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: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(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

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