10/17/2009, 09:54 PM

An important moral question is facing us, my fellow friends of hyper-operations.

As we all know, tetration, having log^n singularites for all n, has an infinite number of branches. When we define pentation as a tetration-iterational (superfunction of tetration), we face ambiguity of definition: WHICH branch of tetration SHOULD we mean? Restricting tet(z) to its principal branch (defined for the complex plane excluding (-infinity, -2]) seems the most "correct" definition, but at the same time the most unnatural.

The same problem goes for hexation since the superlogarithm has two conjugate branch points, penta-logarithm will have a penta-logarithmic singularity at fixed point of tetration. (It is not known whether pentation has any complex fixed points.)

This is a highly subjective question that won't be settled by any amount of attempted rigor, so I propose a split up classification of pentations that we will study. I think we should define "proper pentation" as the function that has pen(z+1)= Tet(pen(z)), where Tet = mainly prinicipal branch of tetration. Or can't we study properties that will apply for all pentations?

As we all know, tetration, having log^n singularites for all n, has an infinite number of branches. When we define pentation as a tetration-iterational (superfunction of tetration), we face ambiguity of definition: WHICH branch of tetration SHOULD we mean? Restricting tet(z) to its principal branch (defined for the complex plane excluding (-infinity, -2]) seems the most "correct" definition, but at the same time the most unnatural.

The same problem goes for hexation since the superlogarithm has two conjugate branch points, penta-logarithm will have a penta-logarithmic singularity at fixed point of tetration. (It is not known whether pentation has any complex fixed points.)

This is a highly subjective question that won't be settled by any amount of attempted rigor, so I propose a split up classification of pentations that we will study. I think we should define "proper pentation" as the function that has pen(z+1)= Tet(pen(z)), where Tet = mainly prinicipal branch of tetration. Or can't we study properties that will apply for all pentations?