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 regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 Gottfried Ultimate Fellow Posts: 873 Threads: 128 Joined: Aug 2007 06/22/2013, 09:55 PM (This post was last modified: 06/22/2013, 10:51 PM by Gottfried.) Well, for the case it is needed, here is some more explanation (see a more general remark at the end). In the "regular tetration" (this is that method, where we use the exponential series wich is recentered around a fixpoint, say the lower (attracting) fixpoint $t_0$ ) we realize the tetration to fractional heights via the "Schröder"-function (see wikipedia), say $\hspace{48} s_0= S (x - t_0)$ where $S(\cdot)$ denotes the Schröder function. After that we calculate the "height"-parameter, say "h" into it, where "h" goes into the exponent of the log of the fixpoint: $\hspace{48} r_0 = s_0 \cdot \log(t_0)^h$ Then we use the inverse Schröder-function to find the value $x_h$ which is the (fractional) h'th iterate "from" $x_0$ : $\hspace{48} x_h = S^{\circ -1} (r_0) + t_0$ Now if we let the "height" h equal zero, thus no iteration, but just change the sign of $r_0 = -s_0$ then we get a "dual" $\tilde x_0$ , where if $x_0$ is between 2 and 4, then the dual is below 2, and if $x_0$ is below 2 then its "dual" is between 2 and 4. This is what I meant with "dual" or "a pair of related numbers". This simple idea reduces to the formula: $\hspace{48} \tilde x_0 = S^{\circ -1} (-s_0) + t_0 = S^{\circ -1} ( - S(x_0 - t_0)) + t_0$ But the effect of changing sign in $s_0$ is alternatively reachable, if we simply introduce an imaginary value for the height-parameter, since $\hspace{48} -s_0 = s_0 \cdot \ln(t_0) ^{i \pi \over \ln \ln t_0}$ Now, starting at some $x_0$ between 2 and 4 we can infinitely iterate and at most approach 2, but we can never arrive with any real height a value below 2 by any number of iterations. We might say, that 2 is the infinite iteration from that $x_0$. We see by this that this concept of "imaginary height" (an imaginary value in the height-parameter h) allows to proceed not only to the value 2 but to values below 2. Thus I said with a sloppy expression: "imaginary height can overstep infinite height". Now, if we take the dual of, say $x_0 = 0$ this gives something above 2. If we iterate this one time towards 4 (which means with one negative height), we get that mentioned value of 2.764... . And its dual is then negative infinity (which itself is $x_0=0$ iterated one time with negative height - a thing which is not possible otherwise because of the occuring singularity). Remark: just to restate it again: "my matrix-method" is nothing else than the regular tetration. I came to this by the (accidentally) rediscovery of the concept of Carleman-matrices (see also wikipedia) not knowing that name, and where I was always working under the assumption of infinite size (no truncation), which has some sophisticated impact against a concept of truncated matrices for instance for the conception of fractional powers 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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