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A system of functional equations for slog(x) ?
#1
Let x be a real number.

Im looking for a real-analytic solution slog that satisfies :

slog(1) = 0.
slog(exp(x))= slog(x)+1.
??slog(??) = ??slog(??)

And by those "??" I mean another functional equation on the real line such that slog is indeed real-analytic.

I tried a few cases but it seems hard.

A suggestion is limiting the range and domain by using

slog(sin(x)^2) = ???

Actually I do not know any system of functional equations that gives a real-analytic nontrivial abel function.

One easily gets contradictions with naive try-outs.

It somewhat reminds of those attempts of making ackermann analogues analytic , and other similar hyperoperator ideas.

slog(sin(x)^2) = slog(f(x)) - 1

however leads to f(x) = exp(sin(x)^2).

Just to show how tricky it is.

Nevertheless Im optimistic although that may be a bit crazy.

regards

tommy1729
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#2
A candidate functional equation is already given here :

http://math.eretrandre.org/tetrationforu...hp?tid=852

although I prefer a simpler one if possible.

I believe in a solution.

regards

tommy1729
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#3
Remind me why the functional equation for superexponentiation (or the superlogarithm) isn't adequate? I seem to recall that there was an issue with apparent inconsistencies?

sexp(x+1) = exp(sexp(x))
slog(exp(x)) = slog(x)+1

Was it just an issue with branches? If so, is that really a problem?
~ Jay Daniel Fox
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#4
(07/28/2014, 05:06 PM)jaydfox Wrote: Remind me why the functional equation for superexponentiation (or the superlogarithm) isn't adequate? I seem to recall that there was an issue with apparent inconsistencies?

sexp(x+1) = exp(sexp(x))
slog(exp(x)) = slog(x)+1

Was it just an issue with branches? If so, is that really a problem?

There are many subtle issues , but it relates to our ignorance.
For instance if exp(exp(v)) = v and v is not the first order fixpoint of exp then equation slog(exp(exp(x))) = slog(x)+2 cannot be both analytic and valid near the point Q with slog(Q) = v.

Despite many posts and progress here , A full understanding of these kind of things is not reached yet.
Functional equations are tricky for complex numbers when functions G are involved such that iterations of G are chaotic.

One simple solution seems to say that Q must lie on another branch with another functional equation.
But it seems not to be solved that easy and intuitive.
Why ? Well because for instance the functional equations on the branches do " not care " about the positions of higher order fixpoints.
And no matter how you choose your branches , this cannot be solved trivially and perfectly due to chaos.

That is just one example.

HOWEVER the point (of the OP) is not an inconsistancy but the fact that EVEN FOR THE REALS these functional equations ALONE do not give uniqueness.

Adding an additional functional equation might be a nice road to another solution for tetration.

That was the intention. In particular with focus on the real line.

regards

tommy1729
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