Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Real and complex behaviour of the base change function (was: The "cheta" function)
#37
(08/21/2009, 10:49 PM)jaydfox Wrote: Given a starting and ending point, it is possible to create a path between them that starts and ends in the principal branch, even if it must sometimes leave the principal branch. I'm working on pictures of what I mean, so in the meantime, hopefully you can picture what I mean.
Normally not, but in this case, yes Wink

Quote:This has me concerned that the base-change formula is truly undefined for non-real values,

I was thinking about a different aproach: I would say all singularities of all are bounded. So out there in the complex plane are points which are distant from all singularities of all . Perhaps one can define the value of each at by using a path that has all singularities to the right side, respectively.
Then one would show that converges uniformly in a neighborhood of . The limit function in the neighborhood of is necesarily again holomorphic there.
From *there* we continue the function to the real line. I guess has only singularities on the real line and at the fixed points of .
Then it would turn out that has an *asymptotic* power series development (for a certain sector of approach) at all points of the (upper) real axis. Which though of course has zero convergence radius.
Reply


Messages In This Thread

Possibly Related Threads...
Thread Author Replies Views Last Post
  Complex Tetration, to base exp(1/e) Ember Edison 7 1,627 08/14/2019, 09:15 AM
Last Post: sheldonison
  Is there a function space for tetration? Chenjesu 0 342 06/23/2019, 08:24 PM
Last Post: Chenjesu
  Can we get the holomorphic super-root and super-logarithm function? Ember Edison 10 2,590 06/10/2019, 04:29 AM
Last Post: Ember Edison
  Degamma function Xorter 0 802 10/22/2018, 11:29 AM
Last Post: Xorter
  b^b^x with base 0<b<e^-e have three real fixpoints Gottfried 1 2,214 11/07/2017, 11:06 AM
Last Post: sheldonison
  An explicit series for the tetration of a complex height Vladimir Reshetnikov 13 11,408 01/14/2017, 09:09 PM
Last Post: Vladimir Reshetnikov
  Taylor series of cheta Xorter 13 11,958 08/28/2016, 08:52 PM
Last Post: sheldonison
  2 real fixpoints again ....... tommy1729 10 8,692 02/23/2016, 10:17 PM
Last Post: tommy1729
  Should tetration be a multivalued function? marraco 17 15,944 01/14/2016, 04:24 AM
Last Post: marraco
  Introducing new special function : Lambert_t(z,r) tommy1729 2 3,461 01/10/2016, 06:14 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)