Quickest way to compute the Abel function on the Shell-Thron boundary
#1
I'm currently writing a protocol to evaluate the modified bennet operators:

$$
\alpha <s>_{\varphi} y = \exp_{y^{1/y}}^{\circ s}\left(\log^{\circ s}_{y^{1/y}}(\alpha) + y + \varphi\right)\\
$$

For \(\varphi\) a complex number--I'm mostly just dealing with real positive values at the moment. The goal is to evaluate the function \(\varphi(\alpha,y,s)\), such that these operators will satisfy Goodstein's equation:

$$
\alpha <s> \left(\alpha <s+1> y\right) = \alpha <s+1> (y+1)\\
$$

But for the moment, I'm just concerning myself with calculating the first function. Everything works great so far, but I'm scratching my head for when \(y^{1/y} = \partial \mathfrak{S}\)-- when it's on the boundary of the Shell-Thron region (equivalently \(|\log(y)| = 1\)). Now I know we can construct a repelling and attracting Abel function about these points--and I know all the theory. But I just realized, I've never actually seen a program that constructs it. I know Sheldon has a program for handling it, but I really don't want to go digging through all the matrix theory. I just want a quick formula. I know if you make a variable change that it becomes pretty elementary.

So for the moment, I can construct \(\alpha <s>_{\varphi} y\), for pretty much the entire complex plane in \(y\) (excluding branch cuts), excluding where \(y^{1/y} \in \mathfrak{S}\). This is primarily because I don't know a fast way to get both abel functions... I could program in a way, but I think it's going to be way too slow. This program is already pretty slow as it is (we have to consistently reinitialize to account for varying bases of the exponential). I don't want to slow it down any more. I was wondering if there's anywhere on this forum that has an easy to read program I can adapt for this.

...I just hope I don't have to write too much just to handle the case \(|\log(y)| = 1\) -_-....

Edit: I thought I'd add that I know how to write in the neutral case but it just nukes the speed of my code. I know how to program in the \(\eta\) case, but I'm wondering what the current fastest way is. For the moment, I'm just returning \(0\) anywhere on the boundary, because it just nukes my code and makes everything so fucking slow for these values of \(y\).
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