Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
tetration base conversion, and sexp/slog limit equations
so, why the wobble??? And how do you convert between constant an extension of sexp to real numbers that has a constant base conversion, and an extension of sexp to real numbers that has all positive odd derivatives?

The limit equation for the conversion I have defined for sexp to real numbers with an exact base changes comes with its own wobble, which becomes noticeable in the third derivative. This violates the requirement that the odd derivatives are positive for all values. In particular, the 5th derivative of this sexp extension will have negative values. The wobble gets smaller as the base approaches (1/e), and gets smaller faster the linear approximation, so the wobble does not seem to effect my earlier convergence error term estimates.

Even though I do not yet understand the wobble, I can characterize it. Lets take Dimitrii's taylor series expansion for sexp_e, and use it to convert to base 1.45, a little bigger than . Compare the converted critical section of this base 1.45 with a wobble free estimate of the critical section of base 1.45. The comparison is a simple subtraction. When converting to bases approaching , the conversion wobble appears to be a perfect sine wave! The relative amplitude of this sine wave stays constant as the base converted to approaches . The phase changes in a predictable way, and the relative height also changes in a predictable way. Also, I know how to lock these down (amplitude, phase, relative height), by requiring the limit use only values for b that are integer multiples, instead of fractional multiples, in the conversion process. One possible source of confusion is that the wobble down converting from base e to base 1.45 is about one 100 times bigger than the wobble inherent to base 1.45, and about 1000 times bigger than the wobble inherent to base 1.44533.

I haven't done it yet, but the upshot is that once I know the conversion sinusoid, I can convert from a base approaching to any other base and get either an sexp with a constant base conversion factor, or Dimitrii's sexp function, where the odd derivatives are positive for all values of x>-2!

[Image: delta_taylor_to_145.gif]

Messages In This Thread
RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/03/2009, 06:15 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 1 617 01/16/2020, 10:08 PM
Last Post: sheldonison
  Complex Tetration, to base exp(1/e) Ember Edison 7 3,313 08/14/2019, 09:15 AM
Last Post: sheldonison
  Is bounded tetration is analytic in the base argument? JmsNxn 0 1,618 01/02/2017, 06:38 AM
Last Post: JmsNxn
  Sexp redefined ? Exp^[a]( - 00 ). + question ( TPID 19 ??) tommy1729 0 1,860 09/06/2016, 04:23 PM
Last Post: tommy1729
  Taylor polynomial. System of equations for the coefficients. marraco 17 18,666 08/23/2016, 11:25 AM
Last Post: Gottfried
  Dangerous limits ... Tommy's limit paradox tommy1729 0 2,136 11/27/2015, 12:36 AM
Last Post: tommy1729
  tetration limit ?? tommy1729 40 53,877 06/15/2015, 01:00 AM
Last Post: sheldonison
  Some slog stuff tommy1729 15 14,410 05/14/2015, 09:25 PM
Last Post: tommy1729
  Totient equations tommy1729 0 2,005 05/08/2015, 11:20 PM
Last Post: tommy1729
  Bundle equations for bases > 2 tommy1729 0 2,042 04/18/2015, 12:24 PM
Last Post: tommy1729

Users browsing this thread: 1 Guest(s)