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 Infinite tetration of the imaginary unit tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 06/24/2011, 12:25 PM sheldon , we dont know if the period is a brjuno number , which is why we are not certain if its a siegel disk. ( else i would have posted it myself ) i dont know much about brjuno numbers but it seems deciding if a algebraic number of degree 3 is a brjuno number is already partially unresolved , hence i doubt period 2.98... is easy. ( i also dont know much about herman rings , feel free to inform me ) although you are correct with probability 1 , we know how " tricky and weird " tetration can be. i would also like to point out that 1.7129i^z is periodic with 2pi i /ln(1.7129 i) = 3.57977339 + 1.22650561 i ( another mysterious number ) , such that 1.7129i^(z + 3.57977339 + 1.22650561 i) ends up in the same orbit as 1.7129i^z and this probably defines the boundary of the siegel disk ( because of the univalent riemann mapping ) and settles my question about large initial values like 1729 + 1729 i. so we have at least 2 types of periods : 2.98... and 3.57977339 + 1.22650561 i that dominate the bahaviour of the superfunction sexp_1.7129i(z) on the main branch. sexp_1.7129i(z) will have its fixpoints (of 1.7129i^z) at +/- oo i. i dare to consider ( not yet conjecture :p ) e§ = 1.7129i^z ln§ = ln(z) / ln(1.7129i) 2§ = e§ - 1/e§ for real x and some suitable real constant c sexp_1.7129i(x) = ln§ ln§ ln§ ... 2§^[x]( e§ e§ e§ ... © ) - with the correct branches - in analogue to my 2sinh method. ( plz dont copy the same arguments as in the sinh thread if irrelevant ) regards tommy1729 « 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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