Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
"Kneser"-like mapping also for complex base?
#3
(11/05/2010, 09:56 AM)mike3 Wrote: Both fixed points are fixed points of the associated logarithm. This behavior suggests that it may be possible to form the tetrational at any complex base out of the two regular iterations of the two fixed points of logarithm.....
I think the case Mike brings up involves both an attracting fixed point and a repelling fixed point. In my earlier post, I was implicitly assuming two repelling fixed points.
Repelling fixed point
-1.135218577668813217726900190 - 1.989607927210830803666191570i
Attracting fixed point
0.5793702071827351250349231804 + 0.6472265454524960204764987313i

If we start with a solution from the repelling fixed point (which is guaranteed to be entire), and then we do a superfunction(z+theta(z)) kneser mapping, to allow sexp(0)=1, but the mapping would only be defined for 1/2 of the complex plane, due to the singularity in theta(z). There are infinitely many such mappings.

So I think the question is, can this mapping equal another theta(z) mapping of the solution from the attracting fixed point? Where the attracting fixed point mapping is defined on the other half of the complex plane, and the two mappings are equal at the real axis, with sexp(0)=1, sexp(1)=B.....
- Sheldon
Reply


Messages In This Thread
RE: "Kneser"-like mapping also for complex base? - by sheldonison - 11/07/2010, 01:37 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Ueda - Extension of tetration to real and complex heights MphLee 2 378 12/03/2021, 01:23 AM
Last Post: JmsNxn
  Arguments for the beta method not being Kneser's method JmsNxn 54 9,429 10/23/2021, 03:13 AM
Last Post: sheldonison
  tommy's singularity theorem and connection to kneser and gaussian method tommy1729 2 675 09/20/2021, 04:29 AM
Last Post: JmsNxn
  Generalized Kneser superfunction trick (the iterated limit definition) MphLee 25 9,160 05/26/2021, 11:55 PM
Last Post: MphLee
  Alternative manners of expressing Kneser JmsNxn 1 1,040 03/19/2021, 01:02 AM
Last Post: JmsNxn
  Questions about Kneser... JmsNxn 2 1,392 02/16/2021, 12:46 AM
Last Post: JmsNxn
  Kneser method question tommy1729 9 10,381 02/11/2020, 01:26 AM
Last Post: sheldonison
  Complex Tetration, to base exp(1/e) Ember Edison 7 10,371 08/14/2019, 09:15 AM
Last Post: sheldonison
  An explicit series for the tetration of a complex height Vladimir Reshetnikov 13 24,955 01/14/2017, 09:09 PM
Last Post: Vladimir Reshetnikov
  Periodic analytic iterations by Riemann mapping tommy1729 1 4,679 03/05/2016, 10:07 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)