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 regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 sheldonison Long Time Fellow Posts: 668 Threads: 24 Joined: Oct 2008 06/24/2013, 04:09 PM (This post was last modified: 06/25/2013, 08:58 AM by sheldonison.) (06/23/2013, 10:20 PM)Gottfried Wrote: ... Unfortunately, that constant has no obvious relation to the asum- zero-height from the other thread - if there were some elegant relation: that were really great. Also whether there might be a relation to the same effect /constant using the upper/repelling fixpoint were an interesting thing: if we perform a handful of integer-height iterations towards the upper fixpoint and find the fixpoint_4-dual and if we could make this somehow consistent, then we had a nice relation over/connecting the whole real line. ...Hi Gottfried, It occurs to me that you might want to use a simpler iteration function to study the asum equations. In particular, the simplest iteration equation I know of is the following iteration function based on the tangent sum equation, which has symmetrical fixed points of +/-1. $f(x)=\frac{x+\tanh(1)}{1+x\tanh(1)}\approx \frac{x+1.5574}{1+1.5574x} \hspace{24} f^{o-1}(x)=\frac{x-\tanh(1)}{1-x\tanh(1)}$ $f^{oz}=\tanh(z) \hspace{24}$ the superfunction agrees with the Schröder function solution generated from both symmetrical fixed points! Generating the asum for this function is more straightforward, since the superfunction for this tangent sum equation is the remarkably simple hyperbolic tangent equation, which is symmetrical and valid everywhere in the complex plane, with simple poles at $\frac{\pi i}{2}+n\pi i$. I think the asum iteration equations are interesting, in that they allow one to generate another different analytic superfunction, but that in the complex plane, the resulting superfunction will never behave as well as the Schroder superfunction or the Kneser superfunction, since it will have additional singularities near imaginary Pi/2, even as x goes to +/- real infinity, whereas tanh(z) is very well behaved near +/- real infinity, where it converges to the +/-1 fixed points for all $\Im(z)$. It might be interesting to generate this asum superfunction for f(x) and compare it to the hyperbolic tangent. The asum of tanh would be an analytic 2-periodic function, with some maximum amplitude. Then the asum superfunction would be generated by taking the $\tanh$ of the inverse of this equation. $\frac{1}{\pi}\sin^{-1}\frac{\text{asum}(\tanh(z))}{\text{amplitude}}$. - Sheldon (06/23/2013, 10:20 PM)Gottfried Wrote: The last point (the only one which I cannot answer myself, perhaps you can look at it): Can we look at your Kneser-method what the dual of, say x_0 = 1 or x_0 = 0 or x_0 = -infty were? I think we need only the appropriate imaginary iteration height to compute the respectively duals. Levenstein- numbers? ;-)What is the definition/equation for a Kneser solution dual? The Kneser solution is not periodic. Would the dual of -infinity, which is sexp(-2) be sexp(2)? - Sheldon « 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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