09/15/2021, 08:15 PM
(This post was last modified: 09/15/2021, 08:48 PM by sheldonison.)
James,
What is the range in analyticity of the tet_beta method? I assume tet_beta has some singularities in the upper half of the complex plane. Here's my heuristic "conjecture", which applies to any alternative tetration. I don't know if Samuel Cogwill and William Paulsen have proven this rigorously or not.
\( \theta(z)=\text{slog}_{k}(\text{tet}_\beta(z))-z \)
Theta is defined at the real axis for z>-2 since both tetrations are increasing there. And theta can be extended to be a 1-cyclic function that is defined at the real axis. In the complex plane, theta might only be defined at the real axis, but then tet_beta is only defined at the real axis. Assuming theta is defined in the complex plane, then theta can either be entire or have singularities in the upper half of the complex plane.
\( \text{tet}_\beta(z)=\text{tet}_k(\theta(z)+z) \)
If theta is entire than z+theta(z) will take on every integer's value an infinite number of times and this introduces an infinite number of new singularities for when z+theta(z)=-2, z-1+theta(z-1)=-3, z-2+theta(z-2)=-4, and these are all new singularities that are in the upper half of the complex plane. The argument is more complex if theta has singularities, but it would seem that theta's singularities also probably lead to singularities for tet_beta(z) .... Moreover theta must also avoid taking on integer values due to the first argument
Iterated exponentiation isn't well defined or increasing in the complex plane unless we are only iterating real values. If Tet(z) gets arbitrarily large positive then somewhere nearby, Tet(z) gets arbitrarily large negative, and exp(Tet(z)) gets arbitrarily close to zero. So the behavior as imaginary z increases would depend on the real value as well, and there will be nearly zero values interspersed with very large values ...
Perhaps James can comment which case he thinks applies to the Beta method.
What is the range in analyticity of the tet_beta method? I assume tet_beta has some singularities in the upper half of the complex plane. Here's my heuristic "conjecture", which applies to any alternative tetration. I don't know if Samuel Cogwill and William Paulsen have proven this rigorously or not.
\( \theta(z)=\text{slog}_{k}(\text{tet}_\beta(z))-z \)
Theta is defined at the real axis for z>-2 since both tetrations are increasing there. And theta can be extended to be a 1-cyclic function that is defined at the real axis. In the complex plane, theta might only be defined at the real axis, but then tet_beta is only defined at the real axis. Assuming theta is defined in the complex plane, then theta can either be entire or have singularities in the upper half of the complex plane.
\( \text{tet}_\beta(z)=\text{tet}_k(\theta(z)+z) \)
If theta is entire than z+theta(z) will take on every integer's value an infinite number of times and this introduces an infinite number of new singularities for when z+theta(z)=-2, z-1+theta(z-1)=-3, z-2+theta(z-2)=-4, and these are all new singularities that are in the upper half of the complex plane. The argument is more complex if theta has singularities, but it would seem that theta's singularities also probably lead to singularities for tet_beta(z) .... Moreover theta must also avoid taking on integer values due to the first argument
Iterated exponentiation isn't well defined or increasing in the complex plane unless we are only iterating real values. If Tet(z) gets arbitrarily large positive then somewhere nearby, Tet(z) gets arbitrarily large negative, and exp(Tet(z)) gets arbitrarily close to zero. So the behavior as imaginary z increases would depend on the real value as well, and there will be nearly zero values interspersed with very large values ...
Perhaps James can comment which case he thinks applies to the Beta method.
- Sheldon