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,

over the domain . They almost agree exactly, with only a slight change in numbers. But, if you've been following my research. Each are holomorphic for,

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 :

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 :

This is to say, as we vary the multiplier in we actually only incite a tiny wobble. And different wobbles correspond to different (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 :

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 and 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 over :

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

over the domain . They almost agree exactly, with only a slight change in numbers. But, if you've been following my research. Each are holomorphic for,

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 :

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 :

This is to say, as we vary the multiplier in we actually only incite a tiny wobble. And different wobbles correspond to different (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 :

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 and 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 over :