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