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 The upper superexponential bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 03/29/2009, 11:23 AM As it is well-known we have for $b the regular superexponential at the lower fixed point. This can be obtained by computing the Schroeder function at the fixed point $a$ of $F(x)=b^x$. More precisely we set $G(x)=F(x+a)-a = b^{x+a}-a=b^a b^x -a = a b^x -a = a(b^x-1)$ This is a function with fixed point at 0, it is the function $F$ shifted that its fixed point is at 0. We compute the Schroeder function $\chi$ of $G$, i.e. the solution of: $\chi(G(x))=c\chi(x)$ where $c=G'(0)=a\ln(b)=\ln(b^a)=\ln(a)$. This has a unique analytic solution with $\chi'(0)=1$. Then we get the super exponential by $\operatorname{sexp}_b(t)=a+\chi^{-1}(c^x \chi(y_0)$ $y_0$ is adjusted such that $1=\operatorname{sexp}_b(0)=a+\chi^{-1}(\chi(y_0))=a+y_0$ i.e. $y_0=1-a$. This procedure can be applied to any fixed point $a$ of $b^x$. The normal regular superexponential is obtained by applying it to the lower fixed point. Now the upper regular superexponential $\operatorname{usexp}$ is the one obtained at the upper fixed point of $b^x$. For this function we have however always $\operatorname{usexp}(x)>a$, so the condition $\operatorname{usexp}(0)=1$ can not be met. Instead we normalize it by $\operatorname{usexp}(0)=a+1$, which gives the formula: $\operatorname{usexp}_b(t)=a+\chi^{-1}\left(\ln(a)^x \chi(1)\right)$ The interesting difference to the normal regular superexponential is that upper on is entire, while the normal one has a singularity at -2 and is no more real for $x<-2$. It is entire because the inverse Schroeder function $\chi^{-1}$ is entire, it can be continued from an initial small disk of radius r around 0 By the equation $\chi^{-1}(c^n x)=G^{[n]}(\chi(x))$ We know that $c>1$ thatswhy we cover the whole complex plane with $c^nx$, $x$ from the initial disc around 0, and we know that $G^{[n}]$ is entire. Here are some pictures of $\operatorname{sexp}$ that are computed via the regular schroeder function as powerseries for our beloved base $b=\sqrt{2}$, $a=2,4$:     and here the upper super exponential base 2 alone: « Next Oldest | Next Newest »

 Messages In This Thread The upper superexponential - by bo198214 - 03/29/2009, 11:23 AM RE: The upper superexponential - by andydude - 03/31/2009, 04:29 AM RE: The upper superexponential - by sheldonison - 04/03/2009, 03:06 PM RE: The upper superexponential - by bo198214 - 04/03/2009, 04:22 PM RE: The upper superexponential - by sheldonison - 04/05/2009, 12:45 PM RE: The upper superexponential - by bo198214 - 04/06/2009, 06:35 AM RE: The upper superexponential - by Kouznetsov - 05/10/2009, 02:13 PM RE: The upper superexponential - by sheldonison - 05/11/2009, 12:55 PM RE: The upper superexponential - by bo198214 - 05/11/2009, 01:21 PM RE: The upper superexponential - by sheldonison - 05/11/2009, 08:12 PM RE: The upper superexponential - by bo198214 - 05/11/2009, 08:31 PM RE: The upper superexponential - by Kouznetsov - 05/12/2009, 08:54 AM RE: The upper superexponential - by bo198214 - 06/01/2009, 07:24 PM RE: The upper superexponential - by tommy1729 - 04/05/2009, 07:05 PM RE: The upper superexponential - by sheldonison - 04/22/2009, 05:02 PM RE: The upper superexponential - by bo198214 - 04/22/2009, 05:34 PM RE: The upper superexponential/eich-value x for h=0? - by Gottfried - 09/18/2009, 04:01 PM

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