The nature of g(exp(f(s))

[tommy1729]

Let be one of those recent compositional asymtotics of tetration.

Let  be its functional inverse.

Now consider the imho interesting equation :



We know that  must be close to the successor function  for large real .

We have that .

I feel like studying this is an important and logical step.

Especially for nonreal s or s being small.

One of the proposed solutions was/is then :



or lim n to oo : 



( for some fixed k , using appropriate ln branches )

Both compute the same function or should (?!)...
But with different practical considerations.

Error terms such as O(exp(-s)) would be usefull too ofcourse.

However do not forget possible singularities of   making things harder or properties only locally.

Regards

tommy1729