• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 tetration base conversion, and sexp/slog limit equations sheldonison Long Time Fellow Posts: 644 Threads: 22 Joined: Oct 2008 02/20/2009, 10:54 AM (This post was last modified: 02/20/2009, 04:32 PM by sheldonison.) The question for sexp/slog is how to define the curve extending sexp from integers to real. One question is does there exist an sexp/slog function for which $\text{slog}_a(\text{sexp}_b(x))-x$ converge to a constant value, or does it converge to a 1-cycle periodic function? And can this be a uniqueness criterion? Consider what happens as b approaches e^(1/e) in the equation $\text{slog}_b(x)$. The curve becomes more and more linear, and there are fewer and fewer degrees of freedom for how to extend the sexp function to real numbers, and still have an increasing "well behaved" function. It must be possible to describe this rigorously in terms of limits. Here is an example. if $b=1.49208... \text{ slog}_b(e)=6$, $\text{slog}_b(2.4989)=5$, $\text{slog}_b(2.2887)=4$, $\text{slog}_b(2.0691)=3$, $\text{slog}_b(1.817)=2$, $\text{slog}_b(1.4923)=1$, $\text{slog}_b(1.0)=0$, In the limit, as b approaches $e^{1/e}$, the continuation of the sexp to real numbers will be a straight line between the second and third terms, or between $\text{log}_b(e)$ and $\text{log}_b(\text{log}_b(e))$. This segment includes the inflection point of the sexp. Then, once the curve for base b is defined, you can convert between base b and base a using $\text{slog}_a(\text{sexp}_b(x))-x$. The trick is to find a value of x, such that the $\text{sexp}_b(x)$ is large enough and equal to a defined integer value for the equation $\text{slog}_a(\text{sexp}_b(x))$. Then you have the conversion factor for large numbers, which can be used to define the sexp/slog curve for base a for all real numbers. This curve for base a assumes $\text{slog}_a(\text{sexp}_b(x))-x$ converges to a constant value as opposed to converging to a 1-cycle periodic function. Moreover, the limit as b approaches $e^{1/e}$ will eliminate any degrees of freedom in defining slog/sexp extension to real numbers for any base a. - Sheldon Levenstein « Next Oldest | Next Newest »

 Messages In This Thread tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/18/2009, 07:01 AM RE: tetration base conversion, questions and results - by sheldonison - 02/19/2009, 12:10 AM tetration base conversion, uniqueness criterion? - by bo198214 - 02/19/2009, 04:24 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/20/2009, 10:54 AM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/20/2009, 01:07 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/20/2009, 02:51 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/21/2009, 12:18 AM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/21/2009, 12:39 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/21/2009, 02:59 PM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/21/2009, 06:36 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/22/2009, 04:41 AM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/22/2009, 04:04 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/24/2009, 08:24 PM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/24/2009, 09:57 PM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/24/2009, 10:21 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/24/2009, 10:54 PM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/24/2009, 11:06 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/26/2009, 11:04 AM RE: tetration base conversion, and sexp/slog limit equations - by bo198214 - 02/26/2009, 12:16 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/26/2009, 02:36 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/28/2009, 05:56 AM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/28/2009, 10:01 AM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/01/2009, 12:18 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/03/2009, 06:15 PM RE: tetration base conversion, and sexp/slog limit equations - by bo198214 - 03/03/2009, 06:46 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/03/2009, 07:27 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/09/2009, 06:34 PM Summay tetration base conversion, and sexp/slog limit equations - by sheldonison - 07/31/2009, 06:55 PM RE: Summay tetration base conversion, and sexp/slog limit equations - by sheldonison - 08/01/2009, 10:32 AM Is it analytic? - by sheldonison - 12/22/2009, 11:39 PM RE: tetration base conversion, and sexp/slog limit equations - by mike3 - 12/25/2009, 08:51 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 12/26/2009, 01:44 AM RE: tetration base conversion, and sexp/slog limit equations - by mike3 - 12/26/2009, 01:54 AM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 12/27/2009, 06:53 AM RE: tetration base conversion, and sexp/slog limit equations - by mike3 - 12/31/2009, 11:45 PM Inherent ringing in tetration, re: base conversion - by sheldonison - 01/02/2010, 05:31 AM RE: Inherent ringing in tetration, base conversion - by mike3 - 01/04/2010, 03:51 AM RE: Inherent ringing in tetration, base conversion - by sheldonison - 01/04/2010, 06:08 AM RE: tetration base conversion, and sexp/slog limit equations - by tommy1729 - 02/26/2013, 10:47 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/27/2013, 07:05 PM

 Possibly Related Threads... Thread Author Replies Views Last Post New Quantum Algorithms (Carleman linearization) Finally Crack Nonlinear Equations Daniel 2 139 01/10/2021, 12:33 AM Last Post: marraco Moving between Abel's and Schroeder's Functional Equations Daniel 1 1,750 01/16/2020, 10:08 PM Last Post: sheldonison Complex Tetration, to base exp(1/e) Ember Edison 7 6,247 08/14/2019, 09:15 AM Last Post: sheldonison Is bounded tetration is analytic in the base argument? JmsNxn 0 2,194 01/02/2017, 06:38 AM Last Post: JmsNxn Sexp redefined ? Exp^[a]( - 00 ). + question ( TPID 19 ??) tommy1729 0 2,430 09/06/2016, 04:23 PM Last Post: tommy1729 Taylor polynomial. System of equations for the coefficients. marraco 17 23,311 08/23/2016, 11:25 AM Last Post: Gottfried Dangerous limits ... Tommy's limit paradox tommy1729 0 2,697 11/27/2015, 12:36 AM Last Post: tommy1729 tetration limit ?? tommy1729 40 67,551 06/15/2015, 01:00 AM Last Post: sheldonison Some slog stuff tommy1729 15 18,833 05/14/2015, 09:25 PM Last Post: tommy1729 Totient equations tommy1729 0 2,585 05/08/2015, 11:20 PM Last Post: tommy1729

Users browsing this thread: 2 Guest(s)