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 Does tetration take the right half plane to itself? sheldonison Long Time Fellow Posts: 664 Threads: 23 Joined: Oct 2008 05/15/2017, 09:00 PM (This post was last modified: 05/15/2017, 11:25 PM by sheldonison.) (05/10/2017, 07:46 PM)JmsNxn Wrote: This question is rather straightforward. Let $1 < \alpha < \eta$. We know that there exists a unique bounded tetration function $F(z) = \alpha \uparrow^2 z$ that is holomorphic for $\Re(z) > 0$. But is $\Re (F(z)) > 0$? Does the bounded tetration function take the right half plane to itself?So are you asking if it is a 1-1 mapping from the right half of the plane to itself?  Certainly there's a singularity at z=-2.   Lets think about base sqrt(2), with two fixed points, 2 and 4. As $\Re(z)$ gets arbitrarily large $\text{sexp}_{\sqrt{2}}(z) \approx 2$  At the imaginary axis, sexp(0)=1.  At half the period, sexp(period/2) the value would have grown to something like 2.468.  In between, imag(z) gets to about +/- 0.7208. So at the imaginary axis, it will roughly "circle" or loop around the fixed point of 2 an infinite number of times, as follows:  This very roughly circular path is what the right half of the plane is mapped to!   Of course, it would be more complicated if I repeated this experiment at the singularity at z=-2 and again mapped one imaginary period, but I don't see the plane mapped to itself at z=-2 either.  The imaginary period is $=\frac{-2\pi i}{\ln(\ln(2))} \approx$ 17.143     - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread Does tetration take the right half plane to itself? - by JmsNxn - 05/10/2017, 07:46 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/10/2017, 08:22 PM RE: Does tetration take the right half plane to itself? - by sheldonison - 05/15/2017, 08:16 PM RE: Does tetration take the right half plane to itself? - by sheldonison - 05/15/2017, 09:00 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/16/2017, 04:09 AM RE: Does tetration take the right half plane to itself? - by sheldonison - 05/16/2017, 03:34 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/16/2017, 08:46 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/16/2017, 04:46 AM

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