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An explanation for this?
#11
(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. Dodgy


(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 Kousnetzov
There 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 , , is that if
and and etc.

This can be used to develop a complex valued entire superfunction such that for all values of z.
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 Tongue
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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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