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 Iterating at fixed points of b^x Gottfried Ultimate Fellow Posts: 757 Threads: 116 Joined: Aug 2007 10/04/2007, 06:53 AM (This post was last modified: 10/04/2007, 11:26 AM by Gottfried.) Maybe I should -at least to restate my view of things- add the following remark. In tetration we do not append exponents to a tower, but bases. So the "partial towers" of an infinite tower are, using a start-value x and a base b (a) x, b^x, b^b^x, ...^ b^b^x and not (b) b, b^b, b^b^... This is crucial, I think. For (a) we get then, for instance for base b=sqrt(2) the two solutions x=2 -> lim h->oo {b,x}^^h =2 x=4 -> lim h->oo {b,x}^^h =4 A supporting argument for this view is also, that if x is already a tower of base b, then the heights are additive... (c.1) {b,x}^^m={b,{b,y}^^n}^^m = {b,y}^^(m+n) (c.2) {b,x}^^m={b,{b,{b,{b,y}^^n}^^n}^^n}^^n = {b,y}^^(4*n) and this is then also coherent with complex fixpoints and real bases as a multisolution problem, even for the limit for infinite heights of towers. This is also, how the matrix-operator-method works, when used for integer-tetration, although, for the finite integer height we may use associativity to change orders of summation and reflect the approach from the opposite direction. It is possibly a bit better expressed in my operators-treatise. ------------------------ Hmm, to avoid confusion, we should possibly talk of "depth" of a powertower instead of "height" to put the mental focus for the problem at the right side Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Iterating at fixed points of b^x - by bo198214 - 09/08/2007, 10:02 AM The fixed points of e^x - by bo198214 - 09/08/2007, 10:34 AM The fixed points of b^x - by bo198214 - 09/08/2007, 11:36 AM RE: Iterating at fixed points of b^x - by jaydfox - 09/12/2007, 06:23 AM RE: Iterating at fixed points of b^x - by bo198214 - 09/12/2007, 09:54 AM RE: Iterating at fixed points of b^x - by GFR - 10/03/2007, 11:03 PM RE: Iterating at fixed points of b^x - by Gottfried - 10/04/2007, 06:53 AM RE: Iterating at fixed points of b^x - by bo198214 - 10/04/2007, 01:30 PM RE: Iterating at fixed points of b^x - by bo198214 - 10/04/2007, 05:54 PM RE: Iterating at fixed points of b^x - by Gottfried - 10/04/2007, 11:05 PM RE: Iterating at fixed points of b^x - by GFR - 01/31/2008, 03:07 PM

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