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 complex iteration (complex "height") Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 03/23/2008, 10:08 PM (This post was last modified: 03/24/2008, 04:34 PM by Ivars.) Some complex value tetration: If I have understood right (if not, tell me please) , basic idea to be able to have some analytics relations is to make $ln(a)^z = 1$ or $z*ln(a) =1$ and if we have chosen complex or real $z$ then we can find $a$ such that : $ln(a) =1/z , b= a^{(1/a)}$ So every time $a^{(1/a)} [4] {1/ln(a)} = 1$ E.g. lower fixed point $a=e^-I=cos(1) -I*sin(1)=0.5403023-I*0.8414098$ we get the base by $b=(e^-I)^{(e^I)}$ if $z=I$ then $(e^{-I})^{(e^I)}[4]I =\exp_b^{\circ I}(1)=\lim_{n\to\infty} \log_b^{\circ n}((e^{-I})*(1-\ln(e^{-I})^I) + \ln(e^{-I})^I \exp_b^{\circ n}(1))$ but : $\ln(e^{-I})=-I$ $\ln(e^{-I})^I=I*\ln(e^{-I})=I*-I=1$ $(e^{-I})^{(e^I)}[4]I =\exp_b^{\circ I}(1)=\lim_{n\to\infty} \log_b^{\circ n}((e^{-I})*(1-1) + 1* \exp_b^{\circ n}(1))$ $(e^{-I})^{(e^I)}[4]I =\exp_b^{\circ I}(1)=\lim_{n\to\infty} \log_b^{\circ n}(\exp_b^{\circ n}(1))$ this seems to equal 1. So with Imaginary height: $(e^{-I})^{(e^I)}[4] I= 1$ Few more interesting outcome : $I^{1/I} = e^{\pi/2}[4] {-I*2/\pi}=1$ $e^{1/e}[4] 1 =1$ Which seems wrong.So where all others, then... Or actually a=e and base $b=e^{1/e}$ seems to be rather unique point in selfroots as it is the only one for a>1 which has only 1 value, so h(e^(1/e))=e and that is the only value ( all other points for a>1 has 2 values for h(a^(1/a))), because 2 different a on different sides of a=e have the same selfroot values). So if we speculate a little to continue with 2 values as a demand for $h(a^{1/a}) a>=1$ , we may propose that: $e^{1/e}[4] 1 =1$ $e^{1/e}[4] 1 =e^{1/e}$ $h( e^{1/e}) =e$ $h( e^{1/e}) =1$ $h(1)=1$ $h(1)=\infty$ Or, if $ln(e) = 1+I*2\pi*k$ $e^{1/e}[4] 1 =e^{1/e}$ $e^{1/e}[4] {1/(1+2\pi*I*k)} =1$ $1^1[4] \infty =1$ $1^1[4] {1/(2\pi*I*k)} =1$ But it was good tex training... 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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