10/17/2009, 09:29 PM
(This post was last modified: 10/17/2009, 09:30 PM by Base-Acid Tetration.)
Not very rigorous argument:
The region around -2 is locally a logarithmic branchpoint. At -2 every branch of tetration, like those of the logarithm, falls to an infinite value. since tet(-3)= log(tet(-2)) and logarithm of infinity, whatever infinity it is, is a kind of essential infinity, so every branch of tetration has these singularities we can repeat this process to get more than one singularities for every branch of tetration, which an entire pentation must avoid, so any entire pentation is trivial, a constant function pen(z) = fixed point of tet(z).
so pentations that we like won't be entire.
The region around -2 is locally a logarithmic branchpoint. At -2 every branch of tetration, like those of the logarithm, falls to an infinite value. since tet(-3)= log(tet(-2)) and logarithm of infinity, whatever infinity it is, is a kind of essential infinity, so every branch of tetration has these singularities we can repeat this process to get more than one singularities for every branch of tetration, which an entire pentation must avoid, so any entire pentation is trivial, a constant function pen(z) = fixed point of tet(z).
so pentations that we like won't be entire.