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Arguments for the beta method not being Kneser's method
Very god damned fascinating, Sheldon. I'm going to have to read back up on Gaussian quadrature and all that nonsense (because I'm pretty sure that's what you're using; just forget the right word for it). I never cared for any of the math behind Gaussian speed ups, and clever integral representations, because I always focused on algebraic representations.

I'm a little dumbfounded by how you are calculating logrho so fast about the singularity--but it makes sense for the most part. We are centering at a value \(\log\rho^n(z_0) = 0\) which has a convergence radius for r; and then we sample a circle about the radius. Then we do some gaussian magic (lol, this is what I need a better explanation of; but it's definitely a symptom of my lack of understanding Gaussian/Riemann quadrature integral speed up black magic!)

I've already started going into salvage mode. I'm looking at how well of an asymptotic approximation this beast really is. And I think I might have an alternative approach which best describes how the asymptotic scenario really works. And I think, much of this is becoming avoidable; but only when \(\lambda\to 0\) and we do it cleverly enough.

I'm going to work on the claim \(\text{tet}_\lambda(s)\) is holomorphic everywhere on \(\mathbb{C}\) upto a set \(\mathbb{E}_\lambda\) in which:

\int_{\mathbb{E}_\lambda} 1\cdot dA = 0\\

Which is a better way of saying my original claim. And second of all; fortifying it as more of an asymptotic solution. Last of all, making sure the same argument works for all bases/multipliers. This still allows for singularity walls; weird fuck ups; and all sorts of taylor series shenanigans. This doesn't affect the first 30 pages of my paper too much; just requires me to choose better language.

I'm still on the fence on the last part of my paper; but I feel that as \(\lambda \to 0\) the set \(\mathbb{E}_\lambda \to (-\infty,-2]\) and \(\text{tet}_\lambda \to \text{tet}_K\). All my numbers and thoughts and proof sketches point towards a normality in the left half plane as we limit \(\lambda \to 0\). This is in tune with the later parts of my paper; but I definitely need to write this out deeper.

Also, I'm beginning to understand why \(\mathbb{R} \subset \mathbb{E}_\lambda\) in a good topological way. I can't really explain this yet; the words are on the tip of my tongue; but I don't have them just yet.

Regards, James

Essentially, I've begun looking at \(q(z) = \text{tet}_1(\log(z))\) which satisfies \(q(e\cdot z)=e^{q(z)}\)--where no such analytic solution can exist on \(\mathbb{R}\). This function is holomorphic almost everywhere for \(\Im(z) > 0\)--but that's all we can say. If we try to compare the difference along the border \(\Im(z) >0 \) and \(\Im(z)< 0\) we get a buch of fractal errors. A similar result holds for \(q_\lambda(e^\lambda \cdot z) = e^{q_\lambda(z)}\); with the normality conditions I have; they must be non-analytic on \(\mathbb{R}\). The only language I can think of that's equivalent; is that we are asking the Schroder functions about the fixed point \(L,L^*\) to magically agree on \(\mathbb{R}\). You and I both know that's nonsense.

But! \(q_\lambda(z)\) will be somewhat holomorphic for \(0 < \Im(z) < 2\pi i/\lambda\); with fractal properties near the boundary.

BUT! as \(\lambda \to 0\) this equation already diverges. And we're asking a divergent Schroder function to equal a divergent Schroder function on the real line. This has much more luck; wayyyyyyy more luck; seeing as this thing still converges.

To such an extent that as \(\Re(z) \to - \infty\) while \(\lambda\to0\) we approach holomorphic functions for \(\Im(z) >0\) and \(\Im(z) < 0 \)--but they agree on the real line. And they look like kneser because as \(|z| \to \infty\) for \(\pi/2 < \arg(z) <\pi\) the function \(\lim_{\lambda \to 0} \text{tet}_\lambda(z) \to L\)--and similarly in the lower half plane. This is a uniqueness condition per Paulsen & Cowgill.

Additionally the more I graph the solutions as \(\lambda \to 0\) we decay to the fixed points \(L,L^*\) geometrically with \(\lambda\). And if you thought the singularities quiet. I suggest looking at mult = 0.001 in my code (making sure to do about 1000 iterations); we have a bunch of fractals near \(\mathbb{R}\) but we get a huge area of convergence towards \(L\). I think it's because we're pushing closer and closer to Kneser.


Messages In This Thread
RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/22/2021, 03:54 AM

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