05/10/2021, 02:26 AM
So I've stumbled across coding this object better. I've managed to make some fairly flawless code; but for some reason I get too many errors when graphed. And as I saw these errors, I realized I've stumbled across numerical evidence that this tetration IS NOT Kneser's tetration.
The key identity you have to recall in Kneser's tetration is that,
\(
\lim_{z\to\infty} \text{tet}_K(z) = L\,\,\text{for}\,\,\pi/2 \le \arg(z) <\pi\\
\)
And further more; that the principal branch of the logarithm; in the upper half plane \( \mathbb{H} = \{z \in \mathbb{C}\,|\,\Im(z) > 0\} \); satisfies,
\(
\lim_{n\to\infty} \log^{\circ n}(\mathbb{H}) = L\\
\)
Now, Pari-gp/Matlab always choose the principal branch. The fact is; my tetration is only stable about the principal branch of \( \log \) near the real axis. Everywhere else, my tetration chooses different logarithms. This is because, as I already suspected; my tetration is not normal in the upper half plane like Kneser's is. This can be summarized as,
\(
\lim_{|\Im(s)| \to \infty} \text{tet}_\beta(s) = \infty\\
\)
So, although my tetration can be calculated perfectly point-wise using,
\(
\log^{\circ n}(\beta(s+n))\\
\)
With the principal branch of \( \log \). It cannot be holomorphic everywhere in \( \mathbb{H} \) if we only use the principal branch of \( \log \). So the anomalies I'm seeing in my code are in fact evidence that this function is NOT Kneser's tetration. And coding this is going to be even harder.
I'm preparing a second paper proving that this tetration is not Kneser's. At this point, I'm certain everything works. But I'm scared this may just develop Kneser's tetration; which my gut says no. And I think I can prove it now.
This is a real valued tetration that is completely different from what we've seen before. There is no special fixed point involved at all.
The key identity you have to recall in Kneser's tetration is that,
\(
\lim_{z\to\infty} \text{tet}_K(z) = L\,\,\text{for}\,\,\pi/2 \le \arg(z) <\pi\\
\)
And further more; that the principal branch of the logarithm; in the upper half plane \( \mathbb{H} = \{z \in \mathbb{C}\,|\,\Im(z) > 0\} \); satisfies,
\(
\lim_{n\to\infty} \log^{\circ n}(\mathbb{H}) = L\\
\)
Now, Pari-gp/Matlab always choose the principal branch. The fact is; my tetration is only stable about the principal branch of \( \log \) near the real axis. Everywhere else, my tetration chooses different logarithms. This is because, as I already suspected; my tetration is not normal in the upper half plane like Kneser's is. This can be summarized as,
\(
\lim_{|\Im(s)| \to \infty} \text{tet}_\beta(s) = \infty\\
\)
So, although my tetration can be calculated perfectly point-wise using,
\(
\log^{\circ n}(\beta(s+n))\\
\)
With the principal branch of \( \log \). It cannot be holomorphic everywhere in \( \mathbb{H} \) if we only use the principal branch of \( \log \). So the anomalies I'm seeing in my code are in fact evidence that this function is NOT Kneser's tetration. And coding this is going to be even harder.
I'm preparing a second paper proving that this tetration is not Kneser's. At this point, I'm certain everything works. But I'm scared this may just develop Kneser's tetration; which my gut says no. And I think I can prove it now.
This is a real valued tetration that is completely different from what we've seen before. There is no special fixed point involved at all.
