• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 03/26/2008, 11:51 AM (This post was last modified: 03/26/2008, 01:15 PM by Ivars.) The character of convergence of the mean to $-\Omega$ $f(x) = \ln(x) \text{ if } x>0$ $f(x) = \ln(-x) \text{ if }x<0$ $\lim_{n\to\infty}\frac{\sum_{n=1}^\infty f^{\circ n}(x)}{n}= -\Omega=-0.567143..=\ln(\Omega)$ can be seen from this graph where first 150 iterations of 100 points x=0.01 step 0.01 = 0.99 are plotted on top of each other. $f(x)_n = \ln(f(x)_{n-1}) \text{ if } f(x)_{n-1}>0$ $f(x)_n = \ln(-f(x)_{n-1}) \text{ if } f(x)_{n-1} <0$     The limited spread (max-min) of each iteration is also visible with such linear choice and size of steps. I guess this is somehow related to tree functions (which are related to Lambert function) and iterated logarithms. Just to remember always that studying $\Omega$ constant is studying Lambert Wo(1)= $\Omega$ so generalizations are possible if things are positioned correctly. So far I do not see them, but I see what to read next. Ivars « Next Oldest | Next Newest »

 Messages In This Thread Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x - by Ivars - 03/25/2008, 10:36 PM RE:Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega for all x - by Ivars - 03/25/2008, 10:45 PM RE: Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x - by Ivars - 03/26/2008, 11:51 AM RE: Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x - by bo198214 - 03/27/2008, 09:29 AM RE: Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x - by Ivars - 04/02/2008, 09:44 PM RE: Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x - by Ivars - 03/27/2008, 03:38 PM RE: Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x - by Ivars - 04/06/2008, 11:45 AM RE: Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x - by Ivars - 04/06/2008, 06:14 PM RE: Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x - by Ivars - 04/06/2008, 06:55 PM RE: Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x - by Ivars - 03/10/2009, 03:34 PM RE: Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x - by tommy1729 - 03/29/2015, 08:02 PM

 Possibly Related Threads... Thread Author Replies Views Last Post [MSE] Shape of orbit of iterations with base b on Shell-Thron-region Gottfried 12 1,237 11/11/2019, 05:05 PM Last Post: sheldonison Math overflow question on fractional exponential iterations sheldonison 4 3,353 04/01/2018, 03:09 AM Last Post: JmsNxn Periodic analytic iterations by Riemann mapping tommy1729 1 2,097 03/05/2016, 10:07 PM Last Post: tommy1729 Dangerous limits ... Tommy's limit paradox tommy1729 0 1,807 11/27/2015, 12:36 AM Last Post: tommy1729 tetration limit ?? tommy1729 40 45,200 06/15/2015, 01:00 AM Last Post: sheldonison Another limit tommy1729 0 1,496 03/18/2015, 06:55 PM Last Post: tommy1729 A limit exercise with Ei and slog. tommy1729 0 1,853 09/09/2014, 08:00 PM Last Post: tommy1729 [MSE] The mick tommy limit conjecture. tommy1729 1 2,463 03/30/2014, 11:22 PM Last Post: tommy1729 regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 Gottfried 7 8,724 06/25/2013, 01:37 PM Last Post: sheldonison tetration base conversion, and sexp/slog limit equations sheldonison 44 52,698 02/27/2013, 07:05 PM Last Post: sheldonison

Users browsing this thread: 1 Guest(s)