Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Iterating at fixed points of b^x
#26
GFR Wrote:Then I decided to use Mathematica and I got:
7#3 = 3.759823526783788538...x 10^695974 = 3759........2343, an integer number with 695975 figures, covering a printout of about 87 DIN A4 pages.
Therefore, yesterday, I decided to follow a lower profile and tried with a more reasonable and "famous" base, i.e.: rho = 4.810477381... . Well, always using Mathematica, I got:
rho#3 = 6.8101069808199648...X10^1304, which is much more civilized. It is not an integer number, but a DIN A4 page will be enought to show it, with a reasonable precision.
Then, this morning, after my breakfast, I tried to imagine how large could be rho#4 and I had to drink four cups of coffee, to recover. Now (perseverare ... diabolicum) I cannot avoid thinking of what could be rho#100 or rho#1000 or, even, [n->+oo]lim (rho#n) !
Its really fun to read your post.
Quote:At the same time, b^x = x (i.e. b^x - x = 0) can be seen as an implicit functional equation, the solutions of which should represent, after infinite iterations, an infinite tetration (infinite tetrates) of base b, which can be written as : y = b#(+oo) = h, the heights of the "infinite tower" with base b.
...
(2) - Kritik of the Mathematical Reason

It is expected that the critical point will be to say that one thing is the limit of rho^n, for n-> oo, and another thing is the determination of the fixpoint of rho^x = x. The countercritical position could be to say that, in this particular case, the two procedures must reach the same non-contradictory results (divine surprise!).

But thats not a critique of *mathematical* reasoning but a critique of *human* reasoning!

Mathematical reason goes like that:
Let be a real (or complex) continuous function and suppose that exists (and this does not include infinity) then .

Proof:
Define the sequence . Then by assumption but we also have by assumption that .
And as is continuous . q.e.d.

Also the opposite implication, that from would follow , is not true! Thats just the human temptation to always revert implications.

Quote:The apparent overlapping of the two strategies is that also +oo seems to be a fixpoint of y = b^y , in fact b^(+oo) = +oo. But it is not quite so, since b^(+oo) is of an infinite order much larger than +oo. In other words, for any b > eta [eta = e^(1/e)], b^x >> x, for x -> +oo, the real plots of y = b^x and y = x will never cross eachother for x -> +oo (I mean, before x = +oo). So, what! They will cross ... after ?? But, after what?! After ... (I don't dare to say) ... infinite?

Yeah the good old infinity. Unfortunately you can not just simply add infinity to the real or complex numbers (it leads to contradictions).
They only thing I can add are two references (from my other forum) to similar ideas which however are rather in an undeveloped state:
Idea stub 1 which evolved from this thread.
(There is especially for Ivars Wink a thread discussing hyperreals on this forum.)
And idea stub 2.

Quote:Please read the attached pdf notes and tell me, if you wish, your reactions. ... Or, please destroy them ... before reading, if you prefer so.
Dunno whether it was a destruction, but I indeed didnt read your attachment yet, but will do it now Smile
Reply


Messages In This Thread
Iterating at fixed points of b^x - by bo198214 - 09/08/2007, 10:02 AM
The fixed points of e^x - by bo198214 - 09/08/2007, 10:34 AM
The fixed points of b^x - by bo198214 - 09/08/2007, 11:36 AM
RE: Iterating at fixed points of b^x - by jaydfox - 09/12/2007, 06:23 AM
RE: Iterating at fixed points of b^x - by GFR - 10/03/2007, 11:03 PM
RE: Iterating at fixed points of b^x - by GFR - 01/31/2008, 03:07 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Are tetrations fixed points analytic? JmsNxn 2 2,670 12/14/2016, 08:50 PM
Last Post: JmsNxn
  Removing the branch points in the base: a uniqueness condition? fivexthethird 0 1,440 03/19/2016, 10:44 AM
Last Post: fivexthethird
  Derivative of exp^[1/2] at the fixed point? sheldonison 10 9,457 01/01/2016, 03:58 PM
Last Post: sheldonison
  [MSE] Fixed point and fractional iteration of a map MphLee 0 1,993 01/08/2015, 03:02 PM
Last Post: MphLee
  iterating x + ln(x) starting from 2 tommy1729 2 2,523 04/29/2013, 11:35 PM
Last Post: tommy1729
  tetration from alternative fixed point sheldonison 21 27,013 12/06/2011, 02:43 PM
Last Post: sheldonison
  attracting fixed point lemma sheldonison 4 9,160 06/03/2011, 05:22 PM
Last Post: bo198214
  cyclic points tommy1729 3 4,222 04/07/2011, 07:57 PM
Last Post: JmsNxn
  iterating non-analytic tommy1729 0 1,713 02/08/2011, 01:25 PM
Last Post: tommy1729
  Branch points of superlog mike3 0 2,194 02/03/2010, 11:00 PM
Last Post: mike3



Users browsing this thread: 1 Guest(s)