(10/18/2022, 07:49 AM)Shanghai46 Wrote: Well I'm not sure that techniques that give different results can both be correct, but you have identified an important step in tetration research, the categorization of the different methods and the reconciliation of their different results.
Whats in your opinion the best extension method? And why?
As for the best method, I will let others more knowledgeable weigh in. I do have my own approach which is one of the simpler attempts which is seen and being an extension of Schroeder's work. My work is also consistent with much if not all of the work Gottfried does with matrices. I suspect that things have been held up for the last fifteen years because of a lack of good convergence proof. Paulsen's work breaks the logjam with it's proof of convergence. It is also based on earlier work while providing better, simpler and cleaner proofs. Paulsen says his work is consistent with Kouznetsov and Trappmann's (Bo) work. There are links at the bottom of my Tetration.org page to Paulsen's and others work.
As far as the best technique I consider that to be mathematical religion. Some folks like me are comfortable with real values being mapped to complex values; others believe real values should map to real values. I'm agnostic - I believe at the different methods should be studied and reconciled.
Daniel