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 Iterability of exp(x)-1 bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 08/11/2007, 09:33 PM (This post was last modified: 08/11/2007, 09:45 PM by bo198214.) Hm, this is strange. I just tried numerically the half iterate of $e^x-1$ and it looks quite convergent. However I read in the article of Erdös and Jabotinsky [1]: Quote:The function $e^z-1$ was shown by I. N. Baker [2] to have no real non-integer iterates. M. Levine [3] showed, using some results of the present paper, that this function and the functions $z+z^2$ and $\frac{z}{1-z^2}$ have no analytic iterate. Now I am confused. If someone wants to compare, I computed the half iterate of $f(x):=e^x-1$ to be $f^{\circ 1/2}(x)=x+{\frac {1}{4}}{x}^{2}+{\frac {1}{48}}{x}^{3}+{\frac {1}{3840}}{x}^{ 5}-{\frac {7}{92160}}{x}^{6}+{\frac {1}{645120}}{x}^{7}+{\frac {53}{ 3440640}}{x}^{8}-{\frac {281}{30965760}}{x}^{9}+O \left( {x}^{10} \right)$ I verified that indeed $f^{1/2}\circ f^{1/2}=f$ and it converges for example for $x=1.0$ Code:1.0,1.250000000,1.270833333,1.270833333,1.271093750,1.271017795,1.271019345,1.271034749,1.271025674,1.271025591,1.271029198,1.271027503 Where is the mistake? [1] P. Erdös and E. Jabotinksy, On analytic iteration, J. Analyse Math. 8, 1960/1961, 361-376. [2] I. N. Baker, Zusammensetzungen ganzer Funktionen, Math. Z 69, 1958, 121-163. [3] M. Levin, MSc. Thesis, Israel Institute of Technology, 1960. « 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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