An explanation for this?

[sheldonison]

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