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 Iterability of exp(x)-1 bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 08/11/2007, 11:42 PM Daniel Wrote:exp(x)-1 definately has non-integer iterated; see heirarchies of height 1/2 at http://tetration.org/Combinatorics/Schro...index.html which is listed in the OEIS as A052122. I don't have access to my computer, but it looks like our results for heirarchies of height 1/2 agree. I also have the general solution which checks with OEIS entries for heirarchies of height -2, -1, 1/2, 1, 2, 3 and 4. The general formula comes from the double binomial expansion, ${f^{\circ s}}_n=\sum_{i=0}^{n-1} (-1)^{n-1-i}\left(s\\i\right)\left(s-1-i\\n-1-i\right){f^{\circ i}}_{n}$ The formula is reliable, I just computed it for the case s=1/2, to exemplify convergence. However I just looked in Baker's Paper and indeed he states (as a German native I just translate it): Quote:Proposition 17. Let $F(z)=e^z-1$; for each real $\sigma$ let $F_\sigma(z)$ be the by $F(F_\sigma(z))=F_\sigma(F(z))$ uniquely determined formal series which has the form $F_\sigma(z)=z+\frac{\sigma}{2}z^2+\sum_{m=3}^\infty a_m(\sigma)z^m$. Then $F_\sigma(z)$ has a positive radius of convergence if and only if $\sigma$ is an integer number. $F_\sigma(z)$ is the $n$-th iterate of $F(z)$ for integer $\sigma=n>0$, hence an entire function. $F_\sigma(z)$ is the inverse series development of $F_{-n}$ for integer $\sigma=n<0$. So instead just of to numerically verify, can we prove that $(e^x-1)^{\circ 1/2}$ converges for some $x>0$? « Next Oldest | Next Newest »

 Messages In This Thread Iterability of exp(x)-1 - by bo198214 - 08/11/2007, 09:33 PM RE: Iterability of exp(x)-1 - by Daniel - 08/11/2007, 09:49 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/11/2007, 11:42 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/12/2007, 09:07 AM RE: Iterability of exp(x)-1 - by jaydfox - 08/12/2007, 04:41 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/13/2007, 08:54 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/11/2007, 10:25 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/13/2007, 09:06 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/13/2007, 09:13 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/13/2007, 09:16 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/13/2007, 09:33 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/13/2007, 10:00 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/13/2007, 10:05 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/13/2007, 10:11 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/13/2007, 10:22 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/13/2007, 10:00 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/13/2007, 10:29 PM RE: Iterability of exp(x)-1 - by andydude - 08/13/2007, 10:30 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/13/2007, 10:39 PM RE: Iterability of exp(x)-1 - by andydude - 08/15/2007, 08:36 AM RE: Iterability of exp(x)-1 - by bo198214 - 08/15/2007, 09:21 AM RE: Iterability of exp(x)-1 - by andydude - 08/15/2007, 08:40 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/15/2007, 08:54 AM RE: Iterability of exp(x)-1 - by jaydfox - 08/15/2007, 08:53 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/15/2007, 09:13 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/20/2007, 04:14 PM RE: Iterability of exp(x)-1 - by andydude - 09/05/2007, 08:15 PM RE: Iterability of exp(x)-1 - by bo198214 - 09/07/2007, 02:45 PM RE: Iterability of exp(x)-1 - by Gottfried - 03/15/2008, 09:13 AM RE: Iterability of exp(x)-1 - by bo198214 - 03/15/2008, 01:14 PM RE: Iterability of exp(x)-1 - by Gottfried - 03/15/2008, 08:25 PM RE: Iterability of exp(x)-1 - by Gottfried - 03/18/2008, 10:14 AM

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