[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. 
>writes in Section §6 of his paper) is "yes".  His solution to f(f(x)) 
= e^x
>is strictly increasing.

  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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