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 An explanation for this? JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 12/30/2010, 06:05 PM (12/30/2010, 12:50 PM)Gottfried Wrote: (12/29/2010, 10:15 PM)JmsNxn Wrote: Thank you. Just one final question, how did you generate this Taylor series? Is there a closed form expression? James, as far as I see this is a graph which was produced by Dimitri Kousnetzov, who also posted here in the forum (you may use this link to find all all posts of him ) and to a certain extend explained his method here. But there is also a published paper of him where he describes this in detail (I've never understood it, btw, because I seem to lack some basic knowledge about cauchy-integrals and riemann-mappings, but for a student of mathematics this may be completely familiar). I think his article is also in our (the tetration-forum's) database of literature (see the related message lit-ref-db in the forum) And have a happy new year - Gottfried I doubt I'll understand it too, I'm only in highschool. (12/30/2010, 02:34 PM)sheldonison Wrote: (12/30/2010, 12:50 PM)Gottfried Wrote: (12/29/2010, 10:15 PM)JmsNxn Wrote: Thank you. Just one final question, how did you generate this Taylor series? Is there a closed form expression?James, as far as I see this is a graph which was produced by Dimitri KousnetzovThere is no known closed form for the Taylor series for tetration. I generated the Taylor series with the kneser.gp program I wrote. I also posted the mathematical equations behind the algorithm here, http://math.eretrandre.org/tetrationforu...hp?tid=487 The basic idea using base e here, where L is the fixed point such that $L=\exp(L)$, $L\approx 0.318+1.317i$, is that if $f(z)=L+\delta$ and $f(z+1)=\exp(f(z))=L\exp(\delta) \approx L+L\delta$ and $f(z+2)=\exp(\exp(f(z))) \approx L+L^2\delta$ etc. This can be used to develop a complex valued entire superfunction such that $\text{superf}(z+1)=\exp(\text{superf}(z))$ for all values of z. $\text{superf}(z) = \lim_{n \to \infty} \exp^{[n]}(L + L^{z-n})$ The problem is that the superf is complex valued, not real valued. A 1-cyclic mapping is used to convert this function to an analytic real valued tetration. The 1-cyclic theta mapping is equivalent to the Riemann mapping in Kneser's algorithm, although convergence is not proven. http://math.eretrandre.org/tetrationforu...hp?tid=487 The Taylor series is generated via a unit circle Cauchy integral. - Sheldon That's a really clever way of extending it. I wish I thought of it « Next Oldest | Next Newest »

 Messages In This Thread An explanation for this? - by JmsNxn - 12/25/2010, 06:07 PM RE: An explanation for this? - by sheldonison - 12/25/2010, 07:06 PM RE: An explanation for this? - by JmsNxn - 12/25/2010, 07:53 PM RE: An explanation for this? - by sheldonison - 12/25/2010, 10:52 PM RE: An explanation for this? - by JmsNxn - 12/26/2010, 03:33 AM RE: An explanation for this? - by sheldonison - 12/26/2010, 12:28 PM RE: An explanation for this? - by JmsNxn - 12/29/2010, 10:15 PM RE: An explanation for this? - by Gottfried - 12/30/2010, 12:50 PM RE: An explanation for this? - by sheldonison - 12/30/2010, 02:34 PM RE: An explanation for this? - by Gottfried - 12/31/2010, 10:42 AM RE: An explanation for this? - by JmsNxn - 12/30/2010, 06:05 PM RE: An explanation for this? - by tommy1729 - 12/26/2010, 09:50 PM RE: An explanation for this? - by Gottfried - 12/31/2010, 10:53 AM RE: An explanation for this? - by bo198214 - 01/06/2011, 03:02 AM RE: An explanation for this? - by mike3 - 01/06/2011, 05:08 AM RE: An explanation for this? - by bo198214 - 01/15/2011, 07:06 AM RE: An explanation for this? - by mike3 - 01/15/2011, 12:26 PM RE: An explanation for this? - by tommy1729 - 01/15/2011, 09:29 PM RE: An explanation for this? - by bo198214 - 01/16/2011, 10:27 AM RE: An explanation for this? - by tommy1729 - 05/03/2014, 08:38 PM

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