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)
#39
(08/13/2009, 07:17 AM)bo198214 Wrote: Walker showed a similar convergence for , where dexp(x) = exp(x)-1.
He showed that the limit is infinitely differentiable on the real axis.
That means that he also wasnt clear about the complex behaviour otherwise he would have shown that the limit is holomorphic as a consequence of local uniform convergence.
But he could prove that local uniform (or compact) convergence only on the real axis, which does not suffice to imply holomorphy (because it could be that during the convergence non-real singularities get dense towads points on the real axis). I will persue this topic in the next days and have still some unexplored ideas at my hands.
Henryk, Do you have a reference for showing that the base change function is infinitely differentiable? Perhaps Walker's paper? I am now able to generate approximations for the Taylor series coefficients for the base change function. This would lead to a way of showing that the function is infinitely differentiable at the real axis, since all of the coefficients converge. But it would also lead to a way to show that the function is nowhere analytic, since the coefficients for large enough n eventually grow faster then any exponential. I wanted to read Walker's paper to see what his approach was, before posting my results.
- Sheldon
Reply


Messages In This Thread
RE: Does the limit converge in the complex plane? - by sheldonison - 08/13/2011, 10:32 AM

Possibly Related Threads...
Thread Author Replies Views Last Post
  New mathematical object - hyperanalytic function arybnikov 4 1,055 01/02/2020, 01:38 AM
Last Post: arybnikov
  Constructing real tetration solutions Daniel 4 995 12/24/2019, 12:10 AM
Last Post: sheldonison
  Complex Tetration, to base exp(1/e) Ember Edison 7 3,027 08/14/2019, 09:15 AM
Last Post: sheldonison
  Is there a function space for tetration? Chenjesu 0 667 06/23/2019, 08:24 PM
Last Post: Chenjesu
  Can we get the holomorphic super-root and super-logarithm function? Ember Edison 10 4,804 06/10/2019, 04:29 AM
Last Post: Ember Edison
  Degamma function Xorter 0 1,123 10/22/2018, 11:29 AM
Last Post: Xorter
  b^b^x with base 0<b<e^-e have three real fixpoints Gottfried 1 2,656 11/07/2017, 11:06 AM
Last Post: sheldonison
  An explicit series for the tetration of a complex height Vladimir Reshetnikov 13 13,616 01/14/2017, 09:09 PM
Last Post: Vladimir Reshetnikov
  Taylor series of cheta Xorter 13 14,401 08/28/2016, 08:52 PM
Last Post: sheldonison
  2 real fixpoints again ....... tommy1729 10 10,243 02/23/2016, 10:17 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)