• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 [Update] Comparision of 5 methods of interpolation to continuous tetration sheldonison Long Time Fellow Posts: 684 Threads: 24 Joined: Oct 2008 10/29/2013, 02:52 PM (This post was last modified: 10/29/2013, 06:26 PM by sheldonison.) (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 $\alpha(z)$ 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 $\Im(z)=\frac{1}{sexp(3.5)}\approx 10^{-78}$. Kneser multiplies this repeating pattern by $2\pi i$ and then takes the exponent of that; $\exp(2\pi i z)$. 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 $\Im(z)=\frac{1}{\text{sexp}(8.5)}$.     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 $\text{sexp}(z)=\alpha^{-1}(z+\theta(z))$, via a 1-cyclic mapping from the inverse Abel function, $\alpha^{-1}(z)$, as well as an sexp(z) Taylor series representation at the real axis. This is because the $\theta(z)$ 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 « Next Oldest | Next Newest »

 Messages In This Thread [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/13/2013, 02:20 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by MikeSmith - 10/14/2013, 10:42 AM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/14/2013, 01:19 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by MikeSmith - 10/14/2013, 08:22 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/15/2013, 12:59 AM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by sheldonison - 10/14/2013, 10:07 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/15/2013, 12:00 AM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/15/2013, 01:09 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by sheldonison - 10/15/2013, 07:14 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/15/2013, 11:14 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/16/2013, 12:54 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by sheldonison - 10/16/2013, 04:12 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/16/2013, 05:07 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 10/22/2013, 12:17 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/22/2013, 01:53 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/27/2013, 11:40 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by sheldonison - 10/28/2013, 04:17 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 10/28/2013, 11:11 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 10/28/2013, 11:29 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by sheldonison - 10/29/2013, 09:37 AM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 10/28/2013, 11:32 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 10/29/2013, 01:11 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by sheldonison - 10/29/2013, 02:52 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 11/01/2013, 02:16 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 02/01/2014, 12:09 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 02/04/2014, 12:31 AM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 02/03/2014, 01:13 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 02/03/2014, 01:44 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 02/03/2014, 01:59 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 02/03/2014, 10:35 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 02/03/2014, 11:07 PM

 Possibly Related Threads… Thread Author Replies Views Last Post Continuous Hyper Bouncing Factorial Catullus 9 117 08/15/2022, 07:54 AM Last Post: JmsNxn Qs on extension of continuous iterations from analytic functs to non-analytic Leo.W 17 819 08/10/2022, 11:34 PM Last Post: JmsNxn Unifying continuous and discrete physics Daniel 0 39 07/31/2022, 01:26 PM Last Post: Daniel A related discussion on interpolation: factorial and gamma-function Gottfried 9 17,758 07/10/2022, 06:23 AM Last Post: Gottfried Question about tetration methods Daniel 17 699 06/22/2022, 11:27 PM Last Post: tommy1729 Small research update MphLee 2 1,104 10/26/2021, 12:22 AM Last Post: MphLee My interpolation method [2020] tommy1729 1 3,244 02/20/2020, 08:40 PM Last Post: tommy1729 Possible continuous extension of tetration to the reals Dasedes 0 3,126 10/10/2016, 04:57 AM Last Post: Dasedes Tribonacci interpolation ? tommy1729 0 3,712 09/08/2014, 10:37 AM Last Post: tommy1729 How many methods have this property ? tommy1729 1 4,928 05/22/2014, 04:56 PM Last Post: sheldonison

Users browsing this thread: 1 Guest(s)