• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 U-tetration or "decremented exponential iteration" andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 01/06/2008, 09:57 PM (This post was last modified: 01/07/2008, 01:52 AM by andydude.) Yes, there are other posts about this, because it is common to slip on this aspect. I think Henryk said it best in this post particularly. I think I have seen something similar in tetration itself as opposed to iter-dec-exp, but I'm not sure how to phrase it. I first saw it when I was following Galidakis' research with his Puiseux series expansions about log(b). Galidakis uses series about the base b, whereas natural iteration uses series expansions about the iterator t in $\exp_b^t(x)$. With the linear approximation to tetration, as well as higher approximations, the table that Galidakis gives for the Puiseux series expansions of ${}^{t}(b)$ will give a discontinuity when the number of times you are differentiating is equal to t. In other words, the function: $(t, k) \mapsto \frac{d}{dt} \left(\frac{d^k}{du^k}\left({}^{t}(e^u)\right)|_{u=0}\right)$ is continuous and differentiable for all $t \ne k$. This can only be seen if you already have a continuous extension of tetration to non-integer heights, if not, then its just a bunch of points. This has been bothering me for quite some time now, but I think it may be related to the iterability of $e^x - 1$. Since tetration and $h^x - 1$ are topologically conjugate, this may help explain this as well... Andrew Robbins « Next Oldest | Next Newest »

 Messages In This Thread U-tetration or "decremented exponential iteration" - by Gottfried - 01/06/2008, 03:06 AM RE: U-tetration or "decremented exponential iteration" - by andydude - 01/06/2008, 09:57 PM RE: U-tetration or "decremented exponential iteration" - by Gottfried - 01/07/2008, 05:46 PM RE: U-tetration or "decremented exponential iteration" - by andydude - 01/07/2008, 06:24 PM RE: U-tetration or "decremented exponential iteration" - by Gottfried - 01/07/2008, 07:10 PM RE: U-tetration or "decremented exponential iteration" - by andydude - 01/09/2008, 02:14 AM

 Possibly Related Threads... Thread Author Replies Views Last Post Math overflow question on fractional exponential iterations sheldonison 4 4,112 04/01/2018, 03:09 AM Last Post: JmsNxn Complaining about MSE ; attitude against tetration and iteration series ! tommy1729 0 1,745 12/26/2016, 03:01 AM Last Post: tommy1729 An exponential "times" table MikeSmith 0 1,905 01/31/2014, 08:05 PM Last Post: MikeSmith exponential baby Mandelbrots? sheldonison 0 2,112 05/08/2012, 06:59 PM Last Post: sheldonison exponential distributivity bo198214 4 6,728 09/22/2011, 03:27 PM Last Post: JmsNxn Iteration series: Different fixpoints and iteration series (of an example polynomial) Gottfried 0 2,979 09/04/2011, 05:59 AM Last Post: Gottfried Base 'Enigma' iterative exponential, tetrational and pentational Cherrina_Pixie 4 10,013 07/02/2011, 07:13 AM Last Post: bo198214 Fractional iteration of x^2+1 at infinity and fractional iteration of exp bo198214 10 17,287 06/09/2011, 05:56 AM Last Post: bo198214 Two exponential integrals Augustrush 2 4,535 11/10/2010, 06:44 PM Last Post: Augustrush [Regular tetration] [Iteration series] norming fixpoint-dependencies Gottfried 11 14,167 08/31/2010, 11:55 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)