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base holomorphic tetration
#5
(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 .

that would be awesome. And this would mean that there is no singularity at ?

Quote:For , we can use this get , which is real, not complex.

with "use this" you mean the powerseries development you just derived at ? But how do you know that it converges and that there is no branchpoint at ?

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 converges -- in order for it to switch to non-real complex values as 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 is not analytic at . However it is currently not clear to me what this states about the regularity of at .

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.
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Messages In This Thread
base holomorphic tetration - by bo198214 - 11/05/2009, 02:12 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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