. Perhaps merely because the derivative goes to zero at integers?

And, I wonder if we'll see more erratic behaviour across a third fix point or a fourth! Is it likely that they'll get more erratic as we get further from the primary fixpoint? Or will it be more chaotic, as to which ones look nice and which ones don't?

Very very interesting.
Reply
#22
(12/05/2011, 07:58 PM)JmsNxn Wrote: That's a very weird looking sexp!

Does this mean we're going to have to clarify uniqueness based on the infinite list of fix points?
James,

Thanks for your comments. The primary fixed point solution obviously looks best at the real axis, but the secondary fixed point solution is very beautiful in the complex plane (see my previous post).

I haven't tried generating the nth fixed point of e^z, beyond n=2, but that would be my intuition, that there are an infinite number of other analytic sexp(z) solutions, one for each fixed point. The functions are hypothesized to all be analytic in the complex plane, except for singularities at the real axis for negative integers less than or equal to -2. As goes to +/- infinity, the function converges to the nth fixed point of exp(z). Also, for the nth fixed point, each loop around the singularity at z=-2 increments by (2n-1)2pi i. Correspondingly, at z=-1, the first 2n-1 taylor series terms are zero.

The existence of such solutions depends on the topology of the Schroder function of the real number line and how the real axis unfolds in the complex plane for the superfunction so that the singularities can cancel out for integers>-2, after the Rieman mapping. Unfortunately, I'm over my head....
- Shel
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Are tetrations fixed points analytic? JmsNxn 2 2,400 12/14/2016, 08:50 PM
Last Post: JmsNxn
  Derivative of exp^[1/2] at the fixed point? sheldonison 10 8,708 01/01/2016, 03:58 PM
Last Post: sheldonison
  [MSE] Fixed point and fractional iteration of a map MphLee 0 1,860 01/08/2015, 03:02 PM
Last Post: MphLee
  alternative fixpoints = branches ? tommy1729 0 1,518 10/11/2014, 08:50 AM
Last Post: tommy1729
  attracting fixed point lemma sheldonison 4 8,758 06/03/2011, 05:22 PM
Last Post: bo198214
  Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation Base-Acid Tetration 19 27,617 10/24/2009, 04:12 AM
Last Post: andydude
  real slog developed at a fixed point bo198214 12 11,097 04/06/2009, 10:10 PM
Last Post: bo198214
  Iterating at fixed points of b^x bo198214 28 23,191 05/28/2008, 07:37 AM
Last Post: Kouznetsov
  Rational sums of inverse powers of fixed points of e jaydfox 14 13,559 11/23/2007, 08:22 AM
Last Post: bo198214
  Continuous iteration from fixed points of base e jaydfox 22 22,457 11/22/2007, 09:08 PM
Last Post: jaydfox



Users browsing this thread: 1 Guest(s)