(11/22/2021, 07:42 AM)Ember Edison Wrote: Your pictures of \( \mu = \lambda = 1+i \), The four slashes inside the image look like the function has crashed.
I also recently tested that in base=\( e^10^24 \) or init(1,1E24,1000), the beta function would crash.
init(1,1E-24,1000) will also crash.
I also created images of a circle scanned in the complex plane with a radius of 1/1E16/1E-16. The images of the real axis still need some time.
I do not believe they crashed for \(\lambda = 1+i, \mu = 1+i\), I believe that's just what the branch cuts are going to look like. It is slanting at the right angle; that's the direction the branch cuts should go in; and the functional equation is still satisfied. You'll see this a lot for \(\lambda\) complex; it'll begin to branch in a chaotic manner. I like to categorize tetration functions as functions taking \(\mathcal{P} \to \mathbb{C}\) where \(\mathcal{B} = \mathbb{C}/\mathcal{P}\) are the set of discontinuities; and so long as:
\[
\int_{\mathcal{B}}\,dA = 0\\
\]
For \(dA\) the standard Lebesgue area measure of \(\mathbb{R}^2\); it's good enough to be a tetration in my boat. So the beta tetrations account for branch cuts and singularities; but are still holomorphic almost everywhere.
I am not surprised base \(e^{10^{24}}\) crashes; sadly I can't think of anyway to fix that. I'll take a look at \(e^{10^{-24}}\), though, there might be a way to fix that, I'll look.