[repost] A nowhere analytic infinite sum for tetration.
#1
In the title I mentioned that this is a repost.
Indeed these ideas are not completely new and have been mentioned before.

However I Will reinvent my own wheel , by a different presentation of  the/my ideas ...

The basic principle is very old : infinite Sum and telescoping.

We want to solve t(exp(x)) = e t(x).

Define 

0 < a < 1.

f(x,a) = exp(x) + a x.

f^[n+1](x,a) = f( f^[n](x,a) ,a).

—-

Now consider

for Every real y there is a solution x to

f(x,a) = y.

Nomatter What a is , but still 0 < a < 1.

——

Now define the Sum from m = - Oo to Oo :

t(z,a) = ... + exp(z + 2)/f^[-2](1/2,a) + exp(z + 1)/f^[-1](1/2,a) + ...

So t(z,a) = sum_m exp(z - m)/f^[m](1/2,a).


Now let Lim a —> 0 :

z = x then :

Lim t(z,0) = t(z) = t(x)

t(exp(x)) = e t(x).

Qed

___

For practical reasons I suggest 

Compute for x < 1 :

t(x) by computing t(exp(x))/e.

——

We can continue

since 1/z + 1/exp(z) + 1/exp(exp(z)) + ...

Is nowhere analytic since exp iterations are chaotic we have :

t(x) is nowhere analytic.

Also

t(exp(x)) = t(x) e.

ln t(  exp(x)  ) = ln t(x) + 1

Thus

ln t(x) = slog(x).

A nowhere analytic slog(x) !

——-

Due to the telescoping infinite Sum , this mzthod is probably the simplest nowhere analytic solution in the spirit of calculus.

I assume it was worthy of Being repeated.

Regards

Tommy1729
Reply


Possibly Related Threads…
Thread Author Replies Views Last Post
  double functional equation , continuum sum and analytic continuation tommy1729 6 184 03/05/2023, 12:36 AM
Last Post: tommy1729
  bounded derivatives and semi-group iso ( repost ?? ) tommy1729 3 101 02/23/2023, 12:12 AM
Last Post: tommy1729
  i^i^i^... repost ? tommy1729 0 73 02/05/2023, 09:46 PM
Last Post: tommy1729
  another infinite composition gaussian method clone tommy1729 2 149 01/24/2023, 12:53 AM
Last Post: tommy1729
  Iteration with two analytic fixed points bo198214 62 7,323 11/27/2022, 06:53 AM
Last Post: JmsNxn
  Qs on extension of continuous iterations from analytic functs to non-analytic Leo.W 18 2,726 09/18/2022, 09:37 PM
Last Post: tommy1729
  Constructing an analytic repelling Abel function JmsNxn 0 443 07/11/2022, 10:30 PM
Last Post: JmsNxn
  Is tetration analytic? Daniel 6 891 07/08/2022, 01:31 AM
Last Post: JmsNxn
  Infinite tetration and superroot of infinitesimal Ivars 129 224,658 06/18/2022, 11:56 PM
Last Post: Catullus
  Interpolating an infinite sequence ? tommy1729 14 1,833 06/17/2022, 10:41 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)