Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Infinite tetration and superroot of infinitesimal
GFR Wrote:Actually, the beautiful formula I propose is:
ssqrt(x) = ln(x) / W(ln(x)), which, for x = 1/2, gives:

ssqrt(1/2) = ln(1/2) / W(ln(1/2)) = 0.26289282802173525.. + 0.4996694356833174.. i

Perfectly coherent with Henryk's formula.

GFR

Since 0^0 = 1 then The reason for 0^0=1 is explained here ( it is not 100% based on real number limits, actually, not at all, its based on symbolic calculations ( binomial theorem):

D.Knuth 2 notes on Notations

Quote:Evidently Libri’s main purpose was to show that unlikely functions can be expressed in algebraic terms, somewhat as we might wish to show that some complex functions can be computed by a Turing Machine. “Give me the function 0^0^x , and I’ll give you an expression for [x divides m].”

Most mathematicians agreed that 0^0 = 1, but Cauchy [5, page 70] had listed 0^0 together with other expressions like 0/0 and ∞−∞ in a table of undefined forms. Libri’s justification for the equation
0^0 = 1 was far from convincing, and a commentator who signed his name simply “S” rose to the attack [45]. August Mobius [36] defended Libri, by presenting his former professor’s reason for believing that 0^0 = 1 (basically a proof that limx→0+ x^x = 1). Mobius also went further and presented a supposed proof that limx→0+ f(x)^g(x) = 1 whenever limx→0+ f(x) = limx→0+ g(x) = 0.
Of course “S” then asked [3] whether Mobius knew about functions such as f(x) = e−1/x and g(x) = x. (And paper [36] was quietly omitted from the historical record when the collected works of Mobius were ultimately published.)
The debate stopped there, apparently with the conclusion that 0^0 should be undefined.

But no, no, ten thousand times no! Anybody who wants the binomial theorem to hold for at least one nonnegative integer n must believe that 0^0 = 1, for we can plug in x = 0
and y = 1 to get 1 on the left and 0^0 on the right.

The number of mappings from the empty set to the empty set is 0^0. It has to be 1.
On the other hand, Cauchy had good reason to consider 0^0 as an undefined limiting form, in the sense that the limiting value of f(x)^g(x) is not known a priori when f(x) and g(x) approach 0 independently. In this much stronger sense, the value of 0^0 is less defined than, say, the value of 0+0. Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth
was on their side.

If we use the formula:



for y=1






Then


Which not what x^x=y as x=0 says. It says

Differentiating at 1 according to l'Hopitals rule gives ( I hope I did not make a mistake):



No big help here. May be if we divide the powerseries for ln(1+x) and W(x) near 0 we get clear 0, since in these new powerseries the coefficient at would be 0, only powers of 0 remain, so the resulting series will equal 0 at x=0.

then as , in this case 3rd superroot of y=1 is NOT=0 , but 1.

4th superroot is again ln(1) =0 , 5th is 1 or ( perhaps) ln(e) etc.

So here even/odd superroots of 1 clearly divide, and it is consistent with formula for ssqrt(1).

Ivars
Reply


Messages In This Thread

Possibly Related Threads...
Thread Author Replies Views Last Post
  [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 1,085 03/20/2018, 12:16 AM
Last Post: tommy1729
  [MO] Is there a tetration for infinite cardinalities? (Question in MO) Gottfried 10 11,548 12/28/2014, 10:22 PM
Last Post: MphLee
  Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 4,997 05/06/2014, 09:47 PM
Last Post: tommy1729
  Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried 5 6,703 07/17/2013, 09:46 AM
Last Post: Gottfried
  Wonderful new form of infinite series; easy solve tetration JmsNxn 1 4,286 09/06/2012, 02:01 AM
Last Post: JmsNxn
  the infinite operator, is there any research into this? JmsNxn 2 5,357 07/15/2011, 02:23 AM
Last Post: JmsNxn
  Infinite tetration of the imaginary unit GFR 40 53,276 06/26/2011, 08:06 AM
Last Post: bo198214
  Infinite Pentation (and x-srt-x) andydude 20 25,297 05/31/2011, 10:29 PM
Last Post: bo198214
  Infinite tetration fractal pictures bo198214 15 22,579 07/02/2010, 07:22 AM
Last Post: bo198214
  Infinite towers & solutions to Lambert W-function brangelito 1 3,728 06/16/2010, 02:50 PM
Last Post: bo198214



Users browsing this thread: 1 Guest(s)