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 elementary superfunctions bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 04/23/2009, 02:23 PM (This post was last modified: 05/11/2009, 10:00 PM by bo198214.) Peeping a bit into the Szekeres-Seminar somewhere on the forum mentioned by Andrew, and rearranging the elementary Schröder functions found by Schröder himself, I can add some more elementary superfunctions: 1. $f(x)=2(x+x^2)$, $F(x)=\frac{e^{2^x}-1}{2}$. Let us check: $F(x+1)=\frac{\left(e^{2^x}\right)^2 -1 }{2}$ and on the other hand $2(F(x)+F(x)^2)=e^{2^x}-1 + \frac{(e^{2^x}-1)^2}{2}=\frac{\left(e^{2^x}\right)^2 -1 }{2}$ So its indeed a superfunction. Edit: It is regular at fixed point $0$: $\lim_{x\to-\infty} \frac{e^{2^{x}}-1}{2} = 0$ 2. $f(x)=4(x+x^2)$, $F(x)=\sinh\left(2^x\right)^2$ Let us check: $F(x+1)=\sinh\left(22^x\right)^2=4\sinh(2^x)^2\cosh(2^x)^2=4\sinh(2^x)^2(1 + \sinh(2^x)^2)=4(F(x)+F(x)^2)$ It is again regular at 0. « 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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