Hey, everyone.
After playing with Sheldon's code, I had to slow down some of his calculations because it was creating a jump discontinuity; which was largely because he wrote,
\(
\beta(s) = \sum_{j=0}^\infty c_j e^{js}\\
\)
But he chose this sum for \( \Re(s) < 0 \) and then iterated the functional equation. This is well and fine in theory; but we lose a tad of accuracy; so I just switched it to \( \Re(s) < -50 \) so we're in a better converging area. This does slow down the code a bit; but it's still leagues faster than my old code.
Secondly, I can no longer hide behind inaccuracy as the cause of the "fractal anomalies" I was seeing. At full accuracy, the fractal anomalies are precisely the branch cuts Sheldon is referring to. Now, thankfully; this is perfectly possible as a result of my paper. It was never something that couldn't happen (though my wording leads away from this). In such a sense, the function \(\text{tet}_1(s)\) is holomorphic almost everywhere on \(\mathbb{C}\). It's still to be seen if the final beta tetration \(\text{tet}_\beta(s)\) suffers the same branching problem; it seems to be a toss up at the moment. This would invalidate the final result of my paper; and I'd have to switch to \(\text{tet}_\beta\) is holomorphic almost everywhere on \(\mathbb{C}\). This would require me to pivot towards the asymptotic thesis; which is these are asymptotic solutions primarily at \(\Re(s) = \infty\).
Sheldon, believes these functions are nowhere analytic on \(\mathbb{R}\); I still am not fully convinced of this. I agree this case does have branch points on the real line, and in the neighborhood; I'm not convinced they are dense yet--especially by the below graph. That being said, if they are analytic; they are a very ugly analytic; with taylor series with horrible radii of convergence caused by points of non-analycity on \(\mathbb{R}\) and in the neighborhood. I especially disagree with nowhere analycity on the real line because at \(\Re(s) = \infty\) we can define an implicit function which is analytic--and showing \(\tau\) is this implicit function isn't too hard. So it may not be the best kind of analytic for \(0 < \Re(s) < 1\); but for \(\Re(s) > R\) I see no reason it won't be. Again though, with horrible radii of convergence.
I've tried to add Ember's protocol of catching underflows/overflows; but I don't think it's possible how he's describing. So instead, any overflow is set to 0 by default. So if it's too small, it's zero. And if it's too big, it's zero. This still helps you discern the shape, but not exactly. This is tetration \(\text{tet}_1(s)\) for \(-2 \le \Re(s) \le 8\) and \( -5 \le \Im(s) \le 5\).
You can see the branch cuts appearing in shifts to the left of where Sheldon's zeroes were found. So if the final beta tetration suffers these same zeroes; we can expect the same phenomena. At the moment, I have no obvious way of adapting sheldon's code to the non-periodic case; but I'm working on it. In that case we do not have the benefit of a exponential series; so we have to be more clever.
The more I've been playing with this though, is that if \(\text{tet}_\beta\) is as nice as I said, it should be Kneser. This is because as we decrease the period the function \(\lim_{\lambda \to 0}\text{tet}_\lambda\) satisfies, on \(S = \{s \in \mathbb{C}\,|\,\pi/2 < |\arg(s)| < \pi\}\), \(\lim_{|s| \to \infty} \lim_{\lambda \to 0} \text{tet}_\lambda(s) = L\), our familiar fixed point. And this is equivalent to being Kneser per Paulsen & Cowgill.
You can also new clusters of singularities too. I believe these happen on a bunch of petals, not just the ones Sheldon chose. Unfortunately I couldn't see how to adapt his code to locate these other petals accurately. So the first batch of petals are most noticeable, but you can notice a similar phenomena happening elsewhere.
Now, in worst case scenario; this is exactly as Sheldon said; and then, we're at a point of question: "Can we salvage this?" For this I've been mulling over Kouznetsov's approach. Say we take \(\tau^{100}(s)\) and we limit the equation:
\[
G(s) = \lim_{n\to\infty} \exp^{\circ n}(\beta(s-n) + \tau^{100}(s-n))
\]
We may be able to still derive an analytic function. Particularly; it'll still be periodic--and by definition will not equal \(\text{tet}_1(s)\). This should converge (? not certain)-- and if you graph \(\text{tet}_1(s)\) way off in the left half plane, the branch cuts quiet themselves to nothing. It may not work for \(n=100\) but a clever use of \(\tau^m\) where \(m = g(n)\) may work. So all is not lost yet!
Anyway, just thought I'd give a quick update on what I've been working on. I'm just gonna fiddle with the code a bit more before I upload it. But it's looking much better.
Regards, James
After playing with Sheldon's code, I had to slow down some of his calculations because it was creating a jump discontinuity; which was largely because he wrote,
\(
\beta(s) = \sum_{j=0}^\infty c_j e^{js}\\
\)
But he chose this sum for \( \Re(s) < 0 \) and then iterated the functional equation. This is well and fine in theory; but we lose a tad of accuracy; so I just switched it to \( \Re(s) < -50 \) so we're in a better converging area. This does slow down the code a bit; but it's still leagues faster than my old code.
Secondly, I can no longer hide behind inaccuracy as the cause of the "fractal anomalies" I was seeing. At full accuracy, the fractal anomalies are precisely the branch cuts Sheldon is referring to. Now, thankfully; this is perfectly possible as a result of my paper. It was never something that couldn't happen (though my wording leads away from this). In such a sense, the function \(\text{tet}_1(s)\) is holomorphic almost everywhere on \(\mathbb{C}\). It's still to be seen if the final beta tetration \(\text{tet}_\beta(s)\) suffers the same branching problem; it seems to be a toss up at the moment. This would invalidate the final result of my paper; and I'd have to switch to \(\text{tet}_\beta\) is holomorphic almost everywhere on \(\mathbb{C}\). This would require me to pivot towards the asymptotic thesis; which is these are asymptotic solutions primarily at \(\Re(s) = \infty\).
Sheldon, believes these functions are nowhere analytic on \(\mathbb{R}\); I still am not fully convinced of this. I agree this case does have branch points on the real line, and in the neighborhood; I'm not convinced they are dense yet--especially by the below graph. That being said, if they are analytic; they are a very ugly analytic; with taylor series with horrible radii of convergence caused by points of non-analycity on \(\mathbb{R}\) and in the neighborhood. I especially disagree with nowhere analycity on the real line because at \(\Re(s) = \infty\) we can define an implicit function which is analytic--and showing \(\tau\) is this implicit function isn't too hard. So it may not be the best kind of analytic for \(0 < \Re(s) < 1\); but for \(\Re(s) > R\) I see no reason it won't be. Again though, with horrible radii of convergence.
I've tried to add Ember's protocol of catching underflows/overflows; but I don't think it's possible how he's describing. So instead, any overflow is set to 0 by default. So if it's too small, it's zero. And if it's too big, it's zero. This still helps you discern the shape, but not exactly. This is tetration \(\text{tet}_1(s)\) for \(-2 \le \Re(s) \le 8\) and \( -5 \le \Im(s) \le 5\).
You can see the branch cuts appearing in shifts to the left of where Sheldon's zeroes were found. So if the final beta tetration suffers these same zeroes; we can expect the same phenomena. At the moment, I have no obvious way of adapting sheldon's code to the non-periodic case; but I'm working on it. In that case we do not have the benefit of a exponential series; so we have to be more clever.
The more I've been playing with this though, is that if \(\text{tet}_\beta\) is as nice as I said, it should be Kneser. This is because as we decrease the period the function \(\lim_{\lambda \to 0}\text{tet}_\lambda\) satisfies, on \(S = \{s \in \mathbb{C}\,|\,\pi/2 < |\arg(s)| < \pi\}\), \(\lim_{|s| \to \infty} \lim_{\lambda \to 0} \text{tet}_\lambda(s) = L\), our familiar fixed point. And this is equivalent to being Kneser per Paulsen & Cowgill.
You can also new clusters of singularities too. I believe these happen on a bunch of petals, not just the ones Sheldon chose. Unfortunately I couldn't see how to adapt his code to locate these other petals accurately. So the first batch of petals are most noticeable, but you can notice a similar phenomena happening elsewhere.
Now, in worst case scenario; this is exactly as Sheldon said; and then, we're at a point of question: "Can we salvage this?" For this I've been mulling over Kouznetsov's approach. Say we take \(\tau^{100}(s)\) and we limit the equation:
\[
G(s) = \lim_{n\to\infty} \exp^{\circ n}(\beta(s-n) + \tau^{100}(s-n))
\]
We may be able to still derive an analytic function. Particularly; it'll still be periodic--and by definition will not equal \(\text{tet}_1(s)\). This should converge (? not certain)-- and if you graph \(\text{tet}_1(s)\) way off in the left half plane, the branch cuts quiet themselves to nothing. It may not work for \(n=100\) but a clever use of \(\tau^m\) where \(m = g(n)\) may work. So all is not lost yet!
Anyway, just thought I'd give a quick update on what I've been working on. I'm just gonna fiddle with the code a bit more before I upload it. But it's looking much better.
Regards, James