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regular tetration base sqrt(2) : an interesting(?) constant 2.76432104
#3
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 ) we realize the tetration to fractional heights via the "Schröder"-function (see wikipedia), say
where 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:



Then we use the inverse Schröder-function to find the value which is the (fractional) h'th iterate "from" :



Now if we let the "height" h equal zero, thus no iteration, but just change the sign of then we get a "dual" , where if is between 2 and 4, then the dual is below 2, and if 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:



But the effect of changing sign in is alternatively reachable, if we simply introduce an imaginary value for the height-parameter, since



Now, starting at some 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 .

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 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 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
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RE: regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 - by Gottfried - 06/22/2013, 09:55 PM

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