(08/14/2018, 10:25 PM)sheldonison Wrote:(08/14/2018, 08:34 PM)Chenjesu Wrote: The claim in the paper J.N. paper appears to be that any complex height z can be represented with a standard analytic series and integral...Why are you so focused on nit-picking one unpublished paper? I have listed peer reviewed published papers and books, applicable to solutions of \( \exp_b^{[\circ z]}(z) \). I'm not saying James Nixon's paper is incorrect, I'm just saying it isn't relevant given that you are interested in peer reviewed rigorous work.
- Dr. William Paulson's two recent papers
- Henryk Trapmann's Uniqueness paper
- Lennart Carleson's book "Complex Dynamics"
- By indirect extension Kneser's 1950 paper
Yet again, you intentionally dodged the discussion. I went on to discuss Trappmann's iterated exponential formula beside the fact that you're the one who suggested a comparison between the two papers.
Let's look at the Trappmann claim for the simple case of the half-exponential function.
\( f(z)= \exp_b^{z}(1)=s \exp_{b}(z)= \underbrace{b^{b^{b^{...}}}}_{\text{z times}} \), this looks interesting, now let us extend it to non-integers:
\( \exp_b^{c}(z)= s \exp_{b}(c+s \log_{b}(z)) \)
We want to know what raising something to the "1/2" height means, at least according to this conjecture, and z in this case is the height of tetration:
\( \exp_b^{ \frac{1}{2}}(1)=s \exp_{b}(\frac{1}{2})= \underbrace{b^{b^{b^{...}}}}_{\text{\frac{1}{2} times}} \)
How should this result be interpreted to yield a final answer in terms of standard mathematical functions for b=e?