[FOM] Analyticity of half-exponentials

[Alasdair Urquhart]

Here is an attempt to answer Joe Shipman's original
question.  Let me say to start with that I am an ignoramus
in this area (as my earlier posting should have made
clear -- I was just quoting stuff I found online).

I have now looked at the three papers that I referenced
in my earlier posting, and I believe that the answer
to the question (about phi(phi(x)) = e^x ) is "yes."

Here is a quote from the 1961 paper by George Szekeres:

"The fractional iteration of e^x and solutions of the
functional equation

(1)		f(f(x)) = e^x

have frequently been discussed in literature.
		 ...
H. Kneser has treated the problem from the point of
view of analytic functions.  The function e^x has no
real fixpoints (i.e. real roots of e^z - z = 0)
and this causes some difficulty in the analytical
treatment of the problem.  However, by applying the
method of Koenigs [paper of 1884] in the neighbourhood
of a complex fixpoint and subsequent conformal transformations
Kneser succeeded in obtaining a real analytic solution of
Abel's equation

(3)			B(e^x) = B(x) + 1

from which he derived by a well known process the fractional
iterates

(4)			f_s(x) = B^{-1}(B(x) + s).

In particular he obtained a real analytic solution of (1),

(5)		phi(x) = f_{1/2}(x) = B^{-1}(B(x) + 1/2)."
					[Szekeres, p. 301]

Here is a quote from the last section of Hellmuth Kneser's
paper (my translation):

"the function

	phi(z) = B^{-1}(B(z) + 1/2)

is analytic and real throughout the whole real axis with
positive derivative and satisfies there the functional
equation

	phi(phi(z)) = e^z." 		 [Kneser p. 66]



I think this answers the original question.

Szekeres points out that the solution is by no means unique,
and his paper is concerned with adding extra conditions
to give a unique solution.  There is a paper immediately
following the paper by Szekeres that I referenced earlier,
written by Morris and Szekeres, giving computational
details of this unique solution (J. Australian Math. Soc.
Vol. 2 (1961), 321-333).  There is a diagram of the solution
on the interval [-3,3] on p. 324.

There is also a monograph more recent than the one that I referenced
in my last posting:

"Iterative Functional Equations" by Kuczma, Choczewski
and Ger (CUP, Encyclopedia of Mathematics), 1990.