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.
Regards.
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.
Regards.