05/22/2022, 12:40 AM
Hey, Tommy I'm curious as to these almost analytic solutions you are suggesting.
The closest I've got is reproducing Kneser--which can be done using the beta method. But this only happens if we assume that:
\[
F(s) \to L\,\,\text{as}\,\,|s|\to\infty\,\,\pi/2 \le |\arg(s)| < \pi\\
\]
Every other solution I've played with, discovered through this iterative formula is pretty much always \(C^\infty\), or it could possibly be analytic, but then it's only analytic on a strip. Actually, all this infinite composition stuff with tetration has me convinced Kneser is the way to go. I'm still convinced that the Gaussian method will actually produce Kneser, because it isolates values in the upper half plane, where iterated logarithms tend to \(L\).
I mean \(L\) as the fixed point \(\exp(z)\) with the smallest imaginary part.
The closest I've got is reproducing Kneser--which can be done using the beta method. But this only happens if we assume that:
\[
F(s) \to L\,\,\text{as}\,\,|s|\to\infty\,\,\pi/2 \le |\arg(s)| < \pi\\
\]
Every other solution I've played with, discovered through this iterative formula is pretty much always \(C^\infty\), or it could possibly be analytic, but then it's only analytic on a strip. Actually, all this infinite composition stuff with tetration has me convinced Kneser is the way to go. I'm still convinced that the Gaussian method will actually produce Kneser, because it isolates values in the upper half plane, where iterated logarithms tend to \(L\).
I mean \(L\) as the fixed point \(\exp(z)\) with the smallest imaginary part.