[FOM] Analyticity of half-exponentials

[Alasdair Urquhart]

> Is there a monotonic real analytic function defined  on the
> non-negative real numbers such that f(f(x)) = 2^x, or f(f(x))=e^x?

These kind of questions have a long history going back to work
of Ernst Schroeder in 1871.  Here is a quote from a 1968 book
that I have copied from an online posting of Professor
Zdislav Kovarik of McMaster University:

Marek Kuczma: Functional Equations in a Single Variable
Monografie Matematyczne 46, Warsaw 1968
In Ch. XV, Sec. 6, he writes [modified for ASCII format]:
 "For the equation
(*)              f^2(x) = e^x,
a real analytic solution has been found by H. Kneser.
This solution, however, is not single-valued (Baker)
and, as pointed out by G. Szekeres, there is no
uniqueness attached to the solution. It seems reasonable
to admit f(x)=F^(1/2)(x), where F^u is the regular
iteration group of  g(x)=e^x, as the "best" solution of
the equation (*) (best behaved at infinity). However,
we do not know whether this solution is analytic for
x>0.

[Kuczma defines and discusses regular iterations at
infinity in Chapter IX, Sec 5.]

References:

Baker, I.N.: The iteration of entire transcendental
 functions and the solution of the functional equation
 f(f(z))=F(z), Math. Ann. 120(1955), pp. 174-180

Kneser, H.: Reele analytische Loesungen der Gleichung
 f(f(x))=e^x und verwandten Funktionalgleichungen,
 J. reine angew. Math. 187(1950), pp. 56-67

Szekeres, G.: Fractional iterations of exponentially
growing functions,
 J. Australian Math. Soc. 2(1961/62), pp. 301-320