(01/04/2016, 12:01 AM)andydude Wrote: It appears that these 3 branches are not connected in the real line, but I would suspect that they are connected in the complex plane (to make a Riemann surface), and that fixing \( {}^0a = 1 \) on the main branch might fix the value of the other branches, at least that's what I would think.[Note: the asymptotes make 5 branches]
If the transition is not singular, then the blue branch should be coincident with the black (main) branch for \( \\[15pt]
{a>e^{e^{-1}}} \). That may give a clue about his better value for °a.
[Edit: the transition for the blue branch must be singular, because it cannot smoothly change from an horizontal asymptote to a parallel line to the main branch, which has a vertical asymptote.
That unless tetration introduces a new kind of numbers which are continuous in the transition.]
They should be coincident, because for the blue branch, any value chosen for °a, is also somewhere in the main branch, so taking iterated logarithms/exponentiations should give the same result.
I have the result, but I do not yet know how to get it.