Hi Daniel,
of course the uniqueness is one essentially important issue about tetration.
Currently we have collected here perhaps 5 different methods to compute an analytic tetration. If they are different, which is 'the correct', 'the right', 'the proper' or 'the best' tetration?
This question (though there were not different computational approaches at his time) already appeared to Szekeres in his article "Fractional iteration of exponentially growing functions" 1961. Where he also mentions that if we have one analytic superexponential sexp, then the function \( f(x)=\text{sexp}(x+\frac{1}{2\pi}sin(2\pi x)) \) is another analytic superexponential, which is even strictly increasing if sexp was.
For overview here are the different established methods to compute an analytic tetration:
In opposition to the numerous computation methods, uniqueness criterions are rare. To my knowledge the only ones existing are the ones I tried to present here on the forum. And I currently prepared an article about uniqueness with Dmitrii.
My vision is that all (or at least most) analytic tetration methods become united by showing that they are equal, by satisfying the same uniqueness criterion.
Only if this is achieved one can properly speak about "the" tetration.
Otherwise its just a bunch of personal taste tetrations.
of course the uniqueness is one essentially important issue about tetration.
Currently we have collected here perhaps 5 different methods to compute an analytic tetration. If they are different, which is 'the correct', 'the right', 'the proper' or 'the best' tetration?
This question (though there were not different computational approaches at his time) already appeared to Szekeres in his article "Fractional iteration of exponentially growing functions" 1961. Where he also mentions that if we have one analytic superexponential sexp, then the function \( f(x)=\text{sexp}(x+\frac{1}{2\pi}sin(2\pi x)) \) is another analytic superexponential, which is even strictly increasing if sexp was.
For overview here are the different established methods to compute an analytic tetration:
- Regular iteration at the lower real fixed point (for bases \( <e^{1/e} \))
- Matrix power method (introduced here by Gottfried, origin unknown?) to compute fractional iterates (The method is explained in this article)
- Walker's method (rediscovered by Robbins) to compute the power series of a super logarithm
- Kouznetsov's method computing directly numerical values of sexp via converging Cauchy integrals along a boundary (for bases \( >e^{1/e} \)).
- Jay's method via a direct limit (for bases \( >^{1/e} \), is it analytic?).
In opposition to the numerous computation methods, uniqueness criterions are rare. To my knowledge the only ones existing are the ones I tried to present here on the forum. And I currently prepared an article about uniqueness with Dmitrii.
My vision is that all (or at least most) analytic tetration methods become united by showing that they are equal, by satisfying the same uniqueness criterion.
Only if this is achieved one can properly speak about "the" tetration.
Otherwise its just a bunch of personal taste tetrations.