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 complex iteration (complex "height") Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 03/23/2008, 07:27 PM (This post was last modified: 03/23/2008, 09:25 PM by Ivars.) I tried to do analytically one case: lower fixed point $a=\Omega=0.567143..$ we get the base by $b={1/e}=\Omega^{1/\Omega}$ if $z={1/\Omega}$ then ${1/e}[4]{1/\Omega}=\exp_{1/e}^{\circ 1/\Omega}(1)=\lim_{n\to\infty} \log_{1/e}^{\circ n}(\Omega*(1-\ln(\Omega)^{1/\Omega}) + \ln(\Omega)^{1/\Omega }\exp_{1/e}^{\circ n}(1))$ but : $\ln(\Omega)=-\Omega$ $\ln(\Omega)^{1/\Omega }=({1/\Omega })*\ln(\Omega)={-\Omega/\Omega }=-1$ ${1/e}[4]{1/\Omega}=\lim_{n\to\infty} \log_{1/e}^{\circ n}(\Omega*(1-(-1)) -1*\exp_{1/e}^{\circ n}(1))$ ${1/e}[4]{1/\Omega}=\lim_{n\to\infty} \log_{1/e}^{\circ n}(\Omega*2 -1*\exp_{1/e}^{\circ n}(1))$ At this point I do not know what to do with limits -how do You apply n to iterating logarithms, and at the same time to iteration of exponentiation of 1/e inside it? If we could move limit inside, than: $\lim_{n\to\infty} \exp_{1/e}^{\circ n}(1))=h({1/e)}=\Omega$ And $2*\Omega-\Omega=\Omega$ $\log_{1/e}(\Omega)= \Omega$ for all n, so ${1/e}[4]{1/\Omega}=\Omega$ which equals ${\Omega^{1/\Omega}[4]{1/\Omega}=\Omega$ I know it is not complex but real... Must be a mistake with the limits, but looks nice still... If someone could explain me on this example how I should have proceeded after the place where I got to limit taking inside iterated logarithm, I promise never to make the same mistake. If we take the same but: $z={-1/\Omega}$ Then I have to go through the whole procedure again. In this case, instead of $2*\Omega$ we will have 0, and + before tower as: $\ln(\Omega)^{-1/\Omega }=(-{1/\Omega })*\ln(\Omega)={-\Omega/-\Omega }=1$ ${1/e}[4]{-1/\Omega}=\lim_{n\to\infty} \log_{1/e}^{\circ n}(1*\exp_{1/e}^{\circ n}(1))$ this seems to equal 1. So with negative height: ${1/e}[4]{-1/\Omega}= 1$ ${\Omega^{1/\Omega}[4]{-1/\Omega}=1$ Ivars « Next Oldest | Next Newest »

 Messages In This Thread complex iteration (complex "height") - by Gottfried - 11/14/2007, 09:32 AM RE: complex iteration (complex "height") - by bo198214 - 03/21/2008, 05:21 PM RE: complex iteration (complex "height") - by Gottfried - 03/22/2008, 01:01 AM RE: complex iteration (complex "height") - by bo198214 - 03/22/2008, 12:10 PM RE: complex iteration (complex "height") - by Gottfried - 03/22/2008, 03:25 PM RE: complex iteration (complex "height") - by bo198214 - 03/23/2008, 01:28 AM RE: complex iteration (complex "height") - by Gottfried - 03/23/2008, 06:43 AM RE: complex iteration (complex "height") - by bo198214 - 03/23/2008, 08:47 AM RE: complex iteration (complex "height") - by Gottfried - 03/23/2008, 10:16 AM RE: complex iteration (complex "height") - by Ivars - 03/22/2008, 09:15 AM RE: complex iteration (complex "height") - by bo198214 - 03/22/2008, 11:59 AM RE: complex iteration (complex "height") - by Ivars - 03/22/2008, 07:45 PM RE: complex iteration (complex "height") - by bo198214 - 03/22/2008, 07:49 PM RE: complex iteration (complex "height") - by Gottfried - 03/22/2008, 08:23 PM RE: complex iteration (complex "height") - by Ivars - 03/22/2008, 08:56 PM RE: complex iteration (complex "height") - by Gottfried - 03/22/2008, 10:36 PM RE: complex iteration (complex "height") - by Ivars - 03/23/2008, 04:06 PM RE: complex iteration (complex "height") - by Gottfried - 03/22/2008, 03:32 PM RE: complex iteration (complex "height") - by Ivars - 03/26/2008, 06:22 PM RE: complex iteration (complex "height") - by bo198214 - 03/26/2008, 08:54 PM RE: complex iteration (complex "height") - by Ivars - 03/26/2008, 10:05 PM RE: complex iteration (complex "height") - by Ivars - 03/27/2008, 07:41 AM RE: complex iteration (complex "height") - by Ivars - 03/28/2008, 11:55 AM RE: complex iteration (complex "height") - by bo198214 - 03/28/2008, 12:56 PM RE: complex iteration (complex "height") - by Ivars - 03/28/2008, 03:13 PM

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