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 New tetration method based on continuum sum and exp-series mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 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 $(1,e^{1/e})$ 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. « Next Oldest | Next Newest »

 Messages In This Thread New tetration method based on continuum sum and exp-series - by mike3 - 04/20/2010, 02:48 AM RE: New tetration method based on continuum sum and exp-series - by bo198214 - 04/20/2010, 10:40 AM RE: New tetration method based on continuum sum and exp-series - by mike3 - 04/20/2010, 09:10 PM RE: New tetration method based on continuum sum and exp-series - by mike3 - 04/22/2010, 10:20 AM RE: New tetration method based on continuum sum and exp-series - by bo198214 - 04/22/2010, 10:35 AM RE: New tetration method based on continuum sum and exp-series - by mike3 - 04/22/2010, 10:57 AM RE: New tetration method based on continuum sum and exp-series - by bo198214 - 04/22/2010, 12:25 PM RE: New tetration method based on continuum sum and exp-series - by mike3 - 04/22/2010, 08:52 PM RE: New tetration method based on continuum sum and exp-series - by tommy1729 - 04/22/2010, 09:36 PM RE: New tetration method based on continuum sum and exp-series - by mike3 - 04/30/2010, 04:36 AM RE: New tetration method based on continuum sum and exp-series - by andydude - 05/02/2010, 09:27 AM RE: New tetration method based on continuum sum and exp-series - by andydude - 05/02/2010, 09:58 AM RE: New tetration method based on continuum sum and exp-series - by tommy1729 - 04/20/2010, 03:26 PM RE: New tetration method based on continuum sum and exp-series - by tommy1729 - 04/20/2010, 07:45 PM RE: New tetration method based on continuum sum and exp-series - by tommy1729 - 04/20/2010, 08:12 PM

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