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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 04/22/2010, 10:20 AM (This post was last modified: 04/22/2010, 10:21 AM by mike3.) (04/20/2010, 10:40 AM)bo198214 Wrote: $\sigma(z)=\sum_{n=0}^{\infty}\sum_{k=-\infty}^{\infty} \eta_n \rho^n_k e^{(\kappa n + 2\pi i k) z}=\sum_{n=0}^{\infty}\sum_{k=-\infty}^{\infty} \sigma_{n,k} e^{(\kappa n + 2\pi i k) z}$ 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). Eh. I'm not sure if this method is going to work. Take what happens when $n = 0$. Then we have coefficients multiplying $e^{2\pi i k z}$ for integer $k$. But this does not continuum-sum under the given method: we get $\frac{e^{2\pi i k z} - 1}{e^{2\pi i k} - 1}$, but the denominator $e^{2\pi i k} - 1$ is 0 when $k \in \mathbf{Z}$. This gives a division by zero. There is no joy trying to use a limit (of, say, $e^{az}$ as $a$ approaches $2\pi i k$) -- this singularity explodes to infinity. Yet it seems we can continuum-sum the tetrational regardless, by using the periodic-approximation method.) What's going on here? (And as an aside, what did you think of the graph of the tetration of a complex base outside the Shell-Thron Region?) « 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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