AHA!
I think I'm getting the hang of making this work. I'm making this as I go; and I've been reminding myself, as I go, how to code in C based languages. I've improved some of the code for this, but I've kept the same structure. I will have more to update on as I progress. But! I thought I'd share a really cool graph I just made. I'm broaching the territory of proving this isn't Kneser's but I'm trying to get there.
Over the domain \( -1 \le \Re(z) \le 4 \) and \( 10 \le \Im(z) \le 12 \), this is \( \text{tet}_\beta(z) \):
The more the chaos, the more I'm right. This tetration is not normal when we apply the principal branch of the logarithm. It behaves like a julia set, and not like a standard Schroder iteration.
Here's a larger portrait over the domain \( -1 \le \Re(z) \le 5 \) and \( 2 \le \Im(z) \le 12 \), this is \( \text{tet}_\beta(z) \),
And another large portrait over the domain \( -1 \le \Re(z) \le 5 \) and \( 10 \le \Im(z) \le 20 \).
I'm still having trouble making fast and accurate code near the real-line--but I'm pretty sure I have the upper and lower half-planes well managed.
So I thought I'd post a graph full of artifacts, and explain why we have these artifacts. Pari-GP always chooses the principal branch of the logarithm. Therefore as we increase the imaginary argument; since this function is not normal on \( \Im(z) > 0 \), then you get a cycle between \( \Im(z) >,<0 \).
No where else is this more obvious than with the recursive definition of the code on the Real line. Everything works great away from the real line; but on the real line the recursive definition fails. Where it's trying to swap between \( Im(z) >,<0 \). This is because, in a neighborhood of the real line, the principal branch of \( \log \) is NOT NORMAL on \( \text{tet}_\beta \). So my pretty Taylor series versions are correct, these recursive graphs are not necessarily. Still this looks similar to what I posted before (I'll add it after).
The purple (resp. green) which appears in the upper/lower half plane; is a swap between \( \pi,-\pi \); which forces all the errors. I'm still in the process of making a "Matrix Add On" for this code; which we'll solve this problem. But nonetheless, here's what a better recursive code looks like.
The recursive definition using the principal branch of \( \log \):
The definition using Taylor series; and where we choose arbitrary logs, not just the principal.
I think I'm getting the hang of making this work. I'm making this as I go; and I've been reminding myself, as I go, how to code in C based languages. I've improved some of the code for this, but I've kept the same structure. I will have more to update on as I progress. But! I thought I'd share a really cool graph I just made. I'm broaching the territory of proving this isn't Kneser's but I'm trying to get there.
Over the domain \( -1 \le \Re(z) \le 4 \) and \( 10 \le \Im(z) \le 12 \), this is \( \text{tet}_\beta(z) \):
The more the chaos, the more I'm right. This tetration is not normal when we apply the principal branch of the logarithm. It behaves like a julia set, and not like a standard Schroder iteration.
Here's a larger portrait over the domain \( -1 \le \Re(z) \le 5 \) and \( 2 \le \Im(z) \le 12 \), this is \( \text{tet}_\beta(z) \),
And another large portrait over the domain \( -1 \le \Re(z) \le 5 \) and \( 10 \le \Im(z) \le 20 \).
I'm still having trouble making fast and accurate code near the real-line--but I'm pretty sure I have the upper and lower half-planes well managed.
So I thought I'd post a graph full of artifacts, and explain why we have these artifacts. Pari-GP always chooses the principal branch of the logarithm. Therefore as we increase the imaginary argument; since this function is not normal on \( \Im(z) > 0 \), then you get a cycle between \( \Im(z) >,<0 \).
No where else is this more obvious than with the recursive definition of the code on the Real line. Everything works great away from the real line; but on the real line the recursive definition fails. Where it's trying to swap between \( Im(z) >,<0 \). This is because, in a neighborhood of the real line, the principal branch of \( \log \) is NOT NORMAL on \( \text{tet}_\beta \). So my pretty Taylor series versions are correct, these recursive graphs are not necessarily. Still this looks similar to what I posted before (I'll add it after).
The purple (resp. green) which appears in the upper/lower half plane; is a swap between \( \pi,-\pi \); which forces all the errors. I'm still in the process of making a "Matrix Add On" for this code; which we'll solve this problem. But nonetheless, here's what a better recursive code looks like.
The recursive definition using the principal branch of \( \log \):
The definition using Taylor series; and where we choose arbitrary logs, not just the principal.
