Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Extension of tetration to other branches
(10/24/2009, 08:13 PM)bo198214 Wrote:
(10/24/2009, 08:01 PM)mike3 Wrote: Actually it does seem to converge. The problem is that it seems to converge to the same value for every z in such cases. I.e., converging to a constant function.


Hmm. Sounds like it's time for a graph... I'll see if I can prepare a 2D one looking at the values of various branches on the real axis (can't do 3D with anything I've got).

(10/24/2009, 08:13 PM)bo198214 Wrote:
Quote: There are uncountably many such limit values, yet as constant functions they are "analytically incompatible" (is that a real term?) with the function (you can't analytically continue a constant function to tetration!)

Well, each constant fixed point of b^x is a tetration! I.e. it satisfies c(z+1)=b^c(z).
Does it converge to fixed points?

But it's a constant function, so it cannot be interpreted as analytic continuation of the specific function to another branch (think about the problem in "reverse": how would you analytically continue from this constant function to a non-constant one? You can't). And not all branches of satisfy . Consider the branch . We get . Note that it takes us back to the principal branch. The equation seems to only hold for all when using the principal branch (though there may be some freedom in the choice of cut of course), if we are interpreting the symbol as a specific single-valued branch. As you can see above, though, if we interpret it in a multivalued sense, that values on some branches of equal for values on some branches of , then it is true, as can be seen from the example I just showed. The situation is similar to that with and : for some branch of log for any given but which branch that is will depend on what is. "Multivalued functions" are funny things, you know?

Messages In This Thread
RE: Extension of tetration to other branches - by mike3 - 10/25/2009, 02:14 AM

Possibly Related Threads…
Thread Author Replies Views Last Post
  On extension to "other" iteration roots Leo.W 31 2,399 Yesterday, 06:45 AM
Last Post: JmsNxn
  Qs on extension of continuous iterations from analytic functs to non-analytic Leo.W 17 831 08/10/2022, 11:34 PM
Last Post: JmsNxn
  Tetration extension for bases between 1 and eta dantheman163 23 35,228 07/05/2022, 04:10 PM
Last Post: Leo.W
  Non-trivial extension of max(n,1)-1 to the reals and its iteration. MphLee 9 8,215 06/15/2022, 10:59 PM
Last Post: MphLee
  Ueda - Extension of tetration to real and complex heights MphLee 4 1,245 05/08/2022, 11:48 PM
Last Post: JmsNxn
  Possible continuous extension of tetration to the reals Dasedes 0 3,128 10/10/2016, 04:57 AM
Last Post: Dasedes
  Andrew Robbins' Tetration Extension bo198214 32 77,450 08/22/2016, 04:19 PM
Last Post: Gottfried
  alternative fixpoints = branches ? tommy1729 0 3,683 10/11/2014, 08:50 AM
Last Post: tommy1729
  Tetration Extension to Real Heights chobe 3 10,638 05/15/2010, 01:39 AM
Last Post: bo198214
  Dmitrii Kouznetsov's Tetration Extension andydude 38 69,899 11/20/2008, 01:31 AM
Last Post: Kouznetsov

Users browsing this thread: 1 Guest(s)