eta as branchpoint of tetrational

[bo198214]

The kneser tetration is real on the real axis , which implies that (conjugation).

So approaching from above or below is just conjugate to each other.
So what one need to show imho is that the imaginary part will not tend to zero when approaching the real axis at .

_Or_ show that it is not conjugate-symmetric there.

Maybe I was not insistent enough on that:

If we have a function that is real-analytic on any interval (a,b) and you continue it through any path in the upper halfplane.
Then for the conintuation in the lower halfplane we have:
,

i.e. in simple words: f is conjugate-symmetric *everywhere*