I've been playing "Elden Ring" for the past few days, so I haven't been on the forums to check the latest news. Anyway, fuck the war, fuck COVID, and peace to all of you.
And then yes congratulations on finally finishing your thesis!
I'm showing the conclusions of my cursory reading here, and I hope I haven't missed key information.
1. The Weak Fatou set is the complement of the Weak Julia set.
2. beta method is holomorphic on Weak Fatou set, and Weak Julia set is not.
3. beta method work well on Shell-Thron region (or at least its interior).
3.1. It can induce a unique Kneser's Tetration.
3.2. And weak Julia set is measure zero.
4. Every periodic solution to the tetration equation which is not the period of the regular iteration must have singularities in the right half plane other than the singularities of the regular iteration
And then I have some questions in case I missed something:
Q1. What happens to beta method if we close to the S-T region from the outside of the S-T region?
Q2. What happens to beta method when imag(base) < 0, In particular, minimum real part for Shell-Thron Region, and -1 ≤ base < 0
Q3. I did not find the beta method Super-logarithm in the paper.
Q4. Is the beta method not sufficiently holomorphic to induce the Super-root?
The last one is purely a question of entertainment:
Q6. What is the behavior of the \( \int \beta (z)dz \) and \( \int tet_{\beta} (z)dz \)?
Oh, I know a lot of people hate solving indefinite integral problems, I'm just kidding. Because all versions of Tetration's indefinite integrals don't look like they would be members of the Mellin/Fox H function.
And then yes congratulations on finally finishing your thesis!
I'm showing the conclusions of my cursory reading here, and I hope I haven't missed key information.
1. The Weak Fatou set is the complement of the Weak Julia set.
2. beta method is holomorphic on Weak Fatou set, and Weak Julia set is not.
3. beta method work well on Shell-Thron region (or at least its interior).
3.1. It can induce a unique Kneser's Tetration.
3.2. And weak Julia set is measure zero.
4. Every periodic solution to the tetration equation which is not the period of the regular iteration must have singularities in the right half plane other than the singularities of the regular iteration
And then I have some questions in case I missed something:
Q1. What happens to beta method if we close to the S-T region from the outside of the S-T region?
Q2. What happens to beta method when imag(base) < 0, In particular, minimum real part for Shell-Thron Region, and -1 ≤ base < 0
Q3. I did not find the beta method Super-logarithm in the paper.
Q4. Is the beta method not sufficiently holomorphic to induce the Super-root?
The last one is purely a question of entertainment:
Q6. What is the behavior of the \( \int \beta (z)dz \) and \( \int tet_{\beta} (z)dz \)?
Oh, I know a lot of people hate solving indefinite integral problems, I'm just kidding. Because all versions of Tetration's indefinite integrals don't look like they would be members of the Mellin/Fox H function.
