Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
[Update] Comparision of 5 methods of interpolation to continuous tetration
#23
(10/29/2013, 01:11 PM)tommy1729 Wrote: I simply asked if I am correct with my 10 step way to do kneser.

I do not ask for the Riemann mapping so I guess its more of a proof question.

But actually I'd say its a " construction " question.

Is the Kneser proof/solution constructed in the 10 steps I posted or is one or more steps wrong ?
Tommy, the sequence is good, but the Riemann mapping step 5) has a lot of sub-steps. For me, the biggest complexity hurdle in Kneser's construction, besides the fact that I don't have a formal math degree, is taking the Abel function, from Schroeder function, of the real axis, which is after the step where he generates the chi-star, but still one or two steps before the Riemann mapping. I don't read German, so have no idea how he proved the infinite region is simply connected, and I wouldn't know how to do so, since the region is increasingly recursively complex.

Here is the rough Abel function of the real axis, showing the repeating pattern; here . Kneser multiplies this repeating pattern by and then takes the exponent of that; . That is the contour that gets wrapped around a unit circle for the Riemann mapping.
   

Here we zoom in on one of the singularities, where sexp(z)=0. The singularity gets ever more complicated as we super-exponentially approach zero. Here, I show what the contour looks like if .
   

My algorithm has a mathematical description, as well as pari-gp code. I don't want to side track too much, but it does something different but equivalent to generate , via a 1-cyclic mapping from the inverse Abel function, , as well as an sexp(z) Taylor series representation at the real axis. This is because the function has a singularity at the real axis, so adequate convergence is not possible with a reasonable number of terms. So my algorithm actually has to iteratively generate two different equivalent representations of sexp(z).
- Sheldon
Reply


Messages In This Thread
RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by sheldonison - 10/29/2013, 02:52 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Possible continuous extension of tetration to the reals Dasedes 0 1,333 10/10/2016, 04:57 AM
Last Post: Dasedes
  Tribonacci interpolation ? tommy1729 0 1,914 09/08/2014, 10:37 AM
Last Post: tommy1729
  How many methods have this property ? tommy1729 1 2,476 05/22/2014, 04:56 PM
Last Post: sheldonison
  (MSE): Comparision of powertowers -.Possibly interesting thread in MSE Gottfried 0 1,906 05/22/2013, 07:02 AM
Last Post: Gottfried
  [UFO] - a contradiction in assuming continuous tetration? Gottfried 18 20,761 08/29/2010, 08:44 PM
Last Post: Gottfried
  Self tetraroot constructed via Newton series interpolation mike3 2 6,510 07/11/2010, 03:38 AM
Last Post: mike3
  Borel summation and other continuation/summability methods for continuum sums mike3 2 5,166 12/30/2009, 09:51 PM
Last Post: mike3
  A false interpolation paradigm (?); a reconsideration Gottfried 4 7,280 09/17/2009, 08:17 AM
Last Post: bo198214
  exponential polynomial interpolation Gottfried 3 7,002 07/16/2008, 10:32 PM
Last Post: andydude
  A related discussion on interpolation: factorial and gamma-function Gottfried 6 10,329 06/27/2008, 06:38 PM
Last Post: Gottfried



Users browsing this thread: 1 Guest(s)