G. F. Romerio
Le Cannet - France ; Saluzzo - Italy
Registered: Sep 2004
Tetration extended to real super-exponents
Hello Mr Dahl !
Thank you for your interest in the problem of extending the validity the tetration operation to real super-exponents. I am writing down these provisional comments during a pause between two trips (Germany and Italy). Despite the fact that, unfortunately, I didn’t have enough time to examine your hypotheses and, obviously, your codes, I should like to congratulate you for the work you have done so far. Some active interest is beginning to focus on this matter, from several horizons.
In Wikipedia, for instance (please see: http://en.wikipedia.org/wiki/Talk:Tetration), Daniel Geisler (the author of the tetration.org Web site) mentioned the postings of the present thread, as well as the Rubtsov/Romerio tetration and slog software, saying (in a rather amusing way) that he can't support the authors’ conclusions but, since he couldn’t really name anyone who support his own conclusions, so it's all good like that (!!!). Moreover, he thinks that the calculator that we proposed does produce a much larger value for e # Pi that he can justify, but that that value is more in line with Andrew Robins' estimate. The solution proposed by Andrew Robins is similar to our simulations and even more accurate.
Indeed, concerning the E-tetra-Pi operation, we have:
for Rubtsov/Romerio: e-tetra-pi = 1.9337456547928 . 10^10
for Ingolf Dahl: e-tetra-pi = 3.71504639065471391171. 10^10
As a provisional conclusion, so to say, our (Rubtsov/Romerio) result is too large according to Daniel Geisler, but near what was obtained by Andrew Robin. Your (Ingor Dahl) value is even larger but, after all, … “only” almost the double of it. Taking into account the order of magnitudes involved in such results [10 ^ 10], this is not a catastrophe. Nevertheless, mathematically speaking, since everybody cannot be right at the same time, probably … somebody (or … everybody) is wrong. I would be happy to tell you that you succeeded and perhaps it is really so. I shall come back to this as soon as possible, also concerning your approximations of Pi-tetra-E.
Actually, the problem of extending tetration (in reality the “tetrational function”) to the real super-exponents x is similar to finding a continuous and derivable function that will coincide will the known values obtained for n = integer(x). Other similar problems are, for example:
(a)– to give a meaning to an expression such as y = b ^ x, for x real, once known
its meaning for x integer (problem solved in the framework of elementary Algebra);
(b)– to give a meaning to y = x ! (the factorial of x, with x real), once known its meaning for x natural (problem solved in the framework of advanced Algebra, with the definition of the Gamma function).
Our general problem here is to give a meaning to an expression such as:
y = b-tetra-x = b # x (with b > 0 and x real > -2)
Unfortunately, we only know how to do it, partially and in some particular cases, by more or less accurate approximations. The general approach should involve the solution of the problem of the analytical continuation of iterated operators. Perhaps fractional calculus and/or the study of functional equations (as you are trying to do, if I am not mistaken) could bring all of us fully “around the corner”. I attach some qualitative simulations obtained by using Mathematica, for various values of the base b. These plots show that, under some conditions, tetration must oscillate.
Giovanni F. Romerio (Gianfranco)
Last edited by G. F. Romerio on 12-31-2006 at 09:02 PM
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