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 [Regular tetration] [Iteration series] norming fixpoint-dependencies Gottfried Ultimate Fellow Posts: 767 Threads: 119 Joined: Aug 2007 08/29/2010, 10:31 PM (This post was last modified: 08/31/2010, 10:05 AM by Gottfried.) Just came across an older subject and thought it would fit into this "norming"-thread. As older fellows here may remember, nearly my first contact with tetration was the question of alternating iteration-series for which I worked out some interesting heuristics. (see [1] and [2]) [update] I should explain, that for convenient ascii-notation of the tetration I "misuse" here the common notation. With z^^h I mean in the context of a given fixed base b, the value of z^^h := $\exp_b^{oh}(z)$ [/update] Using base b=sqrt(2) we have the real-valued interval 2..4 for which we may find iteration heights from -inf to +inf if we start at some value z, say z=3, in this interval. Because in both direction the values of z^^h are finite we can compute a value for the alternating series of that values. So using Pari/Gp we can compute f(z) = sumalt(h=0,(-1)^h*iter(z,h)) + sumalt(h=0,(-1)^h*iter(z,-h) ) - z to evaluate the alternating iteration-series with center at the chosen z. It is clear that this series is periodic for z in the interval z..z^^2 . But what's interesting is, that in general the f(z) is "small" and even we find f(z)=0 Because this is a remarkable result (and matches, for instance, the analogue problem when applied to a doubly-infinite geometric series by analytic continuation) this value z (where f(z)=0) introduces itself gently as candidate for a norm-value, at which the height is defined to be zero or at least an integer. Here is a picture of the sinusoidal curve f(z) when z is moved from z to z^^2 beginning at some arbitrary value z0:     We see that astonishing approximation to a sine-curve, where the amplitude should be normed. Actually the deviance from the sine-curve is of the order of 1e-3 : I mean, if the height-parameter of this curve is compared with the abscissa of the sine-curve after the two curves are matched (for instance by binary search of the same y-values). I'm not experienced with Fourier-analysis, but I think, it would be profitable to try to describe the f(z)-function by a fourier-decomposition. Analoguously this could be done for the other bases 1

 Messages In This Thread [Regular tetration] [Iteration series] norming fixpoint-dependencies - by Gottfried - 07/28/2010, 03:04 PM RE: [Regular tetration] norming fixpoint-dependencies - by Gottfried - 07/28/2010, 09:49 PM RE: [Regular tetration] norming fixpoint-dependencies - by bo198214 - 07/29/2010, 04:53 AM RE: [Regular tetration] norming fixpoint-dependencies - by Gottfried - 07/29/2010, 11:36 AM RE: [Regular tetration] norming fixpoint-dependencies - by Gottfried - 08/29/2010, 10:31 PM RE: [Regular tetration] norming fixpoint-dependencies - by tommy1729 - 08/30/2010, 09:32 AM RE: [Regular tetration] norming fixpoint-dependencies - by Gottfried - 08/30/2010, 10:25 AM RE: [Regular tetration] norming fixpoint-dependencies - by tommy1729 - 08/30/2010, 11:08 AM RE: [Regular tetration] norming fixpoint-dependencies - by Gottfried - 08/30/2010, 12:34 PM RE: [Regular tetration] norming fixpoint-dependencies - by tommy1729 - 08/30/2010, 11:14 PM RE: [Regular tetration] norming fixpoint-dependencies - by Gottfried - 08/31/2010, 05:36 AM RE: [Regular tetration] norming fixpoint-dependencies - by tommy1729 - 08/31/2010, 11:55 PM

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