share with others#plotted #functional_iterations

[JmsNxn]

I also had graphs for sin at 0(complex) tan at 0(complex) and e^x-1 at 0(complex) but in another pc, will append em soon

I thought, I am the one who makes a lot of animated pictures!
Looking forward.

God I tried making animations and just got so god damned frustrated. I wish I knew how to use Sage better; but I couldn't get passed the syntax used to convert Pari-gp code into the Sage shell; and then run Sage's graphing protocol. I tried reading some breakdowns, but to no avail. Nothing helped. Especially because all the tutorials seem to be for the free to use Sage online app that seems to be the status quo now. I would love to make animated plots though. Ember made some animated plots of the beta method as you move \(b\) around.


Here are some graphs that are my favourites because they have cool fractals.

This is tetration base \(\sqrt[3]{3}\) with \(2 \pi i\) period; and centered at the singularities which happen at \(j+\pi i\). This is one of many "candy stripe graphs" I have

   

This is a similar graph, but we've changed to \(\sqrt[3+0.5i]{3}\)--still \(2 \pi i\) period.

   

This is a similar graph, but we're about \(b = e^{-0.3-i}\)

   

This one is really cool because it's for a period that looks like \(2\pi i/(1+i)\) and for base \(b = e^{1+i}\), so it gets pretty whacky. This one isn't a tetration, it's just the asymptotic solution.

   

And then the all-time disappointment. This is when we let the period be \(2\pi i/0.25\) and the base \(b = e\). You can see what sheldon called the clustering of zeroes and branch cuts near the real line. They look like little tornadoes or something. Ultimately led to the proof that the beta method isn't analytic anywhere in \(\mathbb{C}\) when the base is \(b=e\). The only hope would be to let \(0.25\) get smaller and smaller (where the coefficients do get more regular).

   


Also, I have hundreds of these graphs if anyones interested. I have about 1gb of tetration graphs