Looking at the standard tetration for 
I was wondering about something. Taking  = \,^z \alpha)
we first note that F is analytic in 
. As we all know, bounded in z on the right half plane. It is monotone increasing on the real positive line, which leads us to a fixed point, let's call it 
. I could show it, but I assume people also know that  > 0)
.
Algebraically we can characterize by the equations
 = \tau_\alpha)
 \to \tau_\alpha)
for all 
I'm wondering if anyone has any information about the analycity of in . This is rather important because if is analytic then by the functional identity
 = \frac{F_\alpha(\alpha,\tau_\alpha)}{1 - F_x(\alpha,\tau_\alpha)})
and the fact>0)
it follows that
 < 1)
and that the fixed point is geometrically attracting. This would instantly give a solution to pentation, and whats better, a solution to pentation with an imaginary period. Conversely, if then necessarily is analytic in by the implicit function theorem.
All in all, I haven't been able to find results on tetrations fixed points, and whether they are analytic or not. I hope they are, but I can't be sure. This is bugging me because a positive answer would greatly simplify the construction of pentation, and hopefully will shed light on how to show pentations fixed points are geometrically attracting giving a nice solution for hexation, so on and so forth.
Algebraically we can characterize
I'm wondering if anyone has any information about the analycity of
and the fact
and that the fixed point
All in all, I haven't been able to find results on tetrations fixed points, and whether they are analytic or not. I hope they are, but I can't be sure. This is bugging me because a positive answer would greatly simplify the construction of pentation, and hopefully will shed light on how to show pentations fixed points are geometrically attracting giving a nice solution for hexation, so on and so forth.
