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 Infinite tetration of the imaginary unit Gottfried Ultimate Fellow Posts: 757 Threads: 116 Joined: Aug 2007 06/22/2011, 09:28 AM (This post was last modified: 06/22/2011, 10:10 AM by Gottfried.) (Further looking at the (non-) periodicity...) We can write the iteration of b_p more convenient. Remember my convention Code:. p   = Pi/2                                   ~ 1.57079632679 u_p = e^(p*I)  = cos(p) + I*sin(p)           = 1*I t_p = exp(u)   = e^e^(p*I)                   ~ 0.540302305868 + 0.841470984808*I b_p = exp(u/t) = e^(e^(p*I - e^(p*I))     )  ~ 1.98933207608 + 1.19328219947*I With this we can formulate another recursion, simple and based on the angular parameter p: (I swith to tex-notation because of the use of indices) $ \hspace{48} p_1 = p*I - \exp(p*I) = -\cos(p) + I*(p - \sin(p)) \\ \hspace{48} p_h = p_1 + \exp(p_{h-1}) \\$ where always $ \hspace{48} b_h = b^\text{\^\^h } = \exp(\exp(p_h))$ The iteration p_h has here the nested form to the depth h: $ \hspace{48} p_h = p_1 + \exp(p_1 + \exp(p_1 + \exp(p_1+ ...))))$ and it looks very promising, that this can be periodic or even constant only in few and possible exotic cases. Surely, that does not automatically mean, that $b^\text{\^\^h }= \exp(\exp(p_h))$ cannot be periodic, as the example of h and sin(h) shows, but it is a strong hint. Additionally, we can also insert an alternative startexponent z0 as given in my other examples; we have then a small modification: $ \hspace{48} p_1 = p*I - \exp(p*I) \\ \hspace{48} p_2 = p_1 + \log(z0) \\ \hspace{48} p_h = p_1 + \exp(p_{h-1}) \text{ // for h>2 } \\ \\ \hspace{24} \text{ where } \\ \hspace{48} b_2 = b^{z_0} \\ \hspace{48} b_h = \exp(\exp(p_h)) = b^{b_{h-1}}$ Here it seems interesting, that by the recursion the initial value z0 disappears into the tail (notational) of the p1+exp(p1+exp(...))) - expression. Also I doubt by this now, that there could be a substantial difference between the irrational and the rational periods - but well, we have the difference between binomials of integer and non-integer parameters, so.... Hmmm.... Gottfried Added another picture: (Hmm, now it would be interesting to trace the trajectories backwards...) Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 12:09 AM RE: Infinite tetration of the imaginary unit - by Ivars - 02/10/2008, 09:52 AM RE: Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 11:35 AM RE: Infinite tetration of the imaginary unit - by Ivars - 02/10/2008, 01:28 PM RE: Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 08:53 PM RE: Infinite tetration of the imaginary unit - by Ivars - 02/10/2008, 09:10 PM RE: Infinite tetration of the imaginary unit - by Ivars - 02/14/2008, 08:25 PM RE: Infinite tetration of the imaginary unit - by Ivars - 02/15/2008, 05:35 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/19/2011, 09:22 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/20/2011, 05:27 AM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 06:20 AM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 01:36 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/20/2011, 02:46 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/21/2011, 12:13 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/21/2011, 03:02 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/21/2011, 08:00 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/22/2011, 09:28 AM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/22/2011, 03:23 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/23/2011, 08:55 AM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/23/2011, 01:13 PM solved -- they're called Siegel discs - by sheldonison - 06/23/2011, 06:10 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 06:11 AM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/20/2011, 10:22 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/21/2011, 01:58 AM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/21/2011, 11:19 PM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/22/2011, 03:15 PM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/22/2011, 12:25 PM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/22/2011, 07:05 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/24/2011, 08:07 AM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/24/2011, 12:25 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/24/2011, 04:36 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/24/2011, 09:44 PM RE: Infinite tetration of the imaginary unit - by bo198214 - 06/26/2011, 08:06 AM

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