[FOM] Fractional Iteration and Rates of Growth
joeshipman@aol.com
joeshipman at aol.com
Mon May 1 19:59:02 EDT 2006
Taranovsky:
********
By using iteration, we can define natural functions with arbitrarily
high
primitive recursive rate of growth, and an interesting question is
whether these functions have natural analogues defined at all positive
real numbers. Formally, there is no problem: Any f:N->N has an
analytic extension, and if f is sufficiently fast growing, then we can
arrange for all the coefficients of the power series to be positive....
We use analytic continuation (along the real line) to define f for all
positive a and x. There is no non-uniqueness. By using f, we can also
define intermediate rates of growth such as a half-exponential growth
rate, f(2, 1/2, x). Also, f(0, a, x) is best defined as x+a.
*********
How do you know that the resulting power series have nonzero radii of
convergence? It's not clear to me that this gives an algorithm
extending the definition of, say, the "tower function" f(0)=1,
f(i+1)=2^(f(i)) to all real numbers.
Another route might be to calculate the "square root of the exponential
function" as a formal power series, this clearly gives a computable
sequence of coefficients (whatever base you start with, it doesn't have
to be e) without resorting to analysis; then one can try to prove that
the resulting series converges. You may have to start with (e^x - 1)
instead of e^x, and it is then a nontrivial problem to generalize this
to find the natural extension of the true exponential function (and
similarly for the functions (2^x - 1) and 2^x, etc.).
-- JS
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