eta as branchpoint of tetrational

[bo198214]

The kneser tetration \( b\mapsto b[4]p \) is real on the real axis \( b>\eta \), which implies that \( \overline{b} [4] p = \overline{b[4]p} \) (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 \( b<\eta \).

_Or_ show that it is not conjugate-symmetric there.

Maybe I was not insistent enough on that:

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

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