The nature of g(exp(f(s)) tommy1729 Ultimate Fellow Posts: 1,859 Threads: 402 Joined: Feb 2009 05/05/2021, 11:23 PM (This post was last modified: 05/05/2021, 11:26 PM by tommy1729.) 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 « Next Oldest | Next Newest »

 Messages In This Thread The nature of g(exp(f(s)) - by tommy1729 - 05/05/2021, 11:23 PM RE: The nature of g(exp(f(s)) - by JmsNxn - 05/06/2021, 12:50 AM RE: The nature of g(exp(f(s)) - by tommy1729 - 05/12/2021, 12:23 PM RE: The nature of g(exp(f(s)) - by tommy1729 - 05/12/2021, 12:28 PM RE: The nature of g(exp(f(s)) - by Gottfried - 05/12/2021, 02:07 PM RE: The nature of g(exp(f(s)) - by JmsNxn - 05/13/2021, 04:11 AM RE: The nature of g(exp(f(s)) - by tommy1729 - 05/19/2021, 10:06 PM RE: The nature of g(exp(f(s)) - by JmsNxn - 05/19/2021, 10:37 PM RE: The nature of g(exp(f(s)) - by tommy1729 - 05/22/2021, 12:27 PM

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