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 base holomorphic tetration bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 11/06/2009, 12:12 PM (This post was last modified: 11/09/2009, 09:36 AM by bo198214.) (11/06/2009, 04:15 AM)mike3 Wrote: I've also been toying with this, too. It appears, however, that it continues to real values, not complex values, for $b > e^{1/e}$. that would be awesome. And this would mean that there is no singularity at $b=e^{1/e}$? Quote:For $z = 1.5 = \frac{3}{2}$, we can use this get $^{\frac{1}{2}} \left(\frac{3}{2}\right) \approx 1.28087727794$, which is real, not complex. with "use this" you mean the powerseries development you just derived at $z_0$? But how do you know that it converges and that there is no branchpoint at $z=e^{1/e}$? Quote:I'm not sure of a formal proof of the "continuability", though one approach may be to try and differentiate the regular iteration formula, then prove that the limit of the derivative as $b \rightarrow e^{1/e}$ converges -- in order for it to switch to non-real complex values as $b = e^{1/e}$ is passed, that point would have to be some sort of singularity, like a branch point, and so the function would not be differentiable there, and if it is, then that is not the case. *nods* but at least it is already known that the regular iteration $f(z)=\exp_{e^{1/e}}^{\circ t}(z)$ is not analytic at $z=e$. However it is currently not clear to me what this states about the regularity of $f(z)=\exp_z^{\circ t}(1)$ at $z=e^{1/e}$. Quote:I'll see if maybe I can get some graphs on the complex plane but calculating the regular iteration is a bear as it requires lots of numerical precision, at least for the limit formula. Maybe that series formula would be better? Ya I will try it with the series formula (or perhaps a mixture with limit formulas). Actually it seems that no-one posted pictures of tetra-powers yet!!! (yes Bat I mean tetra-powers.) So it will be time that we have some pictures at least, as the theoretic consideration seems utmost complicated to me. « Next Oldest | Next Newest »

 Messages In This Thread base holomorphic tetration - by bo198214 - 11/05/2009, 02:12 PM RE: base holomorphic tetration - by Base-Acid Tetration - 11/05/2009, 10:50 PM RE: base holomorphic tetration - by mike3 - 11/06/2009, 04:15 AM RE: base holomorphic tetration - by mike3 - 11/06/2009, 11:58 AM RE: base holomorphic tetration - by bo198214 - 11/06/2009, 12:12 PM RE: base holomorphic tetration - by mike3 - 11/06/2009, 09:16 PM RE: base holomorphic tetration - by bo198214 - 11/06/2009, 11:29 PM RE: base holomorphic tetration - by mike3 - 11/07/2009, 12:23 AM RE: base holomorphic tetration - by bo198214 - 11/07/2009, 08:17 AM RE: base holomorphic tetration - by mike3 - 11/07/2009, 08:21 AM RE: base holomorphic tetration - by bo198214 - 11/07/2009, 09:55 AM RE: base holomorphic tetration - by bo198214 - 11/07/2009, 04:47 PM RE: base holomorphic tetration - by bo198214 - 11/08/2009, 05:39 PM RE: base holomorphic tetration - by mike3 - 11/08/2009, 08:27 PM RE: base holomorphic tetration - by mike3 - 11/08/2009, 08:25 PM RE: base holomorphic tetration - by bo198214 - 11/08/2009, 08:44 PM RE: base holomorphic tetration - by mike3 - 11/08/2009, 09:51 PM

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