HEY EVERYONE!
I've got some pictures to share!
I've further written my paper (it's still not perfect yet, the next time I share will be the last time); but in the process I ran more numbers. I *ahem* acquired matlab; and familiarized myself with the code. I had some help from a fellow on Stack overflow; where I wasn't calling my recursion properly. And I got a couple of tetration graphs to display.
If we call,
\(
F_\lambda(s) = \lim_{n\to\infty}\log \beta_\lambda(s+n)\\
\)
Which,
\(
F_\lambda(s+1) = e^{F_\lambda(s)}\\
\)
Then, when \( \lambda =\log(2) \) and \( -1\le\Re(s)\le 1,\,-1\le\Im(s)\le1 \); the function \( |F_\lambda(s)| \) looks like, after about 7 iterations,
Where these quick spikes and jumps move further and further together as you do more iterations. And the center of these spikes is Tetration at \( -2 \). We still haven't shifted our argument yet, so that \( F_\lambda(s+x_0) \) is tetration. The sharp drops to infinity is the clustering of singularities about \( \text{tet}(-2) \) in the iteration. But, this graph looks more and more level, as you iterate further and further, excepting where the singularities are. Here is a second graph when \( \lambda = 1/2 + i \), so it models \( |F_\lambda(s)| \) again, but with a complex multiplier (about the logarithmic singularity at \( -2 \)).
Now, the errors occur in one half plane and not the other. This can be related (naively) to \( \beta_\lambda(s) \to \infty \) when \( e^{-\lambda s} \to 0 \). And we can rotate the plane as we move \( \lambda \). Using a different modeling technique, we get convergence in a different way,
This converges to the same function eventually; but is more satisfactory for local values about small numbers. We just take the limit slightly different in this case.
I'm in the process of finalizing this paper; but these graphs confirm, in my mind, that these tetrations are analytic. And one can paste them together. I'll post this paper in maybe a week or so. I'm trying to make sure every nail is hammered at this point.
I think this really is analytic tetration!
I've got some pictures to share!
I've further written my paper (it's still not perfect yet, the next time I share will be the last time); but in the process I ran more numbers. I *ahem* acquired matlab; and familiarized myself with the code. I had some help from a fellow on Stack overflow; where I wasn't calling my recursion properly. And I got a couple of tetration graphs to display.
If we call,
\(
F_\lambda(s) = \lim_{n\to\infty}\log \beta_\lambda(s+n)\\
\)
Which,
\(
F_\lambda(s+1) = e^{F_\lambda(s)}\\
\)
Then, when \( \lambda =\log(2) \) and \( -1\le\Re(s)\le 1,\,-1\le\Im(s)\le1 \); the function \( |F_\lambda(s)| \) looks like, after about 7 iterations,
Where these quick spikes and jumps move further and further together as you do more iterations. And the center of these spikes is Tetration at \( -2 \). We still haven't shifted our argument yet, so that \( F_\lambda(s+x_0) \) is tetration. The sharp drops to infinity is the clustering of singularities about \( \text{tet}(-2) \) in the iteration. But, this graph looks more and more level, as you iterate further and further, excepting where the singularities are. Here is a second graph when \( \lambda = 1/2 + i \), so it models \( |F_\lambda(s)| \) again, but with a complex multiplier (about the logarithmic singularity at \( -2 \)).
Now, the errors occur in one half plane and not the other. This can be related (naively) to \( \beta_\lambda(s) \to \infty \) when \( e^{-\lambda s} \to 0 \). And we can rotate the plane as we move \( \lambda \). Using a different modeling technique, we get convergence in a different way,
This converges to the same function eventually; but is more satisfactory for local values about small numbers. We just take the limit slightly different in this case.
I'm in the process of finalizing this paper; but these graphs confirm, in my mind, that these tetrations are analytic. And one can paste them together. I'll post this paper in maybe a week or so. I'm trying to make sure every nail is hammered at this point.
I think this really is analytic tetration!
