(09/24/2021, 04:52 PM)Ember Edison Wrote:(09/21/2021, 07:22 PM)sheldonison Wrote:(07/23/2021, 04:05 PM)JmsNxn Wrote: What I was pointing out is that Samuel Cogwill and William Paulsen proved a uniqueness condition....
This implies that ... is Kneser's solution; unless it has singularities in the upper half plane.
http://myweb.astate.edu/wpaulsen/tetration2.pdf
James, thanks for pointing out that Cowgill/Paulsen have proven this uniqueness criteria! Very nice.
I talked with James on a zoom call, and I hope to understand Jame's Beta method well enough to generate a Taylor series for \( \lambda=1 \) case, first for James \( 2\pi i \) periodic \( \beta \) function, and then for his Tetration solution generated from \( \beta(\lambda=1) \). This function is very interesting to me all by itself! Also, if I understand Cowgill's proof, then if Jame's solution can be arbitrarily extended to arbitrarily larger then 2pi i imaginary periods with other different values of \( 0<\lambda<1 \), then in the limit it must be Kneser's solution otherwise Jame's solution must have singularities in the upper half of the complex plane.
Can this method solve the base that theta-mapping cannot solve?
See my thread here; I run a quick toy model for a \( 2 \pi i \) periodic tetration base \( b = 1/2 \). As far as I can tell this should work on the real positive line; and should work in the complex plane, but I'm not sure. I think the real trouble would be \( b <0 \).