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 elementary superfunctions bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 04/23/2009, 01:25 PM (This post was last modified: 05/11/2009, 09:56 PM by bo198214.) Triggered by the interesting finding of Andrew I open this thread for the further investigation of elementary superfunctions, i.e. functions $F$ that are expressible with elementary functions and operations such that $F(x+1)=f(F(x))$ for a given elementary function $f$. Our first example is: $f(x)=2x^2-1$ with a superfunction $F(x)=\cos(2^x)$. Now the $\cosh$ has the same property $\cosh(2x)=2\cosh(x)^2 -1$ as the $\cos$. Hence $F(x)=\cosh(2^x)$ is another superfunction of $2x^2-1$. Indeed $f^{[t]}(x)=F(t+F^{-1}(x))$ exists and is differentiable at $x=1$. But it does not exist at the other fixed point $-\frac{1}{2}$, because $\operatorname{arccosh}\left(-\frac{1}{2}\right)$ is not defined. Edit: both are regular super-functions at fixed point 1. $\lim_{x\to-\infty} F(x)=1$. So if we are at polynomials $f$, we can also give an elementary superfunction for $f(x)=x^a$, i.e. $F(x)=c^{a^x}$. Because $F(x+1)=c^{a^xa}=F(x)^a$. Edit: these are the regular super-exponentials at 1. $\lim_{x\to-\infty} F(x)=1$. Generally for Chebyshev polynomials, these are the polynomials $T_n$ such that $\cos(nx)=T_n(\cos(x))$ - for example above we used $T_2(x)=2x^2-1$ -, we know already two elementary superfunctions of $f(x)=T_n(x)$, these are $F(x)=cos(n^x)$ and $F(x)=\cosh(n^x)$. « 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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