[FOM] Analyticity of half-exponentials
joeshipman@aol.com
joeshipman at aol.com
Thu Apr 19 11:53:59 EDT 2007
>
>It is, and the answer to the original question (according to what H.
Kneser
>writes in Section §6 of his paper) is "yes". His solution to f(f(x))
= e^x
>is strictly increasing.
>
>Tjark
OK, that's good. So now I would like Urquhart to elaborate on the
following comment he made:
"This solution, however, is not single-valued (Baker)
and, as pointed out by G. Szekeres, there is no
uniqueness attached to the solution."
If Kneser created his increasing real analytic solution to f(f(x))= e^x
by a process involving arbitrary choices, that's not very satisfactory,
and the question remains how to define the "best" solution (by which
*I* mean a solution constructed by a process which generalizes to find
g such that g(g(x))= f(x), h such that h(h(x))=g(x), etc., with all
functions increasing and preferably analytic).
-- JS
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