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 New tetration method based on continuum sum and exp-series bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 04/20/2010, 10:40 AM (This post was last modified: 04/22/2010, 10:31 AM by bo198214.) (04/20/2010, 02:48 AM)mike3 Wrote: $\sum_{n=0}^{z-1} f(n) = \sum_{n=-\infty}^{\infty} a_n \frac{b^{nz} - 1}{b^n - 1}$ Really really interesting. Quote:with the term at $n = 0$ interpreted as $a_0 z$, so, $\sum_{n=0}^{z-1} f(n) = \left(\sum_{n=-\infty}^{-1} \frac{a_n}{b^n - 1}\right) + \left(\sum_{n=1}^{\infty} \frac{a_n}{b^n - 1}\right) + a_0 z + \left(\sum_{n=-\infty}^{-1} \frac{a_n}{b^n - 1} b^{nz}\right) + \left(\sum_{n=1}^{\infty} \frac{a_n}{b^n - 1} b^{nz}\right)$. However, when viewed in the complex plane, we see that exp-series are just Fourier series $f(z) = \sum_{n=-\infty}^{\infty} a_n e^{i \frac{2\pi}{P} n z}$. which represent a periodic function with period $P$. Thus it would seem that only periodic functions can be continuum-summed this way. Tetration is not periodic, so how could this help? 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. If I consider Kneser's approach for $b>e^{1/e}$ we can obtain a related representation. Kneser's approach also starts with the regular iteration at a complex fixed point. Lets call the corresponding superfunction $\sigma_0$ which is not real on the real axis. If we call the real superfunction $\sigma$ then we get that $\theta(z)=\sigma^{-1}_0(\sigma(z))-z$ is a 1-periodic function, i.e. $\sigma(z)=\sigma_0(z+\theta(z))=\eta(s e^{\kappa(z+\theta(z))})+\lambda$ This is not periodic, nonetheless we can expand it similar to a periodic function. $\sigma(z)=\lambda+\sum_{n=1}^\infty \eta_n e^{n\kappa z} e^{n\kappa \theta(z)}$ (for simplicity consdier $s^n$ contained in $\eta_n$). The last factor is a 1-periodic function: $e^{n\kappa\theta(z)}=\rho(z)^n=\sum_{k=-\infty}^{\infty} \rho^n_k e^{2\pi i k z}$ If we insert this into our previous equation: $\sigma(z)-\lambda=\sum_{n=1}^{\infty}\sum_{k=-\infty}^{\infty} \eta_n \rho^n_k e^{(\kappa n + 2\pi i k) z}=\sum_{n=1}^{\infty}\sum_{k=-\infty}^{\infty} \sigma_{n,k} e^{(\kappa n + 2\pi i k) z}$ So we have a double exponential series instead of a single series, but nevertheless you again can apply your exponential summation. Though I in the moment have not the time to carry it out myself (so either you do it or I do it later). « 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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