A first hires look at tetration \(\lambda = 1\) and \(b = e\)

[JmsNxn]

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.

Do you have any opinion about \(  b \to \pm \infty \)? Like the folklore about \( tet_0(z) / tet_{\infty}(z) \) inside the forum, in the images they clearly show some kind of symmetry

I apologize, but my knowledge of this forum doesn't go as far back as I wished it did. There are many posts; even if I was a member; I was 16 at the time and was barely grasping calculus, lol. If you could explain what you mean exactly, I'd be happy to comment.

And good news. I have figured out how to make init(1,1E24) to work; I'm just thinking of the best way to program it at the moment, such that it works conducively with the rest of the program. Give me 5 days or so, and I think I can get it to work. It's a small fix, but I want to make sure it doesn't screw up anything major. But just so you know, the value \(\beta(1)\) will already be an overflow to the best of my calculations. The proposed solution will successfully grab the taylor series; but the recursive protocol may fail because it requires searching in a radius of \(|z| \le 10^{-24}\) and applying \(\exp\) a total of \(10^{24}\) times to get the next value; and this typically results in a recursion error on pari's part. I'm seeing if I can bypass this some how. Worst comes to worst. I can definitely produce the series:

\[
\beta_{1,10^{24}}(z) = \sum_{j=1}^\infty a_j e^{zj}\\
\]

But then trying to push forward I imagine may cause difficulties.  Again, give me five days, I'll see if I can integrate the code. I imagine you'll have to take a larger sample than a 1000 too, because this series will converge much slower than when \(b\) is a reasonable value.

EDIT: This is the final edit of this post.  I've got it to work at a beta level with previous code. Just let me debug at this point. We have to add another function which behaves like beta, but works for absurdly large values. I need to install more ram for the most part. Luckily I recently bought another 32 gb stick so I can let pari-gp get ridiculous. I'll update in 5 days or so. Again, I see how to make it work; it's just a matter of making it compatible with the previous programming. Also, there's a tad bit of arbitrary; so it fails for crazy values; but you can tweak it to make it work. I use the sqrt function but it isn't necessary; I just pulled it from a hat.