Let \( f(s) \) be one of those recent compositional asymtotics of tetration.
Let \( g(s) \) be its functional inverse.
Now consider the imho interesting equation :
\( f(h(s))=\exp(f(s)) \)
We know that \( h(s) \) must be close to the successor function \( s+1 \) for large real \( s \).
We have that \( h(s)=g(\exp(f(s)) \).
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 :
\( tet(s+k) = f(h^{[s]}(g(1))) \)
or lim n to oo :
\( [tex]tet(s+k) =ln^{[n]}(f(h^{[s]}(g(exp^{[n]}(1))))) \)
( 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 \( tet(s),f(s),g(s),h(s) \) making things harder or properties only locally.
Regards
tommy1729
Let \( g(s) \) be its functional inverse.
Now consider the imho interesting equation :
\( f(h(s))=\exp(f(s)) \)
We know that \( h(s) \) must be close to the successor function \( s+1 \) for large real \( s \).
We have that \( h(s)=g(\exp(f(s)) \).
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 :
\( tet(s+k) = f(h^{[s]}(g(1))) \)
or lim n to oo :
\( [tex]tet(s+k) =ln^{[n]}(f(h^{[s]}(g(exp^{[n]}(1))))) \)
( 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 \( tet(s),f(s),g(s),h(s) \) making things harder or properties only locally.
Regards
tommy1729