A compilation of graphs for the periodic real valued tetrations

[JmsNxn]

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 \( x \in [-1,2] \):

   

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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.

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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

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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 \( x \in [-1,1] \):


   

[JmsNxn]

Okay, I've been compiling a bunch of graphic information. And if we call,

\(
F_\lambda(s) = \lim_{n\to\infty} \log^{\circ n}\beta_\lambda(s+x_\lambda+n)\\
F_\lambda(0) = 1\,\,\text{for all}\,\, \Re(\lambda)>0\\
F_\lambda(s+1) = \exp(F_\lambda(s))\\
F_\lambda(s) : \mathbb{R}^+ \times \mathbb{R}^+ \to \mathbb{R}^+\\
\)

Then,

\(
\text{tet}_\beta(s) = \lim_{\lambda\to 0^+} F_\lambda(s)\\
\)

What this is essentially saying is that we can interchange the limits. The construction I had was,

\(
\text{tet}_\beta(s) = \lim_{n\to\infty} \lim_{\lambda \to 0^+} \beta_\lambda(s+n+x_\lambda)\,\,\text{while}\,\,\lambda = \mathcal{O}(n^{-\epsilon})\,\,\text{for}\,\,0 < \epsilon < 1\\
\)

But we can do one better by writing,

\(
\text{tet}_\beta(s) =\lim_{n\to\infty} \lim_{\lambda \to 0^+} \beta_\lambda(s+n+x_\lambda)\,\,\text{while}\,\,\lambda = \mathcal{O}(n^{-\epsilon})\,\,\text{for}\,\,0 < \epsilon < 1\\
= \lim_{\lambda \to 0^+}\lim_{n\to\infty} \beta_\lambda(s+n+x_\lambda)
= \lim_{\lambda \to 0^+} F_\lambda(s)\\
\)

And that my construction was existence/construction; but once that's shown--we can use alternative ways of representing. We can just limit the multiplier to zero. We don't need anything too fancy.

This is to entice the reader into thinking that we are really just expanding the strip of holomorphy. Where,

\(
F_1(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < \pi\,\,\text{excluding}\,\,(-\infty,-2]\\
F_{0.5}(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < 2\pi\,\,\text{excluding}\,\,(-\infty,-2]\\
F_{0.25}(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < 4\pi\,\,\text{excluding}\,\,(-\infty,-2]\\
F_{0.125}(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < 8 \pi\,\,\text{excluding}\,\,(-\infty,-2]\\
F_{0.0625}(z)\,\,\text{is holomorphic for}\,\, |\Im(z)| < 16 \pi\,\,\text{excluding}\,\,(-\infty,-2]\\
\vdots\\
\text{tet}_\beta(z)\,\,\text{is holomorphic for}\,\,z\in\mathbb{C}\,\,\text{excluding}\,\,(-\infty,-2]\\
\)
 

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I think this is the revelation I've been looking for. I'm able to make graphs of say \( F_{0.1}(z) \) much better than I can make a graph of \( \lim_{n\to\infty} \log^{\circ n} \beta(z+n) \); however, on the real line they barely disagree--about an error of 1E-10. So, as I see it; this cracks the code of how to map the Riemann sphere so that \( \lim_{\Im z \to \infty} \text{tet}_\beta(z) = \infty \). We've literally just shrunk the multiplier to zero.

So the periodic function \( \lambda(z) \) is actually just \( \lambda = 0 \); how cool is that!?!?!?!?

Guys, this is the cracked code of the beta method! The trivial solution to the equation,

\(
\text{tet}_\beta(z) = F_{\lambda(z)}(z)\\
\lambda(z+1) = \lambda(z)\\
\lim_{\Im z \to \infty} \lambda(z) = 0\\
\)

IS THE CORRECT SOLUTION! I can't believe I was always "normalizing" at the end. The trick is to include the function \( x_\lambda \) in the construction.  All I have to do is show that the interchange of limits is perfectly legal; which isn't much. It's just ensuring on compact sets the limit stays stable. I'm so god damn excited! Can't believe I didn't think of this !

Regards, James

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Here's a graph of \( F_{\log(1.5)}(z) \) over the domain \( -2 \le \Re(z) \le 3 \) and \( 0 \le \Im(z) \le 5 \); which is almost \( \text{tet}_\beta \); there's about a \( 1 E -3 \) discrepancy on the real line. This function is holomorphic for about \( |\Im(z)| \le 7 \) excluding the branch cut.