An explanation for this?

[JmsNxn]

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.

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 \( L=\exp(L) \), \( L\approx 0.318+1.317i \), 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