09/21/2021, 07:22 PM
(This post was last modified: 09/21/2021, 07:24 PM by sheldonison.)
(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.
- Sheldon