(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.