05/31/2021, 07:34 AM
I managed to make the following graph by storing a matrix of Taylor series. I'm working on making a separate program intended to work through a matrix method. I'm kind of reverse engineering how Sheldon was storing Taylor series in a matrix to make fatou.gp; and I think I'm getting the hang of it. Again, this is not anything like Sheldon's process though; I'm just appreciating how he stored matrices to make such a great calculator.
Here is my tetration \( \text{tet}_\beta(z) \) for \( -1 \le \Re(z) \le 4 \) and \( |\Im(z)| \le 5 \). The graph isn't perfect, by no measure at all. I only used a 100 term Taylor Series; and my Taylor series converge very very slowly; and I sparsed out my data points too much, I think. Especially near the real line, and especially near the singularity at \( z=-2 \). I'm hoping Sheldon may have some light to shed on a more efficient way to make a matrix method approach.
The main reason I am thinking this isn't Kneser's is because this tetration appears to slowly diverge as we increase the imaginary argument in either direction. I'm still working on a proof that this tetration is not Kneser's; but I've developed a couple heuristic arguments by this point. It also, doesn't look exactly like Kneser's.
I've kind of categorized this solution as the solution about the fixed point at infinity of \( \exp(z) \). Where the fixed point only exists as \( \exp(z) : \mathbb{C}_{\Re(z) > 0} \to \mathbb{C} \) where this restricted exponential map satisfies \( \exp(\infty) = \infty \) in a well enough manner. This is then, very god damn similar to Kneser's iteration. But it isn't about a fixed point; unless you count \( \infty \) as a weird kind of fixed point. It's structured on solving Schroder's equation at infinity.
Here is my tetration \( \text{tet}_\beta(z) \) for \( -1 \le \Re(z) \le 4 \) and \( |\Im(z)| \le 5 \). The graph isn't perfect, by no measure at all. I only used a 100 term Taylor Series; and my Taylor series converge very very slowly; and I sparsed out my data points too much, I think. Especially near the real line, and especially near the singularity at \( z=-2 \). I'm hoping Sheldon may have some light to shed on a more efficient way to make a matrix method approach.
The main reason I am thinking this isn't Kneser's is because this tetration appears to slowly diverge as we increase the imaginary argument in either direction. I'm still working on a proof that this tetration is not Kneser's; but I've developed a couple heuristic arguments by this point. It also, doesn't look exactly like Kneser's.
I've kind of categorized this solution as the solution about the fixed point at infinity of \( \exp(z) \). Where the fixed point only exists as \( \exp(z) : \mathbb{C}_{\Re(z) > 0} \to \mathbb{C} \) where this restricted exponential map satisfies \( \exp(\infty) = \infty \) in a well enough manner. This is then, very god damn similar to Kneser's iteration. But it isn't about a fixed point; unless you count \( \infty \) as a weird kind of fixed point. It's structured on solving Schroder's equation at infinity.
