[FOM] Fractional Iteration and Rates of Growth

[joeshipman@aol.com]

If, as you say, there is a unique real analytic function g(x) = 
f(1,x,0) defined by analytic continuation along the (positive) real 
axis, such that g(1)=1, g(2)=2, g(3)=4, g(4)=16, g(5)=65536, etc., can 
you

1) Prove that g and all of its derivatives are monotonically increasing?

2) Prove that g satisfies the functional equation g(x+1)=2^g(x) 
everywhere it is defined? (Obviously its domain cannot be extended to 
negative integers if this is the case.)

3) Calculate g(3.5)  and g(4.5) to the nearest integer?

4) Specify a maximal subdomain of the complex plane to which g can be 
extended?

If either 1) or 2) is not possible, then analytic continuation does not 
appear to give a "natural" way of extending the "Tower" function to  
real arguments.

-- JS

-----Original Message-----
From: Dmytro Taranovsky <dmytro at mit.edu>

  In the previous posting, I defined a function f meant to represent
natural primitive recursive rates of growth:
f(1, 1, x) = 2*x
f(n, a, x) = ath iterate of f(n, 1, x)
f(n+1, 1, x) = f(n, x, x)
with f(0, a, x) = x + a, with f analytic (using analytic continuation
along the real axis), and with f(0)=0 and f'(0)>0 for n>0 (n is an
integer) and any real a.

At integral arguments, f is just an analogue of the Ackermann function.
For all real a and b, f(n, a+b, x) = f(n, a, f(n, b, x)) (x>0).  f is
analytic in both x and a.  f(1, a, x) = 2^a*x.