06/10/2023, 06:59 PM
I have delivered the first release of my Mathematica software as a function for performing fractional iteration. I plan on using this as a basis for my future research. But before I model tetration through software, I need to model it mentally so that I have some expectations of what I will see.
The type of complex tetration I do is based on fixed points. There seems to be general agreement as to the value of expressions like \(^zb\) when evaluated with \(b\) within the Shell-Thron region. Since my work is based on fixed points and there are a countable infinity of fixed points, I argue that complex tetration should be considered as an infinite ensemble with one tetration function per fixed point.
What I expect to see is an infinite set of tetration functions that agree in the tetration of whole numbers, but vary from one another for fractional values. The fixed points imaginary values grows linearly while their real values grow much slower. Considered in polar coordinates from the fixed point, as the radius grows, the arc length from one whole number tetration to the next becomes smaller. I'm expecting to see real tetration as a limit of complex tetration as the fixed points approach imaginary infinity. I expect to see the variance of imaginary values to be inverse to the distance of the fixed point from the real numbers.
The type of complex tetration I do is based on fixed points. There seems to be general agreement as to the value of expressions like \(^zb\) when evaluated with \(b\) within the Shell-Thron region. Since my work is based on fixed points and there are a countable infinity of fixed points, I argue that complex tetration should be considered as an infinite ensemble with one tetration function per fixed point.
What I expect to see is an infinite set of tetration functions that agree in the tetration of whole numbers, but vary from one another for fractional values. The fixed points imaginary values grows linearly while their real values grow much slower. Considered in polar coordinates from the fixed point, as the radius grows, the arc length from one whole number tetration to the next becomes smaller. I'm expecting to see real tetration as a limit of complex tetration as the fixed points approach imaginary infinity. I expect to see the variance of imaginary values to be inverse to the distance of the fixed point from the real numbers.
Daniel