Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
tetration base conversion, and sexp/slog limit equations
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 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 . 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 , , , , , , ,

In the limit, as b approaches , the continuation of the sexp to real numbers will be a straight line between the second and third terms, or between and . 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 . The trick is to find a value of x, such that the is large enough and equal to a defined integer value for the equation . 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 converges to a constant value as opposed to converging to a 1-cycle periodic function. Moreover, the limit as b approaches will eliminate any degrees of freedom in defining slog/sexp extension to real numbers for any base a.
- Sheldon Levenstein

Messages In This Thread
RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/20/2009, 10:54 AM
Is it analytic? - by sheldonison - 12/22/2009, 11:39 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)