04/20/2010, 09:10 PM

(04/20/2010, 10:40 AM)bo198214 Wrote: This is not completely true. The regular tetration in the base range has the form

where is a holomorphic function with and (this is the inverse of the Schröder function), is the fixed point and (and s is some arbitrary constant which you would choose to ascertain that .

So it is periodic.

This is true, but currently it is only a hypothesis, not a proven theorem, that the regular iteration satisfies the continuum sum equation. Though numerically it looks good. But if one could find an explicit form (either as a closed form or a series with explicit terms or something) for the coefficients in its exp/Fourier expansion, we could take its (canonical) continuum sum and perh. that could help in the finding of the proof if this hypothesis is really right or not.