I got inspired by this
http://math.eretrandre.org/tetrationforum/showthread.php?tid=1149
Where Tommy assumes problems for tetration.
I was intrested in a closed form for when the fixpoint is on the edge of a new branch of ln or equivalent on the edge of the univalent zone of the exponential.
So i write ( with $a,q$ real variables and $q>1$)
$$ exp( q ( a + \frac {2 \pi}{q} i) ) = a + \frac{2\pi }{q} i$$
This reduces
$$ exp(q a) = a + \frac {2 \pi }{q} i ?? $$
Or maybe
$ exp ( q a) = a $ ?
Im confused.
Maybe im confused because of branches.
What about closed forms for $a,b,q$ ?
Does that fixpoint even exist ??
How does the LambertW function handle this ?