08/14/2022, 06:20 AM
So this is a fun one, that I think is extraordinarily interesting. This is related directly into "semi-operators" or "half-operators" as we can call them. If we take:
\[
x \langle s\rangle y = \exp^{\circ s}_{y^{1/y}}\left(\log^{\circ s}_{y^{1/y}}(x) + y\right)
\]
Then this function interpolates \(+,\times,\exp\). And nearly satisfies the equation:
\[
x\langle s \rangle (x\langle s+1\rangle y ) \approx x \langle s+1 \rangle (y+1)\\
\]
I always wanted to make an animation of these continuous interpolations, because they have all the properties of half exponentials.
Anyway, here is \(3 \langle 1.9\rangle y\), which is almost periodic like \(3^y\), but just misses the criteria. These iterations are done entirely using the repelling fixed point iteration (about \(4\) rather than \(2\)).
And to imagine what the animation would look like, here is \(3 \langle 1.5 \rangle y\):
This one is an example of a super function of exponentiation when the base is \(1/2\) and, yes the period is \(2 \pi i\). You can see a lot of chaos, but in this instance, this function is pretty much holomorphic everywhere--save a few singularities and branch cuts, lol. And yes, IT IS TETRATIONAL. The thing is, there are countably infinite \(x_0\) to choose as a normalization point
.
This one is a graph that is trying to construct Kneser's iteration. This is done using the conjectured formula:
\[
\text{tet}_{K}(s) = \lim_{n\to\infty} \log^{\circ n} \beta_{1/\sqrt{1+s+n + s_0}}(s+n+s_0)\\
\]
This graph is still incorrect though, and isn't converged perfectly yet; as this was a low recursion graph (still probably took me 3 days to compile, lol).
This is another graph which attempted to show the regularity of the \(\beta\) method when you let \(\lambda \to 0\):
Despite the niceness of this graph, we have fairly poor Taylor series data. Not disproving the result, just slow. But it looked a good amount like Kneser's Taylor data. Just converging poorly.
\[
x \langle s\rangle y = \exp^{\circ s}_{y^{1/y}}\left(\log^{\circ s}_{y^{1/y}}(x) + y\right)
\]
Then this function interpolates \(+,\times,\exp\). And nearly satisfies the equation:
\[
x\langle s \rangle (x\langle s+1\rangle y ) \approx x \langle s+1 \rangle (y+1)\\
\]
I always wanted to make an animation of these continuous interpolations, because they have all the properties of half exponentials.
Anyway, here is \(3 \langle 1.9\rangle y\), which is almost periodic like \(3^y\), but just misses the criteria. These iterations are done entirely using the repelling fixed point iteration (about \(4\) rather than \(2\)).
And to imagine what the animation would look like, here is \(3 \langle 1.5 \rangle y\):
This one is an example of a super function of exponentiation when the base is \(1/2\) and, yes the period is \(2 \pi i\). You can see a lot of chaos, but in this instance, this function is pretty much holomorphic everywhere--save a few singularities and branch cuts, lol. And yes, IT IS TETRATIONAL. The thing is, there are countably infinite \(x_0\) to choose as a normalization point
.This one is a graph that is trying to construct Kneser's iteration. This is done using the conjectured formula:
\[
\text{tet}_{K}(s) = \lim_{n\to\infty} \log^{\circ n} \beta_{1/\sqrt{1+s+n + s_0}}(s+n+s_0)\\
\]
This graph is still incorrect though, and isn't converged perfectly yet; as this was a low recursion graph (still probably took me 3 days to compile, lol).
This is another graph which attempted to show the regularity of the \(\beta\) method when you let \(\lambda \to 0\):
Despite the niceness of this graph, we have fairly poor Taylor series data. Not disproving the result, just slow. But it looked a good amount like Kneser's Taylor data. Just converging poorly.