07/18/2014, 12:23 PM

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

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