08/20/2007, 04:14 PM

I just got an answer from Peter Walker

Peter L. Walker Wrote:The difference between Noel's paper and mine is as follows. His is concerned with a particular iterate of e^x - 1, namely

(e^x - 1)^[1/2], while mine is concerned with an 'exponential of iteration' (to use Szekeres' notation, for which see the paper of mine in the Proc. AMS 1990 which I mentioned before, and the references there), i.e. a solution F of the functional equation, in this case

F(x + 1) = h(F(x)) = e^(F(x)) - 1,

so the two things are not meant to be the same.

Of course the solution F can be used to construct iterates via the definition

h_a (x) = F(G(x) + a)

where G is F inverse, provided that a suitable inverse exists.

In this case the Proc AMS paper shows that F is entire (with however some very bizarre properties). It does map the real axis strictly monotonically onto (0,infinity) so that there is a well-defined real-analytic inverse G on a neighbourhood of (0,infinity). The difficulty is that G has a very nasty singularity at the origin so that the above definition for h_1/2 makes no sense at 0, and it is not surprising that the solution obtained from equating coefficients in the power series has radius of convergence zero.

Another way of looking at this is to say that e^x - 1 can be iterated for non-integers, but only for x > 0.