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 Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 03/27/2008, 09:29 AM Ivars Wrote:$f(x) = \ln(x) \text{ if } x>0$ $f(x) = \ln(-x) \text{ if }x<0$ This can be easier expressed as $f(x)=\ln(|x|)$ Quote:$\lim_{n\to\infty}\frac{\sum_{n=1}^\infty f^{\circ n}(x)}{n}= -\Omega$ That seems indeed true. Basicly it has to do with that $\Omega$ is a (the only) fixed point of $x\to\ln|x|$. Proof: $x=\ln|x|$ $e^x=|x|$ this can not be satisfied for $x>0$ as $e^x$ has no fixed point. So we consider the case $e^x=-x$, which is equivalent to $1=(-x)e^{-x}$ which is by your definition $-x=\Omega$. However that fixed point is repelling $|\ln(|x|)'(-\Omega)|=|\frac{1}{x}(-\Omega)|=|-\frac{1}{\Omega}|>1$, thatswhy $f^{\circ n}(x)$ does not converge. On the other hand the iterated value is always thrown back into the negative domain by $\log(x)$ for positve $x$. And there it oscillates around the fixed point until it is again repelled into the positive domain. So most time it is oscillating around $-\Omega$, thatswhy the average is just this. « 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

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