Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Derivative of exp^[1/2] at the fixed point?
I wanted to post what I've found out so far; which is not complete, and explain why only four derivatives are defined for Kneser's half iterate at the fixed point of L. First, let S(z) be the Schroeder function, is the Abel function, and be the formal half iterate which is not real valued at the real axis.

Then let be the real valued Kneser half iterate. Here's the closest I've gotten, where is a new 1-cyclic function whose coefficients can be derived from the 1-cyclic mapping used for the .

The constant term for , so that

Since , then the
, a taylor series in (z-L).
So what we have for the individual theta(z) terms is

p is the pseudo period of sexp

multiplier for S(z); also a function of (z-L)

update I think this is a complete form for the Kneser half iterate.

I left out many details including all of details about how to derive from . I haven't yet used these equation to calculate values for Kneser's half iterate in terms of the formal half iterate, and verify they match the expected values generated through other means. But the first 4 derivatives are zero at L, and the 5th derivative of has a singularity at L. That's all for now; next I would like to calculate some of the terms to test this equation out.
- Sheldon

Messages In This Thread
RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/29/2015, 10:25 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  tetration from alternative fixed point sheldonison 22 34,021 12/24/2019, 06:26 AM
Last Post: Daniel
  Semi-exp and the geometric derivative. A criterion. tommy1729 0 1,611 09/19/2017, 09:45 PM
Last Post: tommy1729
  How to find the first negative derivative ? tommy1729 0 1,726 02/13/2017, 01:30 PM
Last Post: tommy1729
  Are tetrations fixed points analytic? JmsNxn 2 3,573 12/14/2016, 08:50 PM
Last Post: JmsNxn
  A calculus proposition about sum and derivative tommy1729 1 2,311 08/19/2016, 12:24 PM
Last Post: tommy1729
  Derivative of E tetra x Forehead 7 10,462 12/25/2015, 03:59 AM
Last Post: andydude
  [MSE] Fixed point and fractional iteration of a map MphLee 0 2,428 01/08/2015, 03:02 PM
Last Post: MphLee
  A derivative conjecture matrix 3 5,085 10/25/2013, 11:33 PM
Last Post: tommy1729
  attracting fixed point lemma sheldonison 4 10,610 06/03/2011, 05:22 PM
Last Post: bo198214
  Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation Base-Acid Tetration 19 33,730 10/24/2009, 04:12 AM
Last Post: andydude

Users browsing this thread: 1 Guest(s)