Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
tetration base conversion, and sexp/slog limit equations
bo198214 Wrote:Are you sure that it is 1-periodic?

I mean it is well known that
must be 1-periodic
for two superexponentials f and g.

This implies that

does not look 1-periodic?
You are correct, but it turns out not to matter that much. First, the two bases I'm comparing are the same. One is the ideal , for b a little bigger than , and the other is converted from base e to base b. So slog(sexp(x))-x would have to be 1-cyclic. Since the base is approaching , and since I'm comparing sexp in the critical section, which is very linear, so I can get away with a short cut of just subtracting the two sexp functions, since I don't have an slog function in my spreadsheet. But for an sexp with a linear approximation over the critical section, its not that big a deal. One of the two waves is a line segment, and the other is a line segment plus a sinusoid with an amplitude of 0.0004 times the slope of the line segment. The difference is too small to be seen in the rough graphs I made.

Mostly, I'm looking for a curve fit to a sine wave for the 1-periodic transfer function, as opposed to a 1-cyclic wave with higher order terms. If it is a sine wave, then there is hope for calculating it theoretically, as opposed to empirically. Once we have the delta sinusoid, in theory the sine wave can be applied to the critical section for a base approaching , and then used to generate sexp for another base with all positive odd derivatives. I realize that this is pretty hypothetical, and that the only thing I'm basing this on is how good a curve fit there is between the graph, and an ideal sine wave, but an ideal sine wave makes the math a lot cleaner.

Maybe Dimitrii could verify my results, by graphing
for b approaching .
If I'm right, the result will approach a perfect sine wave, as b approaches . The amplitude of the sine wave will converge on about 0.0004 as the conversion base approaches approaches .

new prediction: its even possible that all base conversions for Dimitrii's extension of sexp to real numbers will converge on 1-periodic sine waves, but surely someone would've noticed this already. Its all possible that it will be a sine wave in the sexp domain, (at the critical section, as the limit approaches e^(1/e)), but only an approximate sine wave in the slog domain.

Messages In This Thread
RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/03/2009, 07:27 PM
Is it analytic? - by sheldonison - 12/22/2009, 11:39 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Moving between Abel's and Schroeder's Functional Equations Daniel 1 941 01/16/2020, 10:08 PM
Last Post: sheldonison
  Complex Tetration, to base exp(1/e) Ember Edison 7 4,179 08/14/2019, 09:15 AM
Last Post: sheldonison
  Is bounded tetration is analytic in the base argument? JmsNxn 0 1,804 01/02/2017, 06:38 AM
Last Post: JmsNxn
  Sexp redefined ? Exp^[a]( - 00 ). + question ( TPID 19 ??) tommy1729 0 2,025 09/06/2016, 04:23 PM
Last Post: tommy1729
  Taylor polynomial. System of equations for the coefficients. marraco 17 20,204 08/23/2016, 11:25 AM
Last Post: Gottfried
  Dangerous limits ... Tommy's limit paradox tommy1729 0 2,304 11/27/2015, 12:36 AM
Last Post: tommy1729
  tetration limit ?? tommy1729 40 58,113 06/15/2015, 01:00 AM
Last Post: sheldonison
  Some slog stuff tommy1729 15 15,882 05/14/2015, 09:25 PM
Last Post: tommy1729
  Totient equations tommy1729 0 2,160 05/08/2015, 11:20 PM
Last Post: tommy1729
  Bundle equations for bases > 2 tommy1729 0 2,208 04/18/2015, 12:24 PM
Last Post: tommy1729

Users browsing this thread: 1 Guest(s)