08/23/2021, 11:54 PM
As we're posting our own conjectures here, I thought I'd add mine to this list.
In William Paulsen and Samuel Cowgill's paper; they outline the following uniqueness condition:
\(
F(z)\,\,\text{is holomorphic on}\,\,\mathbb{C}/(-\infty,-2]\\
F(0) = 1\\
F(\overline{z}) = \overline{F(z)}\\
\forall x \in \mathbb{R}\, \,\lim_{y\to\infty} F(x+iy) = L\\
\text{where}\,L\,\text{is the fixed point of}\, \exp\, \text{with minimal imaginary argument}\\
F(z+1) = e^{F(z)}\\
\text{Then necessarily,}\\
F(z)\,\,\text{is Kneser's Tetration}\\
\)
Which they prove, as I believe, completely satisfactorily (Kouznetzov seemed to have doubts).
This question is in two parts:
A). Does there exist a tetration function \( G \) which has all these properties except,
\(
\lim_{y\to\infty} G(x+iy) = \infty\\
\)
B.) Does it satisfy the same uniqueness condition that William Paulsen and Samuel Cowgill proposed? (Albeit, with the different behaviour \( \Im(z) = \infty \)).
For A.)--See related threads about the beta method (https://math.eretrandre.org/tetrationfor...p?tid=1314, https://math.eretrandre.org/tetrationfor...p?tid=1334), which seems to point to the beta method not being kneser (I've proved the beta-method converges, but not that it isn't still Kneser's method). But numbers clearly show a discrepancy...
For B.) I don't know at all, but there's a hunch--by looking at Tommy's Gaussian method and the beta method they seem to be one and the same. See: https://math.eretrandre.org/tetrationfor...p?tid=1339 . Furthermore, this conjecture was made by how well different manners of coding the beta method all still gave the same numbers; when I would use different asymptotic shortcuts the same function appears.
I believe I can sketch a proof of A.), but I'd need oversight before it ever became a proof. As for B.), I can't even think of a line of attack.
Regards, James
In William Paulsen and Samuel Cowgill's paper; they outline the following uniqueness condition:
\(
F(z)\,\,\text{is holomorphic on}\,\,\mathbb{C}/(-\infty,-2]\\
F(0) = 1\\
F(\overline{z}) = \overline{F(z)}\\
\forall x \in \mathbb{R}\, \,\lim_{y\to\infty} F(x+iy) = L\\
\text{where}\,L\,\text{is the fixed point of}\, \exp\, \text{with minimal imaginary argument}\\
F(z+1) = e^{F(z)}\\
\text{Then necessarily,}\\
F(z)\,\,\text{is Kneser's Tetration}\\
\)
Which they prove, as I believe, completely satisfactorily (Kouznetzov seemed to have doubts).
This question is in two parts:
A). Does there exist a tetration function \( G \) which has all these properties except,
\(
\lim_{y\to\infty} G(x+iy) = \infty\\
\)
B.) Does it satisfy the same uniqueness condition that William Paulsen and Samuel Cowgill proposed? (Albeit, with the different behaviour \( \Im(z) = \infty \)).
For A.)--See related threads about the beta method (https://math.eretrandre.org/tetrationfor...p?tid=1314, https://math.eretrandre.org/tetrationfor...p?tid=1334), which seems to point to the beta method not being kneser (I've proved the beta-method converges, but not that it isn't still Kneser's method). But numbers clearly show a discrepancy...
For B.) I don't know at all, but there's a hunch--by looking at Tommy's Gaussian method and the beta method they seem to be one and the same. See: https://math.eretrandre.org/tetrationfor...p?tid=1339 . Furthermore, this conjecture was made by how well different manners of coding the beta method all still gave the same numbers; when I would use different asymptotic shortcuts the same function appears.
I believe I can sketch a proof of A.), but I'd need oversight before it ever became a proof. As for B.), I can't even think of a line of attack.
Regards, James