Basics of Fractionally Iterated Exponential Functions

I already wrote about fractionally iterated exponential functions (“FIE”) a year and a half ago, but now want to go over it again in more detail. Not too much detail, just enough to explain what they are, briefly how anyone can calculate them, and give some reasons why FIEs are interesting.

What is a Fractionally Iterated Exponential Function?

Iteration of a function is to compute it repeatedly, to feed a function its own output some number of times.

If the function has an inverse, at least over some useful domain, we define

Fractional iteration is when we ask what happens if n is not an integer. Let’s turn the ’n’ upside down to ‘u’ when it’s a real number: f_u(x)

A half-iteration would work like this:

A Fractionally Iterated Exponential function (FIE) is defined by:

the inverse case being

(We won’t use the overbarred exp notation much)

This does not pin down a unique function, but allows infinitely many. Given one specific FIE, infinitely many others may be created from that using an arbitrary periodic function of period one.

We pick out one unique definition by demanding certain conditions of smoothness. This one smoothest solution was found by Australian mathematician George Szekeres and seems most likely to have some future use in physics, statistics…