10/25/2009, 09:33 AM
(10/25/2009, 07:43 AM)bo198214 Wrote:(10/25/2009, 02:14 AM)mike3 Wrote: But it's a constant function, so it cannot be interpreted as analytic continuation of the specific function \( \mathrm{tet}_b(z) \) to another branch
Yes you are right, its no more the regular tetration.
Can you say which branch sequence you used to produce the constant function?
I tried \( [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...] \) (see the obvious pattern), \( b = \sqrt{2} \) as the "infinite branch code". This seems to make it converge onto the constant function with everywhere-value ~9.21189016 + 3.20965748i. It seems any
code will work. E.g. \( [0, 1, 1, 1, 1, 1, 1, 1, ...] \) produces a different constant, ~9.09072986 + 3.37850978i.
I suspect the Riemann surface has a interesting nested structure. Do you agree?
(10/25/2009, 07:43 AM)bo198214 Wrote: Indeed I noticed that interpretation already in other contexts. If you have a multivalued function satisfying a certain functional equation it satisfies this equation only if you choose the suitable branches.
Easiest example is log with the functional equation log(ab)=log(a)+log(b).
I remember also this kind of description in [Kuzma: iterative functional equations] for regular holomorphic Abel functions. Particularly I noticed this behaviour with Dmitrii's superlog, and now you add regular tetration.
Yeah, also consider \( \exp^u(\exp^v(z)) = \exp^{u+v}(z) \) for fractional/real/complex u and v (i.e. any but natural numbers) done via, say, Kouznetsov's tetration.