Thread Rating:
  • 1 Vote(s) - 5 Average
  • 1
  • 2
  • 3
  • 4
  • 5
pentation and hexation
#1
Xorter and I have been exchanging some emails.  Here's the question.  Start with tetration, base e.  If you want to extend to analytic tetration, download fatou.gp from this website.  Fatou was recently updated to do some really really cool stuff, but that's for another post.
sexpinit(exp(1));
sexp(-1)=0
sexp(0)=1
sexp(1)=e

Now onto Pentation.  The latest version of fatou.gp also has built in support for pentation for real bases, but lets start with the integer values for Pentation.
genpent(exp(1));
pent(-2)=-1 /* slog(pent(-1))=slog(0)=-1 */
pent(-1)=0  /* slog(pent(0))=slog(1)=0 */
pent(0)=1 /* pent(0,1) are by definition 1 and the base=e */
pent(1)=e

We don't yet have analytic hexation, but if it existed ....
hex(-3)=-2 /* pent^{-1}(hext(-2))=pent^{-1}(-1)=-2 */
hex(-2)=-1 /* pent^{-1}(hext(-1))=pent^{-1}(0)=-1 */
hex(-1)=0 /* pent^{-1}(hext(0))=pent^{-1}(0)=-1 */
hex(0)=1 /* hex(0,1) are by definition, 1 and the base=e  */
hex(1)=e

If one uses the simple linear extension between hex<-2,-1>, then there would be a singularity somewhere near hex(-3.86) since hex(-2.86)~=-1.85, and the fixed point for tetration base(e) is ~=-1.85.  Near that value, hex(z) would go to minus infinity.  One could imagine generating an analytic hexation function by using a bipolar theta mapping using the two primary complex conjugate fixed points of pentation.  This would require a fair amount of work.
-2.2597543772948619217694227344490 +/- 1.3844243840414798988939409459371*I

So does this pattern really hold? would hept(-4,-3,-2,-1,0,1)=(-3,-2,-1,0,1,e)?  Could it be analytic? Hexation probably has a real fixed point....
What about oct?  Oct(-5,-4,-3,-2,-1,0,1)=(-4,-3,-2,-1,0,e)?

Here is what hexation base(e) would look like from <-3.85,1.5> with the linear approximation on <-2,-1>.  I also show the line y=x for reference.
[attachment=1282]
   
- Sheldon
Reply


Messages In This Thread
pentation and hexation - by sheldonison - 08/07/2017, 07:33 PM
RE: pentation and hexation - by JmsNxn - 08/21/2017, 08:05 PM
RE: pentation and hexation - by sheldonison - 08/22/2017, 02:03 PM
RE: pentation and hexation - by JmsNxn - 08/22/2017, 10:38 PM
RE: pentation and hexation - by sheldonison - 09/03/2017, 10:11 PM
RE: pentation and hexation - by JmsNxn - 09/03/2017, 11:52 PM
RE: pentation and hexation - by sheldonison - 09/04/2017, 03:04 AM
RE: pentation and hexation - by JmsNxn - 09/04/2017, 04:07 AM
RE: pentation and hexation - by Ember Edison - 09/18/2019, 06:34 AM
RE: pentation and hexation - by sheldonison - 09/18/2019, 02:34 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Tetration is pentation. This deserve more thinking. marraco 2 3,893 03/30/2015, 02:54 PM
Last Post: marraco
  [2015] 4th Zeration from base change pentation tommy1729 5 6,471 03/29/2015, 05:47 PM
Last Post: tommy1729
  Mizugadro, pentation, Book Kouznetsov 41 51,017 03/02/2015, 08:13 PM
Last Post: sheldonison
  Infinite Pentation (and x-srt-x) andydude 20 28,102 05/31/2011, 10:29 PM
Last Post: bo198214
  Regular "pentation"? mike3 12 21,969 04/04/2011, 03:16 AM
Last Post: BenStandeven
Smile Pentation roots self but please you do... nuninho1980 2 7,275 11/03/2010, 12:54 PM
Last Post: nuninho1980
  Pentation's definitional ambiguity Base-Acid Tetration 14 23,036 12/15/2009, 11:23 PM
Last Post: Base-Acid Tetration
  Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation Base-Acid Tetration 19 32,347 10/24/2009, 04:12 AM
Last Post: andydude
  Exploring Pentation - Base e jaydfox 22 40,403 03/03/2008, 08:04 PM
Last Post: Ivars



Users browsing this thread: 1 Guest(s)