User blog:Username5243/My attempt at continuous UNAN

This is my attempt to define UNAN for non-integer values of the base and iterator. (I'm not going to try with numbers inside the brackets.)

I was partly inspired by PsiCubed's letter notation, which some of the definitions I used are based on.

note: a[#]b is always increasing only for a > 2. For a <= 2, there will be odd behaviors sometimes, as explained below.

Basic arrays
In normal UNAN, the three main rules are:


 * a[0]b = a*b
 * a[n]1 = a
 * a[n+1]b+1 = a[n]a[n+1]b

Generalizing them to nonintegers is easy. NOte that I explicitly state that a[1]b = a^b, rather than it following the standard rules - this is just to preserve the standard definition of continuous exponentiation, whereas if we did it the other wy, we'd get 10[1]1.5 = 10[0]1[1]0.5 = 10[0]10[0]0.5 = 50, which I'd rather not have.


 * a[0]b = a*b, a[1]b = a^b
 * a[n]b = a[1]b for 0 < x <= 1
 * a[n+1]b+1 = a[n]a[n+1]b for b > 1

Note: The function a[n]b only increases without bound for x > e1/e. For smaller values of a, the tetration is generally known to either converge or alternate between 0 and 1. I haven't looked into what happens for pentation and higher, but I'm fairly sure for a > 1 it will still be an increasing function, just converging to one number.