Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Brute force tetration A_k(s) is analytic !
#1
I propose an analytic tetration here.

One very similar to other ideas from this year.

Imo very natural and just requiring alot of simple computions. I dare say brute force.

Consider for real k > 5 and Re(s) > 0 :



We can solve this by infinite composition : 
( yes to be formal we need a " z parameter " too, but that is a detail )

Now consider the functions

The limit k to +oo gives us A_oo(s) = A(s).

Now A(s) satisfies for Re(s) > 0 :

A(0) = exp( -exp(-oo) + exp(-oo + 0)) = exp(0 + exp(-oo)) = exp(0) = 1.

and 

A(s+1) = exp(A(s) + 0) = exp(A(s))

So assuming the limit exists this is analytic tetration on the halfplane Re(s) > 1.

By analytic continuation ( taking log's ) this is analytic tetration !!

In fact I think it has no singularities for Re(s) > 1 making it perhaps satisfy a uniqueness condition ??

The limit is designed to ( when it exists ) grow slowly on the complex plane (in the imaginary direction ) for Re(s) > 1 therefore probably remaining analytic.

We could in principle even approximate this in say excel.

regards

Tom Marcel Raes

tommy1729
Reply


Messages In This Thread
Brute force tetration A_k(s) is analytic ! - by tommy1729 - 03/18/2021, 01:08 AM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Nixon-Banach-Lambert-Raes tetration is analytic , simple and “ closed form “ !! tommy1729 11 3,026 02/04/2021, 03:47 AM
Last Post: JmsNxn
  [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 2,770 03/20/2018, 12:16 AM
Last Post: tommy1729
Question Analytic matrices and the base units Xorter 2 5,027 07/19/2017, 10:34 AM
Last Post: Xorter
  Non-analytic Xorter 0 2,822 04/04/2017, 10:38 PM
Last Post: Xorter
  A conjectured uniqueness criteria for analytic tetration Vladimir Reshetnikov 13 21,018 02/17/2017, 05:21 AM
Last Post: JmsNxn
  Is bounded tetration is analytic in the base argument? JmsNxn 0 2,839 01/02/2017, 06:38 AM
Last Post: JmsNxn
  Are tetrations fixed points analytic? JmsNxn 2 5,891 12/14/2016, 08:50 PM
Last Post: JmsNxn
  The bounded analytic semiHyper-operators JmsNxn 2 6,432 05/27/2016, 04:03 AM
Last Post: JmsNxn
  Periodic analytic iterations by Riemann mapping tommy1729 1 4,273 03/05/2016, 10:07 PM
Last Post: tommy1729
  Bounded Analytic Hyper operators JmsNxn 25 36,840 04/01/2015, 06:09 PM
Last Post: MphLee



Users browsing this thread: 1 Guest(s)