[FOM] Analyticity of half-exponentials
Alasdair Urquhart
urquhart at cs.toronto.edu
Mon Apr 16 10:19:52 EDT 2007
> 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
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