Rank-Wise Approximations of Hyper-Operations

[Catullus]

First of all, a^^z is always solvable. It's which solution that matters. Why would you choose a^^1/2 is complex when you can find one that's real. This doesn't mean one is better than the other; this means we are searching for different criteria. This forum is not about the one and only solution of tetration; it's about the many ways to solve tetration. So yes, what is written here is a solution. It's a solution solved by Schroder though, and ultimately is not that helpful, other than with iterated exponentials near a complex fixed point \(a^L = L\). This isn't a problem. I was choosing \(e\) as an example of where all of this would fail. Any \(a>\eta\) suffers the problems this has when \(a=e\).

A real valued tetration like that would not agree with the limit formula from https://math.eretrandre.org/tetrationfor...42#pid4442. However there might be a complex valued one that does! How would that tetration behave? Also the main point of this thread was going to be about approximation(s) of convergent hyperoperations that become better and better as the rank goes up, like how when 1<a<η a^^x can be approximated better and better as x becomes bigger and bigger by LambertW(-ln(a))/ln(a)-b*LambertW(-ln(a))^x. Where b=lim x → ∞ (LambertW(-ln(a))/ln(a)-a^^x)/LambertW(-ln(a))^x. It might be that as x grows larger and larger for some a a[x]b converges exponentially to a for any b. sqrt(2)+2~3.414. sqrt(2)*2~2.828. sqrt(2)^2=2. sqrt(2)^^2~1.633. sqrt(2)^^^2~1.520. They do seem to converge. How would we approximate the convergence as x grows larger and larger of a[x]b as a function continuous in terms of the x variable that becomes better and better as x grows larger and larger, when a[x]b converges? Even if a>η (Resulting in a non-real valued tetration.) a[x]b, for any b may converge with larger and larger x.