Revisting my accelerated slog solution using Abel matrix inversion

[sheldonison]

Wow, thanks for the analysis!  I still haven't attempted to get into the details of your slog solution and how you calculate the Kneser mapping, because I'm still trying to understand my solution.  I mean, I understand it at a high level, but I keep seeing patterns and trying to tie them together.

But I think I have an answer to your question above, and unfortunately, it's not a very exciting answer.  Brace yourself:

The first term in my Taylor series is inaccurate.

.... The algebraic singularity, with it's complex periodicity, is what allows the Kneser solution to have non-trivial branches (where the value in one branch is not merely a constant difference from the value of the same point in a different branch).  

Lets keep the focus on your Jay's slog for now.  I have some pretty pictures; along with some equations, and later I'll make a new post consolidating other posts and explaining how Sheldon's fatou.gp works.  This thread is helpful because it clearly shows that very good solutions that are self consistent to a very high degree can actually be \( \theta \) mappings of Kneser's solution which is really really cool to me and very exciting, so I disagree with your statement, "its not a very exciting answer."  Someday when I write a published paper I will use this thread in my explanation of Sheldon's slog and what is required to mach Kneser's slog.

I like the term "algebraic singularity"; it is also really cool, but complicated.  The Period is \( \frac{2\pi i}{L} \), and when I investigated it, the algebraic singularity magnitude tracked Period^n.  That's from memory and I don't remember the equations; I look forward to your proof.

So, now we have Jay's slog, and we can convert it to a very good approximation of Kneser's slog as follows:

\( \text{KneserSlog}(z)=\alpha(z)+\theta_s(\alpha(z))\approx\text{JaySlog}(z)+\theta_\R(\text{JaySlog}(z)) \)

Here \( \theta_\R \) is the theta mapping at the real axis with complex conjugate pairs required to keep the JaySlog real valued at the real axis, and \( \theta_s \) is the unique theta mapping of the Schroder/Abel function which generate Kneser's slog. 

The rest of this is my speculative musings ... If \( \theta_\R \) has n complex conjugate pair terms, then can we use those n unknown terms to force n terms of y^-1, y^-2, ... y^-n to be zero in the \( \theta_s \) mapping???? ...  hmmmm ..... perhaps this is a fixer-upper equation???? is it a linear equation or something more complicated????  A matrix????    Can we use for example the 60 sample points using  \( \alpha^{-1}(z_0..z_0+1) \) from my previous thread in terms of the n unknown terms in \( \theta_\R \) to write a matrix equation????