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 elementary superfunctions bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 04/23/2009, 03:46 PM (This post was last modified: 05/11/2009, 10:04 PM by bo198214.) Next function $f(x)=cx^a$, the natural numbered iterates are: $f^{[1]}(x)=cx^a$ $f^{[2]}(x)=c(cx^a)^a=c^{a+1} x^{a^2}$ $f^{[3]}(x)=c\left(c^{a+1}x^{a^2}\right)^a = c^{a^2+a+1} x^{a^3}$ $f^{[n]}(x)=c^{\sum_{k=0}^{n-1}a^k} x^{a^n}=c^{\frac{a^n-1}{a-1}} x^{a^n}$ So a super-function would be $F(t)=c^{\frac{a^t-1}{a-1}}\exp(a^t)=\exp(a^t+\ln( c)\frac{a^t-1}{a-1})$ (*) $F(t)=\exp\left(\left(1+\frac{\ln( c)}{a-1}\right)a^t -\frac{\ln( c)}{a-1}\right)$ For the regular iteration we need to find a fixed point $\lambda$ $c\lambda^a = \lambda$ $c\lambda^{a-1}=1$ or $\lambda=0$ $\lambda=c^{-\frac{1}{a-1}}=c^{\frac{1}{1-a}}$, $a\neq 1$. Then (*) looks like: $F(x) = (e /\lambda)^{a^x} \lambda$ If we translate $F$ along the x-axis we can get $F(x)=\exp(a^x)\lambda$ Check: $cF(x)^a=\exp(a^x)^a c\lambda^a=\exp(a^{x+1})\lambda=F(x+1)$ Edit: It is regular at $\lambda$. Summary: $F(x)=\exp(a^x)c^{\frac{1}{1-a}}$ is the at $\lambda=c^{\frac{1}{1-a}}$ regular superfunction of $f(x)=cx^a$, for $a\neq 1$. « Next Oldest | Next Newest »

 Messages In This Thread elementary superfunctions - by bo198214 - 04/23/2009, 01:25 PM RE: elementary superfunctions - by bo198214 - 04/23/2009, 02:23 PM RE: elementary superfunctions - by bo198214 - 04/23/2009, 03:46 PM RE: elementary superfunctions - by tommy1729 - 04/27/2009, 11:16 PM RE: elementary superfunctions - by bo198214 - 04/28/2009, 08:33 AM RE: elementary superfunctions - by bo198214 - 03/27/2010, 10:27 PM RE: elementary superfunctions - by bo198214 - 04/18/2010, 01:17 PM RE: elementary superfunctions - by tommy1729 - 04/18/2010, 11:10 PM RE: elementary superfunctions - by bo198214 - 04/25/2010, 08:22 AM RE: elementary superfunctions - by Kouznetsov - 04/25/2010, 09:11 AM RE: elementary superfunctions - by bo198214 - 04/25/2010, 09:23 AM RE: elementary superfunctions - by bo198214 - 04/25/2010, 10:48 AM RE: elementary superfunctions - by Kouznetsov - 04/25/2010, 11:35 AM RE: elementary superfunctions - by bo198214 - 04/25/2010, 12:12 PM RE: elementary superfunctions - by Kouznetsov - 04/25/2010, 12:42 PM RE: elementary superfunctions - by bo198214 - 04/25/2010, 01:10 PM RE: elementary superfunctions - by Kouznetsov - 04/25/2010, 01:52 PM Super-functions - by Kouznetsov - 05/11/2009, 02:02 PM [split] open problems survey - by tommy1729 - 04/25/2010, 02:34 PM RE: [split] open problems survey - by bo198214 - 04/25/2010, 05:15 PM

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