[FOM] Fractional Iteration and Rates of Growth

[joeshipman@aol.com]

Dmytro,

I misread your definition. The "tower" function g(1)=1, g(2)=2, g(3)=4, 
g(4)=16, g(5)=65536 is f(1,x,0) if the "base function" f(1,1,x)=2^x, 
not 2*x as you originally defined.

But your original procedure of "analytic continuation to the positive 
real axis" ought to work just as well if you start with the 2^x 
function. You said "We use analytic continuation (along the real line) 
to define f for all positive a and x.  There is no non-uniqueness." so 
why would there be non-uniqueness in the case where your base function 
is 2^x instead of 2*x?

If there is a real analytic function defined on the positive real axis, 
with f(1)=1 and f(x+1)=2^f(x), and if, as you say, such a function 
could not have all of its derivatives be non-negative at any point, 
then it is not a satisfactory extension of the "tower" function, since 
we surely want such an extension to have the property of "all 
derivatives positive everywhere".

If it the case that the function whose values at 0,1,2,3,4,5,.... are 
0,1,2,4,16,65536,... has unique real analytic extension, how do you 
calculate f(3.5) and f(4.5)?

It would be very desirable to have a canonical way to take a 
"functional square root", because then we could extend functions like 
"iterated exponentiation base e" and "tower" (= iterated exponentiation 
base 2) to all real arguments. But analytic continuation seems like too 
cheap a way to do this. Another way would be to find a formal power 
series f(x) =  a0 + a1x + a2(x^2) + ... such that the formal power 
series f(f(x)) equals the ordinary power series for e^x or 2^x.  Is 
there a formal power series with this property, and where does it 
converge?

-- JS