The Different Fixed Points of Exponentials

[bo198214]

Have you read Paulsen's paper on Complex tetration?

He claims you can construct holomorphic tetration \(\text{tet}_b(z)\) in \(b\) everywhere except a branching problem at \(\eta\) and \(0\). His paper is very enlightening, as the majority of it is analysing how the fixed points behave within and outside the Shell-Thron region.

https://www.researchgate.net/profile/Wil...-bases.pdf

I was reading his paper about the double dagger method, but not this one yet. When skimming through however, I don't see any proofs for the construction (convergence and equality to Kneser). Ok, but if I understand that properly, he says that numerically his solution (which is supposed to be Kneser's) can be continued into the Shell-Thron region except \(\eta\) is a branch point.
(Or does he give a proof that Kneser can be (without any numerical assumptions) continued through the Shell-Thron boundary?)
That's cool, and actually my experiments with the regular iteration look similar.

EDIT: Also, are you using pari to create these animations, if so you have to tell me how, I've wanted to animate things for so long...

No, I, since always, use sage (online even, on cocalc.com) for any computational needs.