This will be a somewhat short thread, as I'm trying to showcase a crazy cool feature of tetration. This is a note on the absolute chaos of how these functions work. To begin, here is a graph:

Although this may look like a linear graph, this is actually three tetrations being graphed. These would be,

\(

F_{\log(2)}(x)\\

F_{\log(5)}(x)\\

F_{\log(7)}(x)\\

\)

over the domain \( x \in [-0.001,0.001] \). They almost agree exactly, with only a slight change in numbers. But, if you've been following my research. Each are holomorphic for,

\(

F_{\log(2)}(x)\,\,\text{is holomorphic for}\,\,|\Im(z)| < \pi/\log(2)\,\,\Re(z) > 0\\

F_{\log(5)}(x)\,\,\text{is holomorphic for}\,\,|\Im(z)| < \pi/\log(5)\,\,\Re(z) > 0\\

F_{\log(7)}(x)\,\,\text{is holomorphic for}\,\,|\Im(z)| < \pi/\log(7)\,\,\Re(z) > 0\\

\)

So they are not the same holomorphic functions. There are singularities on the boundaries of these domains too--so these domains are maximal in a sense. There's an almost fuzz between them which make them each unique, yet, absolutely undistinguishable to the eye. Here is a zoomed out version of this graph at \( x \in [-0.25,0.25] \):

We can see a tad more straying here, but no hint that these are actually 3 different analytic functions! It just looks like a weird graph of one function! The craziest picture is a zoomed out full picture. Here are the above three tetrations graphed over \( x \in [-1,2] \):

This is to say, as we vary the multiplier \( \lambda \) in \( F_\lambda \) we actually only incite a tiny wobble. And different wobbles correspond to different \( \lambda \) (almost like a frequency), but for the most part it's undetectable up to at least 3 digits.

Now to make these graphs I stuck to 9 digit precision, But at 9 digits.... for \( x \in [-0.000000001,0.000000001] \):

We can finally start to see that these are different tetrations. Please note that the boxy nature of this graph is because it's made of only a couple sample points. If you went full hi-res though, this would be a 9 digit accurate result.

I just thought this was really cool. And that, it's nearly impossible to distinguish between \( F_{\log(2)} \) and \( F_{\log(5)} \) on the real-line. But they are holomorphic on different domains.

I thought this was really cool, figured I'd share.

Regards, James

I thought I'd post another graph to fill the quota. Here is the graph of \( F_{\log(2)}(x) - F_{\log(5)}(x) \) over \( x \in [-1,1] \):

Although this may look like a linear graph, this is actually three tetrations being graphed. These would be,

\(

F_{\log(2)}(x)\\

F_{\log(5)}(x)\\

F_{\log(7)}(x)\\

\)

over the domain \( x \in [-0.001,0.001] \). They almost agree exactly, with only a slight change in numbers. But, if you've been following my research. Each are holomorphic for,

\(

F_{\log(2)}(x)\,\,\text{is holomorphic for}\,\,|\Im(z)| < \pi/\log(2)\,\,\Re(z) > 0\\

F_{\log(5)}(x)\,\,\text{is holomorphic for}\,\,|\Im(z)| < \pi/\log(5)\,\,\Re(z) > 0\\

F_{\log(7)}(x)\,\,\text{is holomorphic for}\,\,|\Im(z)| < \pi/\log(7)\,\,\Re(z) > 0\\

\)

So they are not the same holomorphic functions. There are singularities on the boundaries of these domains too--so these domains are maximal in a sense. There's an almost fuzz between them which make them each unique, yet, absolutely undistinguishable to the eye. Here is a zoomed out version of this graph at \( x \in [-0.25,0.25] \):

We can see a tad more straying here, but no hint that these are actually 3 different analytic functions! It just looks like a weird graph of one function! The craziest picture is a zoomed out full picture. Here are the above three tetrations graphed over \( x \in [-1,2] \):

This is to say, as we vary the multiplier \( \lambda \) in \( F_\lambda \) we actually only incite a tiny wobble. And different wobbles correspond to different \( \lambda \) (almost like a frequency), but for the most part it's undetectable up to at least 3 digits.

Now to make these graphs I stuck to 9 digit precision, But at 9 digits.... for \( x \in [-0.000000001,0.000000001] \):

We can finally start to see that these are different tetrations. Please note that the boxy nature of this graph is because it's made of only a couple sample points. If you went full hi-res though, this would be a 9 digit accurate result.

I just thought this was really cool. And that, it's nearly impossible to distinguish between \( F_{\log(2)} \) and \( F_{\log(5)} \) on the real-line. But they are holomorphic on different domains.

I thought this was really cool, figured I'd share.

Regards, James

I thought I'd post another graph to fill the quota. Here is the graph of \( F_{\log(2)}(x) - F_{\log(5)}(x) \) over \( x \in [-1,1] \):