• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Derivative of exp^[1/2] at the fixed point? sheldonison Long Time Fellow Posts: 633 Threads: 22 Joined: Oct 2008 12/25/2015, 04:05 PM (This post was last modified: 12/25/2015, 04:24 PM by sheldonison.) (12/24/2015, 03:25 AM)sheldonison Wrote: Numerical experiments suggest that the derivative of Knesser's real valued half iterate is continuous at L, and the first derivative at the singularity at L is $\sqrt{L}$, and that the 2nd derivative may also match the formal half iterate. I think the key equations in understanding this behavior are the slog, the sexp, the Schroeder equation at L, and the Abel equation generated from the Schroeder equation, and the theta mapping from the Abel equation to the slog. Kneser's exp^{0.5}: $\exp^{0.5}(z) = \text{sexp}(\text{slog}(z) + 0.5)$ Here is the formal exp^{0.5} at L, generated from the formal Schroeder equation developed at the fixed point L $S(z)$ where: $S(\exp(z)) = S(z)\cdot L\;\;\;S(z+L)=z+a_2\cdot z^2 + a_3 \cdot z^3 ...$ Then the formal exp^{0.5} at L is exactly the same as: $S^{-1}\left( S(z) \cdot \sqrt{L} \right)$ The next step is to show the Abel equation, developed from the Schroeder equation; where $\alpha\left(\exp(z)\right) = \alpha(z)+1$ $\alpha(z) = \log_L\left(S(z)\right)$ And Kneser's slog(z) developed from the Abel equation is: $\text{slog}(z) = \alpha(z) + \theta\left(\alpha(z)\right)\;\;$ where $\theta(z)$ is a 1-cyclic function decaying to zero at $\Im \infty$ - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/23/2015, 04:39 PM RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/24/2015, 03:25 AM RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/25/2015, 04:05 PM RE: Derivative of exp^[1/2] at the fixed point? - by andydude - 12/27/2015, 11:15 AM RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/27/2015, 11:40 PM RE: Derivative of exp^[1/2] at the fixed point? - by andydude - 12/29/2015, 10:51 PM RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/29/2015, 10:25 PM RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/31/2015, 11:02 AM RE: Derivative of exp^[1/2] at the fixed point? - by tommy1729 - 12/31/2015, 01:25 PM RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 01/01/2016, 03:58 PM RE: Derivative of exp^[1/2] at the fixed point? - by tommy1729 - 12/30/2015, 01:27 PM

 Possibly Related Threads... Thread Author Replies Views Last Post tetration from alternative fixed point sheldonison 22 28,705 12/24/2019, 06:26 AM Last Post: Daniel Semi-exp and the geometric derivative. A criterion. tommy1729 0 1,305 09/19/2017, 09:45 PM Last Post: tommy1729 How to find the first negative derivative ? tommy1729 0 1,424 02/13/2017, 01:30 PM Last Post: tommy1729 Are tetrations fixed points analytic? JmsNxn 2 2,948 12/14/2016, 08:50 PM Last Post: JmsNxn A calculus proposition about sum and derivative tommy1729 1 1,956 08/19/2016, 12:24 PM Last Post: tommy1729 Derivative of E tetra x Forehead 7 9,328 12/25/2015, 03:59 AM Last Post: andydude [MSE] Fixed point and fractional iteration of a map MphLee 0 2,108 01/08/2015, 03:02 PM Last Post: MphLee A derivative conjecture matrix 3 4,453 10/25/2013, 11:33 PM Last Post: tommy1729 attracting fixed point lemma sheldonison 4 9,592 06/03/2011, 05:22 PM Last Post: bo198214 Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation Base-Acid Tetration 19 30,457 10/24/2009, 04:12 AM Last Post: andydude

Users browsing this thread: 1 Guest(s)