New tetration method based on continuum sum and exp-series

[mike3]

This is not completely true. The regular tetration in the base range \( (1,e^{1/e}) \) has the form
\( \sigma(z)=\eta(se^{\kappa z}) + \lambda \)
where \( \eta \) is a holomorphic function with \( \eta(0)=0 \) and \( \eta'(0)=1 \) (this is the inverse of the Schröder function), \( \lambda \) is the fixed point and \( \kappa=\ln(f'(\lambda)) \) (and s is some arbitrary constant which you would choose to ascertain that \( \sigma(0)=1 \).

So it is \( 2\pi i/\kappa=2\pi i/\ln(\ln \lambda) \) 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.