Are tetrations fixed points analytic? JmsNxn Ultimate Fellow     Posts: 1,214 Threads: 126 Joined: Dec 2010 12/12/2016, 10:56 PM (This post was last modified: 12/12/2016, 10:59 PM by JmsNxn.) Looking at the standard tetration for $1 < \alpha < \eta$ I was wondering about something. Taking $F(\alpha,z) = \,^z \alpha$ we first note that F is analytic in $\alpha$. 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 $\tau_\alpha$. I could show it, but I assume people also know that $F_\alpha(\alpha,x) > 0$. Algebraically we can characterize $\tau_\alpha$ by the equations $F(\alpha,\tau_\alpha) = \tau_\alpha$ $F(\alpha,F(\alpha,...k\,times ...F(\alpha,x) \to \tau_\alpha$ for all $0 \le x \le \tau_\alpha$ I'm wondering if anyone has any information about the analycity of $\tau_\alpha$ in $\alpha$. This is rather important because if $\tau_\alpha$ is analytic then by the functional identity $\tau'(\alpha) = \frac{F_\alpha(\alpha,\tau_\alpha)}{1 - F_x(\alpha,\tau_\alpha)}$ and the fact $F_\alpha(\alpha,x)>0$ it follows that \( 0

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