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 regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 Gottfried Ultimate Fellow Posts: 787 Threads: 121 Joined: Aug 2007 06/22/2013, 12:52 PM (This post was last modified: 06/22/2013, 01:15 PM by Gottfried.) In the regular tetration we use the Schröder-function for the linearization of the fractional heights-computation beginning at some x0. Experimenting with it, say tetration with the base $b=\sqrt 2$ we have a lower fixpoint $t_0=2$. Here the values of the Schröder-function for $x \gt t_0$ are positive and that of $x \lt t_0$ are negative. Since the effect of changing sign of the Schröder-value is the same as using an imaginary component to the height-parameter, we can say, that iterations from the region above the fixpoint $t_0$ down to that below that fixpoint can be achieved by an imaginary height - so, in some sense, the imaginary oversteps the infinite iteration-height. But this allows to define a pairwise relation between x-values, whose Schröder-values have opposite signs. So $x=1$(below the fixpoint) has the negative schröder-value from $x=2.46791405...$ (above the fixpoint). Let's call the two related points "duals" of each other. One can find that the dual of $x=- \infty$ for base $b=\sqrt 2$ is about $x_w \approx 2.76432104000012572327981201783...$ Does someone "know" this value and knows (or with some seriousness guesses) more properties of this value? For instance, what is then the dual of that $\log_b(x_w) \approx 2.93385035151$ : is this $\log_b(-\infty)$ ? Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 - by Gottfried - 06/22/2013, 12:52 PM RE: regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 - by tommy1729 - 06/22/2013, 08:43 PM RE: regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 - by Gottfried - 06/22/2013, 09:55 PM RE: regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 - by sheldonison - 06/23/2013, 11:13 AM RE: regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 - by Gottfried - 06/23/2013, 10:20 PM RE: regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 - by sheldonison - 06/24/2013, 04:09 PM RE: regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 - by Gottfried - 06/25/2013, 10:45 AM RE: regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 - by sheldonison - 06/25/2013, 01:37 PM

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